Minimum supports of eigenfunctions of graphs: a survey
MMinimum supports of eigenfunctions of graphs: asurvey (cid:63)
Ev Sotnikova a , Alexandr Valyuzhenich a, ∗ a Sobolev Institute of Mathematics, Ak. Koptyug av. 4, Novosibirsk 630090, Russia
Abstract
In this work we present a survey of results on the problem of finding the mini-mum cardinality of the support of eigenfunctions of graphs.
Keywords: eigenfunction, eigenfunctions of graphs, eigenspace, minimumsupport, trade, bitrade, 1-perfect bitrade, weight distribution bound
1. Introduction
The eigenvalues of a graph are closely related to its structural propertiesand invariants (see the monographs [23, 30, 32]). Eigenfunctions (equivalently,eigenvectors) of graphs, in contrast to their eigenvalues, have received only spo-radic attention of researchers. In particular, basic properties of eigenfunctionsof graphs can be found in the work of Merris [70]. Among the most famousresults we can recall the theory around Perron-Frobenius vector [76, 42, 5] withits applications to a variety of problems including ranking, population growthmodels, Markov chains behavior and many other [75, 56, 67, 84]; and the resultsabout Fiedler vector [39, 33, 34] and its connection to the problems of spectralgraph partitioning and clustering [81, 74], graph coloring [4], graph drawing[61] and other (for example, [86, 85]). In addition, it is worth noting a series ofworks [16, 17, 18, 29, 36, 37, 43, 82, 96] devoted to various discrete versions ofCourant’s nodal domain theorem. We refer the reader to [19], [31, Chapter 9]and [83] for more details about eigenfunctions of graphs.In this work we consider undirected graphs without loops and multiple edges.The eigenvalues of a graph are the eigenvalues of its adjacency matrix. Let G = ( V, E ) be a graph with vertex set V = { v , . . . , v n } and let λ be an (cid:63) The study was carried out within the framework of the state contract of the SobolevInstitute of Mathematics (project no. 0314-2019-0016). ∗ Corresponding author
Email addresses: [email protected] (Ev Sotnikova), [email protected] (Alexandr Valyuzhenich)
Preprint submitted to ADAM February 23, 2021 a r X i v : . [ m a t h . C O ] F e b igenvalue of G . The set of neighbors of a vertex x is denoted by N ( x ). Afunction f : V −→ R is called a λ -eigenfunction of G if f (cid:54)≡ λ · f ( x ) = (cid:88) y ∈ N ( x ) f ( y ) (1)holds for any vertex x ∈ V . Note that if f is a λ -eigenfunction of G , then A −→ f = λ −→ f , where A is the adjacency matrix of G and −→ f = ( f ( v ) , . . . , f ( v n )) T ,i.e. −→ f is an eigenvector of the matrix A with eigenvalue λ . The set of functions f : V −→ R satisfying (1) for any vertex x ∈ V is called a λ -eigenspace of G .Denote by U λ ( G ) the λ -eigenspace of G . The support of a function f : V −→ R is the set S ( f ) = { x ∈ V | f ( x ) (cid:54) = 0 } . A λ -eigenfunction of G is called optimal if it has the minimum cardinality of the support among all λ -eigenfunctions of G . In this work we focus on the following extremal problem for eigenfunctionsof graphs. Problem 1 (MS-problem) . Let G be a graph and let λ be an eigenvalue of G .Find the minimum cardinality of the support of a λ -eigenfunction of G . In what follows, in this work we will use the abbreviation MS-problem insteadof Problem 1. Now we discuss the deep connection between MS-problem and theintersection problem of two combinatorial objects and the problem of findingthe minimum size of trades.Many combinatorial objects (equitable partitions, completely regular codes,Steiner systems S ( k − , k, n ), 1-perfect codes, etc.) can be defined as eigen-functions of graphs with some discrete restrictions. The study of such objectsoften leads to the problem of finding the minimum possible difference betweentwo objects from the same class (for example, see [38, 41, 49, 78, 79]). Sincethe symmetric difference of such two objects is also an eigenfunction of thecorresponding graph, this problem is directly related to MS-problem.Trades of different types are used for constructing and studying the struc-ture of different combinatorial objects (combinatorial t -designs, codes, Latinsquares, etc.). Trades are also studied independently as some natural gen-eralization of objects of the corresponding type (trades can exist even if thecorresponding complete objects do not exist). Roughly speaking, trades reflectpossible differences between two combinatorial objects from the same class: if C (cid:48) and C (cid:48)(cid:48) are two combinatorial objects with the same parameters, then the pair( C (cid:48) \ C (cid:48)(cid:48) , C (cid:48)(cid:48) \ C (cid:48) ) is a trade (for more information on trades see [15, 24, 46, 66]).Many types of trades ( T ( k − , k, v ) Steiner trades, q -ary T q ( k − , k, v ) Steinertrades, 1-perfect trades, extended 1-perfect trades, latin trades, etc.) can berepresented as eigenfunctions of the corresponding graphs with some additionaldiscrete restrictions (for example, see [65, Section 2.4]). So, for such trades theproblem of finding the minimum size can be reduced to MS-problem for thecorresponding graphs (see, for example, [66, 94, 98]).In particular, MS-problem has appeared as a natural generalization of thefollowing results. 2 Let C and C be two distinct binary perfect codes of length n = 2 m − C ∩ C is 2 n − m − n − . Equivalently, they found theminimum possible cardinality of their symmetric difference C (cid:52) C . Thisresult can be proved by applying the so-called weight distribution boundfor the Hamming graph H ( n,
2) and its eigenvalue − • In [49] Hwang proved that the minimum size of a T ( t, k, v ) trade is 2 t +1 andobtained a characterization of T ( t, k, v ) trades of size 2 t +1 . In particular,the minimum size of a T ( t, k, v ) Steiner trade was found in [49]. For t = k − J ( v, k ) and its eigenvalue − k (see Subsection2.2 and Section 4). It is interesting that Frankl and Pach [41] also foundthe minimum size of a T ( t, k, v ) trade. They formulated their results interms of null t -designs. In Section 8 we will meet null designs again duringour discussion about optimal eigenfunctions of the Grassmann graph.MS-problem was first formulated by Krotov and Vorob’ev [98] in 2014 (theyconsidered MS-problem for the Hamming graph). During the last six years, MS-problem has been actively studied for various families of distance-regular graphs[8, 45, 64, 66, 90, 91, 93, 94, 95, 98, 99] and Cayley graphs on the symmetricgroup [53]. In particular, MS-problem is completely solved for all eigenvaluesof the Hamming graph [94, 95] and asymptotically solved for all eigenvalues ofthe Johnson graph [99]. Note that for eigenfunctions of distance-regular graphsa lower bound for its support cardinality is known. This bound is called theweight distribution bound and we will discuss it in details in Section 4. In thiswork we give a survey of results on MS-problem. We also discuss constructionsof optimal eigenfunctions and the main ideas of the proofs of the results.Now we would like to consider the following problem. Problem 2.
Let G = ( V, E ) be a graph and let λ be an eigenvalue of G . Find min f ∈ U λ ( G ) ,f (cid:54)≡ |{ x ∈ V | f ( x ) ≥ }| . Note, that the statements of MS-problem and Problem 2 are similar. Ananalogue of Problem 2 for association schemes was first formulated in 1984 byBier [10]. Later, Bier and Delsarte [12, 13] and Bier [11] studied the sameproblem for eigenvectors belonging to the direct sum of several eigenspaces ofan association scheme. Bier and Manickam [14], Manickam and Mikl´os [68] andManickam and Singhi [69] initiated the study of Problem 2 for the second largesteigenvalue of Johnson and Grassmann graphs. In particular, the following twoconjectures were formulated in 1988.
Conjecture 1 (Manickam, Mikl´os and Singhi [68, 69]) . Let x , . . . , x n be realnumbers such that x + . . . + x n = 0 . If n ≥ k , then there are at least (cid:0) n − k − (cid:1) k -element subsets of the set { x , . . . , x n } with nonnegative sum. V be an n -dimensional vector space over a finite field F q . Let (cid:2) Vk (cid:3) q denote thefamily of all k -dimensional subspaces of V and let (cid:2) nk (cid:3) q denote the q -Gaussianbinomial coefficient. For each 1-dimensional subspace v ∈ (cid:2) V (cid:3) q , assign a real-valued weight f ( v ) ∈ R so that the sum of all weights is zero. For a generalsubspace S ⊂ V , define its weight f ( S ) to be the sum of the weights of all the1-dimensional subspaces it contains. Conjecture 2 (Manickam and Singhi [69]) . Let V be an n -dimensional vectorspace over F q and let f : (cid:2) V (cid:3) q → R be a weighting of the -dimensional sub-spaces such that (cid:80) v ∈ [ V ] q f ( v ) = 0 . If n ≥ k , then there are at least (cid:2) n − k − (cid:3) q k -dimensional subspaces with nonnegative weight. Conjecture 1 is still open. However, there are several relatively recent works[2, 28, 40, 77] with polynomial bounds. In particular, Alon, Huang and Sudakov[2] verified Conjecture 1 for n ≥ k . A linear bound n ≥ k was obtainedby Pokrovskiy [77]. In 2014 Chowdhury, Sarkis and Shahriari [28] and Huangand Sudakov [48] independently showed that Conjecture 2 holds for n ≥ k .Using the technique of the work [28], Ihringer [50] proved that Conjecture 2 istrue for n ≥ k and large q . Some new results on Problem 2 for the third largesteigenvalue of the Johnson graph can be found in [73]. It seems very intriguingto establish the interconnection between Problem 2 and MS-problem.The paper is organized as follows. In Section 2, we give two examples ofcombinatorial problems that are closely related to MS-problem. In Section 3,we introduce basic definitions and notations. In Section 4, we discuss what theweight distribution bound is and how it can be calculated from the intersectionarrays of the distance-regular graphs. We complete this section with severalintuitive examples. In Sections 5-11, we give a survey of results on MS-problemfor the Hamming graph, the Doob graph, the Johnson graph, the Grassmanngraph, the bilinear forms graph, the Paley graph and the Star graph respectively.In Section 12, we present some observations on optimal eigenfunctions of graphs.In Section 13, we formulate several open problems.
2. Eigenfunctions in combinatorial configurations and MS-problem
In this section, we recall that equitable 2-partitions, 1-perfect codes and T ( k − , k, v ) Steiner trades can be defined as eigenfunctions of graphs withsome discrete restrictions. We also discuss the connections of MS-problem withthe intersection problem of two 1-perfect codes of a given graph and the problemof finding the minimum size of Steiner trades. -perfect codes Let G = ( V, E ) be a graph. An ordered r -partition ( C , . . . , C r ) of V iscalled equitable if for any i, j ∈ { , . . . , r } there is S i,j such that any vertex of C i has exactly S i,j neighbors in C j . The matrix S = ( S i,j ) i,j ∈{ ,...,r } is called the4 uotient matrix of the equitable partition. A set C ⊆ V is called a 1 -perfect code in G if every ball of radius 1 contains one vertex from C . For more informationon equitable partitions and perfect codes we refer the reader to [9], [44, Chapter5] and [1, 47, 88, 89].Let G be a k -regular graph and let ( C , C ) be an equitable 2-partition of G with the quotient matrix S = (cid:18) a bc d (cid:19) . The eigenvalues of S are k and a − c . We define the function f ( C ,C ) on thevertices of G by the following rule: f ( C ,C ) ( x ) = (cid:40) b, if x ∈ C ; − c, if x ∈ C .One can verify that f ( C ,C ) is an ( a − c )-eigenfunction of G . So, any equitable2-partition can be represented as an eigenfunction of the corresponding graph.Suppose that C is a 1-perfect code in G . Then the partition ( C, C ) is equitablewith the quotient matrix (cid:18) k k − (cid:19) . Therefore, the function f ( C,C ) is a ( − G . So, if C and C are 1-perfect codes in G , then the function f = f ( C ,C ) − f ( C ,C ) is also a( − G . Moreover, we have the equality | S ( f ) | = | C (cid:52) C | . Thus, the problem of finding the minimum cardinality of the symmetric dif-ference of two distinct 1-perfect codes of a regular graph can be reduced toMS-problem for this graph and eigenvalue − T ( k − , k, v ) Steiner trades
Let v , k , t be positive integers such that v > k > t and let X be a set of size v . A pair ( T , T ) of disjoint collections of k -subsets (blocks) of X is called a T ( t, k, v ) trade if every t -subset of X is included in the same number of blocks of T and T . The size of a T ( t, k, v ) trade ( T , T ) is | T | + | T | . A T ( t, k, v ) tradeis called Steiner if every t -subset of X is included in at most one block of T ( T ). For further details on T ( t, k, v ) trades we refer the reader to [15, 46, 57].Suppose that ( T , T ) is a T ( k − , k, v ) Steiner trade. The Johnson graph J ( v, k ) can be defined as follows. The vertices of J ( v, k ) are k -subsets of X ,and two vertices are adjacent if they have exactly k − f ( T ,T ) on the vertices of J ( v, k ) by the following rule: f ( T ,T ) ( x ) = , if x ∈ T ; − , if x ∈ T ;0 , otherwise.5or a ( k − A of X denote by C ( A ) the set of vertices of J ( v, k ) con-taining the set A (these vertices form a clique of size v − k + 1 in J ( v, k )). Wenote that C ( A ) either contains one element from T and one element from T or does not contain elements from T ∪ T . Using this fact, one can easily checkthat f ( T ,T ) is a ( − k )-eigenfunction of J ( v, k ). Moreover, we have the equality | S ( f ( T ,T ) ) | = | T | + | T | . Thus, the problem of finding the minimum size of T ( k − , k, v ) Steiner tradescan be reduced to MS-problem for the Johnson graph J ( v, k ) and its eigenvalue − k .
3. Basic definitions
Recall that a distance d G ( v, u ) = d ( u, v ) between two vertices v and u ina graph G = ( V, E ) is the length of the shortest path that connects them.The largest distance between any pairs of vertices is called the diameter D . Aconnected graph G = ( V, E ) is called distance-regular if it is regular of degree k and for any two vertices v, u ∈ V at distance i = d ( v, u ) there are precisely c i neighbors of u which are at distance i − v and precisely b i neighbors of u which are at distance i +1 from v ; where c i and b i do not depend on the choice ofvertices u and v but depend only on d ( u, v ). Numbers b i , c i , a i = k − b i − c i arecalled the intersection numbers and a set { b , b , . . . , b D − ; c , . . . , c D } is calledan intersection array of a distance-regular graph G . For more details aboutdistance-regular graphs, the reader is referred to a classical monograph [22] anda recent survey [35].Let G = ( V , E ) and G = ( V , E ) be simple graphs. The Cartesianproduct G (cid:3) G of graphs G and G is defined as follows. The vertex set of G (cid:3) G is V × V ; and any two vertices ( x , y ) and ( x , y ) are adjacent if andonly if either x = x and y is adjacent to y in G , or y = y and x isadjacent to x in G .Suppose G = ( V , E ) and G = ( V , E ) are two graphs. Let f : V −→ R and f : V −→ R . Denote G = G (cid:3) G . We define the tensor product f · f onthe vertices of G by the following rule:( f · f )( x, y ) = f ( x ) f ( y )for ( x, y ) ∈ V ( G ) = V × V . We will use the tensor product of functionsfor constructing optimal eigenfunctions of the Hamming and Doob graphs inSubsection 5.1 and Section 6.Let Sym(X) denote the symmetric group on a finite set X and let Sym n denote the symmetric group on the set { , . . . , n } .Let Σ q = { , , . . . , q − } . Let f ( x , . . . , x n ) be a function defined on theset Σ nq , let π ∈ Sym n and let σ , . . . , σ n ∈ Sym(Σ q ). We define the functions f π and f π,σ ,...,σ n as follows: f π ( x , . . . , x n ) = f ( x π (1) , . . . , x π ( n ) )6nd f π,σ ,...,σ n ( x , . . . , x n ) = f ( σ ( x π (1) ) , . . . , σ n ( x π ( n ) )) . We will use the functions f π and f π,σ ,...,σ n in Subsections 5.1 and 5.2.Let G = ( V, E ) be a graph. A set C ⊆ V is called a completely regular code in G if the partition ( C (0) , . . . , C ( ρ ) ) is equitable, where C ( d ) is the set of verticesat distance d from C and ρ ( the covering radius of C ) is the maximum d forwhich C ( d ) is nonempty. In other words, a subset of V is a completely regularcode in G if the distance partition with respect to the subset is equitable. Formore information on completely regular codes see [21], [44, Chapter 11.7] and[59, 60]. We will use completely regular codes in Section 12.
4. Weight Distribution Bound
In this section we recall what a weight distribution bound is and how it canbe used as a lower bound for MS-problem in case of distance-regular graphs.Weight distribution bound is well known and has appeared in several papersunder different disguise (for more details see [63], [66]). In order for this surveyto be self-contained we would like to provide the full proof and equip the readerwith several intuitive examples.Let A be the adjacency matrix of some distance-regular graph G = ( V, E ).Now consider the distance- t graph G t = ( V, E t ) defined as follows: two vertices v and u are adjacent in G t if and only if they are at distance t in G . In otherwords, { v, u } ∈ E t ⇐⇒ d G ( v, u ) = t . By A t we denote the adjacency matrixof G t .Considering the combinatorial definition of distance regularity from the ma-trix point of view, we obtain the following recurrence (see, for example, equation(1) in [35]): A t A = a t A t + b t − A t − + c t +1 A t +1 , (2)for t = 0 , , . . . , D where b − A − = c D +1 A D +1 = 0.From the above we can show that there exist polynomials P t of degree t suchthat: A t = P t ( A ) , t = 0 , , . . . , D. It is well known that in case of Hamming graphs these polynomials areactually Kravchuk polynomials (up to some linear change of variables) and thoseare Eberlein polynomials in case of Johnson graphs.But how can we make use of it in finding the lower bound for our MS-problem? Suppose f is a λ -eigenfunction of our graph G . Since A i −→ f = λ i −→ f ,we get the following equations: A t −→ f = P t ( A ) −→ f = P t ( λ ) −→ f . In other words, f is a P t ( λ )-eigenfunction of graph G t . As an immediateconsequence we obtain: 7 t ( λ ) f ( v ) = (cid:88) u ∈ V,d ( u, v )= t f ( u ) . In other words, in distance-regular graphs the sum of the eigenfunction val-ues on the vertices at distance t from a fixed vertex v depends only on f ( v )and the corresponding eigenvalue. Without lost of generality we can consider f ( v ) = 1. The array [1 , P ( λ ) , . . . , P D ( λ )] is called the weight distribution of a λ -eigenfunction.Thus from (2) we can write the following recurrence: P ( λ ) = 1 ,P ( λ ) = λ,P t ( λ ) = λP t − ( λ ) − b t − P t − ( λ ) − a t − P t − ( λ ) c t , where t = 2 , . . . , D. Now we are just one step away from obtaining the lower bound we are lookingfor. Let w be such a vertex that | f ( w ) | = max u ∈ V | f ( u ) | . The trick is the following:instead of λ -eigenfunction f we consider a function g = 1 f ( w ) f. Thus g is also a λ -eigenfunction and S ( f ) = S ( g ). Moreover, g ( w ) = 1 and | g ( u ) | ≤ g ( w ) = 1 for all u ∈ V . From the weight distribution we obtain P t ( λ ) = (cid:88) d ( u, w )= t g ( u ) . Therefore, g has at least | P t ( λ ) | non-zero values at distance t from a vertex w . This proves the next lemma. Lemma 1 ([66], Corollary 1) . Let f be a λ -eigenfunction for a distance-regulargraph G of diameter D , then the following bound takes place: | S ( f ) | ≥ D (cid:88) i =0 | P i ( λ ) | . In case of irrational eigenvalues this bound can be refined:
Lemma 2.
Let f be a λ -eigenfunction for a distance-regular graph G of diam-eter D , then the following bound takes place: | S ( f ) | ≥ D (cid:88) i =0 (cid:100)| P i ( λ ) |(cid:101) . { ,
2; 1 , } and itseigenvalues are {− (4) , (5) , (1) } . Calculating the weight distribution we obtain[1 , λ, λ − • For λ = 1 it gives us the lower bound 4. An optimal 1-eigenfunctionachieves this bound. A subgraph induced on non-zero vertices can bedescribed as two non-incident edges. An example is presented below (Fig-ure 1). Figure 1: Optimal 1-eigenfunction of the Petersen graph. • For λ = − − H -graph. See Figure 2 and Figure 3. Figure 2: Optimal ( − − As a quick illustration of bound refinement, let us consider the Heawoodgraph, a distance-regular graph on 14 vertices. Its intersection array is { , ,
2; 1 , , } and its spectrum is {± (1) , ±√ (6) } . The weight distribution is [1 , λ, λ − , ( λ − λ )]. Thus for λ = ±√ √
2, while the refined bound is 6. Fig. 4 presents an example of an optimal √ Figure 4: Optimal √ More examples can be found in [90], where MS-problem is solved togetherwith a characterisation of such functions for 10 out of 13 cubical distance-regulargraphs for all their eigenvalues.Thus for any distance-regular graph a lower bound on a cardinality of a λ -eigenfunction support can be calculated directly from the intersection array ofa graph with respect to the corresponding eigenvalue λ . However this bound isnot necessary feasible. We will see in the next sections that a weight distributionbound is achieved for: • an eigenvalue − • a minimum eigenvalue of the Paley graphs of square order (since this graphis self-complementary of diameter 2, this property also holds for a secondnon-principal eigenvalue); • a minimum eigenvalue of the Johnson graphs; • a minimum eigenvalue of the Grassmann graphs; • a minimum eigenvalue of strongly regular bilinear forms graphs over aprime field.However this is not the case for a minimum eigenvalue of bilinear forms graphsof larger diameter, for example.We can see from the list above that the minimum eigenvalue λ D attractedsome special attention. This is because we can say something about non-zeros ofoptimal λ D -eigenfunctions. Indeed for every distance-regular graph admittinga so-called Delsarte pair the existence of a λ D -eigenfunction f achieving the10eight distribution bound is equivalent to the existence of an isometric distance-regular subgraph induced on the non-zero values of f (see Corollary 2 from [66]).We will explore this property more in the Section 9, but actually this result isa partial case of a more general theory which also demonstrates the strongconnection between eigenfunctions and bitrades. For a deeper dive into thetopic the interested reader is referred to Sections 2 and 3 of [66].
5. Hamming graph
In this section, we give a survey of results on MS-problem and its gener-alizations for the Hamming graph. The
Hamming graph H ( n, q ) is defined asfollows. Let Σ q = { , , . . . , q − } . The vertex set of H ( n, q ) is Σ nq , and twovertices are adjacent if they differ in exactly one position. This graph is adistance-regular graph. The Hamming graph H ( n, q ) has n + 1 distinct eigen-values λ i ( n, q ) = n ( q − − q · i , where 0 ≤ i ≤ n . Denote by U i ( n, q ) the λ i ( n, q )-eigenspace of H ( n, q ). The direct sum of subspaces U i ( n, q ) ⊕ U i +1 ( n, q ) ⊕ . . . ⊕ U j ( n, q )for 0 ≤ i ≤ j ≤ n is denoted by U [ i,j ] ( n, q ). We say that a function f ∈ U [ i,j ] ( n, q ), where f (cid:54)≡
0, is optimal in the space U [ i,j ] ( n, q ) if | S ( f ) | ≤ | S ( g ) | forany function g ∈ U [ i,j ] ( n, q ), g (cid:54)≡ U [ i,j ] ( n, q ). In [64] Krotov based on the approach of work [80] proved thatthe minimum cardinality of the support of a λ i ( n, H ( n,
2) ismax(2 i , n − i ). In [98] Krotov and Vorob’ev showed that the cardinality of thesupport of a λ i ( n, q )-eigenfunction of H ( n, q ) is at least2 i · ( q − n − i for iq n ( q − > q n · ( 1 q − i/ · ( in − i ) i/ · (1 − in ) n/ for iq n ( q − ≤
2. In [93] Valyuzhenich for q ≥ λ ( n, q )-eigenfunction of H ( n, q ) is 2 · ( q − · q n − and obtained a characterization of optimal λ ( n, q )-eigenfunctions. Later in[94, 95] the following generalization of MS-problem for the Hamming graph wasconsidered. Problem 3.
Let n ≥ , q ≥ and ≤ i ≤ j ≤ n . Find the minimum cardinalityof the support of functions from the space U [ i,j ] ( n, q ) . In [95] Valyuzhenich and Vorob’ev found the minimum cardinality of thesupport of a function from the space U [ i,j ] ( n, q ) for arbitrary q ≥ q = 3 and i + j > n . Moreover, in [95] a characterization of functionsthat are optimal in the space U [ i,j ] ( n, q ) was obtained for q ≥ i + j ≤ n and q ≥ i = j , i > n . In [94] Valyuzhenich found the minimum cardinality of thesupport of a function from the space U [ i,j ] ( n, q ) for q = 2 and q = 3, i + j > n .Thus, Problem 3 is completely solved for all n ≥ q ≥
2. As a consequence,MS-problem for the Hamming graph is also solved for all eigenvalues.In what follows, in this section we will consider in detail Problem 3. InSubsection 5.1, we present constructions of functions that are optimal in thespace U [ i,j ] ( n, q ). In Subsection 5.2, we give a survey of results on Problem 3 anddiscuss the main ideas of the proof of these results. In particular, we carefullyexplore Lemma 3 which is a key tool for solving Problem 3. In Subsection 5.3,we focus on a connection between Problem 3 and the problem of finding theminimum size of 1-perfect bitrades in the Hamming graph. In this subsection, we discuss constructions of functions that are optimalin the space U [ i,j ] ( n, q ). It is interesting that in all cases such functions areconstructed as a tensor product of several elementary optimal functions definedon the vertices of the Hamming graph of diameter not greater than three.Firstly, we define five sets of elementary optimal functions.For k, m ∈ Σ q we define the function a q,k,m on the vertices of the Hamminggraph H (2 , q ) by the following rule: a q,k,m ( x, y ) = , if x = k and y (cid:54) = m ; − , if y = m and x (cid:54) = k ;0 , otherwise.The function a , , is shown in Figure 5. We note that a q,k,m is optimal in thespace U (2 , q ) for any k, m ∈ Σ q . Denote A q = { a q,k,m | k, m ∈ Σ q } .We define the function ϕ on the vertices of the Hamming graph H (2 ,
3) bythe following rule: ϕ ( x, y ) = , if x = y = 0; − , if x = 1 and y = 2;0 , otherwise.For a, b ∈ Σ denote by a ⊕ b the sum of a and b modulo 3. We define thefunction ϕ on the vertices of the Hamming graph H (3 ,
3) by the following rule: ϕ ( x, y, z ) = ϕ ( x, y ) , if z = 0; ϕ ( x ⊕ , y ⊕ , if z = 1; ϕ ( x ⊕ , y ⊕ , if z = 2.The function ϕ is shown in Figure 6. We note that ϕ is optimal in the space U (3 , B = { ϕ π,σ ,σ ,σ | π ∈ Sym , σ , σ , σ ∈ Sym(Σ ) } . k, m ∈ Σ q and k (cid:54) = m we define the function c q,k,m on the vertices of theHamming graph H (1 , q ) by the following rule: c q,k,m ( x ) = , if x = k ; − , if x = m ;0 , otherwise.The function c , , is shown in Figure 7. We note that c q,k,m is optimal in thespace U (1 , q ) for any k, m ∈ Σ q and k (cid:54) = m . Denote C q = { c q,k,m | k, m ∈ Σ q , k (cid:54) = m } .For k ∈ Σ q we define the function d q,k on the vertices of the Hamming graph H (1 , q ) by the following rule: d q,k ( x ) = (cid:40) , if x = k ;0 , otherwise.The function d , is shown in Figure 7. We note that d q,k is optimal in the space U [0 , (1 , q ) for any k ∈ Σ q . Denote D q = { d q,k | k ∈ Σ q } .Let e q : Σ q −→ R and e q ≡
1. The function e is shown in Figure 7. Wenote that e q is optimal in the space U (1 , q ). Denote E q = { e q } . Figure 5: Function a , , in H (2 , igure 6: Function ϕ ( x, y, z ) in H (3 , c , , , d , and e in H (1 , Now, we define four classes of functions that are optimal in the space U [ i,j ] ( n, q )for the corresponding cases.Let i + j ≤ n . We say that a function f defined on the vertices of H ( n, q )belongs to the class F ( n, q, i, j ) if f = c · i (cid:89) k =1 g k · n − i − j (cid:89) k =1 h k · j − i (cid:89) k =1 v k , where c is a real non-zero constant, g k ∈ A q for k ∈ [1 , i ], h k ∈ E q for k ∈ [1 , n − i − j ] and v k ∈ D q for k ∈ [1 , j − i ].14et i + j > n . We say that a function f defined on the vertices of H ( n, q )belongs to the class F ( n, q, i, j ) if f = c · n − j (cid:89) k =1 g k · i + j − n (cid:89) k =1 h k · j − i (cid:89) k =1 v k , where c is a real non-zero constant, g k ∈ A q for k ∈ [1 , n − j ], h k ∈ C q for k ∈ [1 , i + j − n ] and v k ∈ D q for k ∈ [1 , j − i ].Let i + j ≤ n and i + j > n . We say that a function f defined on the verticesof H ( n,
3) belongs to the class F ( n, i, j ) if f = c · n − i − j (cid:89) k =1 g k · i + j − n (cid:89) k =1 h k · j − i (cid:89) k =1 v k , where c is a real non-zero constant, g k ∈ A for k ∈ [1 , n − i − j ], h k ∈ B for k ∈ [1 , i + j − n ] and v k ∈ D for k ∈ [1 , j − i ].Let i + j > n . We say that a function f defined on the vertices of H ( n, F ( n, i, j ) if f = c · n − j (cid:89) k =1 g k · i +2 j − n (cid:89) k =1 h k · j − i (cid:89) k =1 v k , where c is a real non-zero constant, g k ∈ B for k ∈ [1 , n − j ], h k ∈ C for k ∈ [1 , i + 2 j − n ] and v k ∈ D for k ∈ [1 , j − i ].We note that functions from F ( n, q, i, j ) and F ( n, q, i, j ) are optimal in thespace U [ i,j ] ( n, q ) for q ≥ i + j ≤ n and q ≥ q (cid:54) = 3), i + j > n respectively.We also note that functions from F ( n, i, j ) and F ( n, i, j ) are optimal in thespace U [ i,j ] ( n,
3) for i + j ≤ n , i + j > n and i + j > n respectively. In this subsection, we discuss Problem 3. The following theorem is a com-bination of the results proved in [94, 95] (see [95, Theorems 1 and 3] and [94,Theorems 3-6]).
Theorem 1. Let f ∈ U [ i,j ] ( n, q ) , where q ≥ , i + j ≤ n and f (cid:54)≡ . Then | S ( f ) | ≥ i · ( q − i · q n − i − j and this bound is sharp. Moreover, for q ≥ the equality | S ( f ) | = 2 i · ( q − i · q n − i − j holds if and only if f π ∈ F ( n, q, i, j ) for somepermutation π ∈ Sym n . Let f ∈ U [ i,j ] ( n, q ) , where q ≥ , q (cid:54) = 3 , i + j > n and f (cid:54)≡ . Then | S ( f ) | ≥ i · ( q − n − j and this bound is sharp. Moreover, for i = j and q ≥ the equality | S ( f ) | = 2 i · ( q − n − i holds if and only if f π ∈ F ( n, q, i, i ) for somepermutation π ∈ Sym n . Let f ∈ U [ i,j ] ( n, , where i + j ≤ n , i + j > n and f (cid:54)≡ . Then | S ( f ) | ≥ n − j ) − i · i + j − n and this bound is sharp. Let f ∈ U [ i,j ] ( n, , where i + j > n and f (cid:54)≡ . Then | S ( f ) | ≥ i + j − n · n − j and this bound is sharp. Now, we discuss the main ideas of the proof of Theorem 1.Let f be a real-valued function defined on the vertices of the Hamminggraph H ( n, q ) and let k ∈ Σ q , r ∈ { , . . . , n } . We define a function f rk on thevertices of H ( n − , q ) as follows: for any vertex y = ( y , . . . , y r − , y r +1 , . . . , y n )of H ( n − , q ) f rk ( y ) = f ( y , . . . , y r − , k, y r +1 , . . . , y n ) . One of the important points in the proof of Theorem 1 is the following.
Lemma 3 ([95], Lemma 4) . Let f ∈ U [ i,j ] ( n, q ) and r ∈ { , , . . . , n } . Then thefollowing statements are true: f rk − f rm ∈ U [ i − ,j − ( n − , q ) for k, m ∈ Σ q . (cid:80) q − k =0 f rk ∈ U [ i,j ] ( n − , q ) . f rk ∈ U [ i − ,j ] ( n − , q ) for k ∈ Σ q . Lemma 3 is a very useful tool for studying of eigenfunctions of the Hamminggraph. It shows the connection between eigenspaces of the Hamming graphs H ( n, q ) and H ( n − , q ). In particular, this lemma allows to apply inductionon n , i and j (we can use the induction assumption for the functions f rk − f rm , (cid:80) q − k =0 f rk and f rk ). Moreover, we suppose that Lemma 3 can be useful not onlyfor the MS-problem but also for other problems. For example, recently in [72]Mogilnykh and Valyuzhenich used Lemma 3 for investigation of equitable 2-partitions of the Hamming graph with the eigenvalue λ ( n, q ). One interestinggeneralization of Lemma 3 for the products of graphs can be found in [92,Theorem 3.11]. In this subsection, we discuss one interesting application of Theorem 1 forthe problem of finding the minimum size of 1-perfect bitrades in the Hamminggraph.Let us recall some definitions. Let G = ( V, E ) be a graph. For a vertex x ∈ V denote B ( x ) = N ( x ) ∪ { x } . Let T and T be two disjoint nonempty subsets of V . The ordered pair ( T , T ) is called a 1 -perfect bitrade in G if for any vertex x ∈ V the set B ( x ) either contains one element from T and one element from T or does not contain elements from T ∪ T . The size of a 1-perfect bitrade( T , T ) is | T | + | T | . 16 xample 1. Let T = { , } and T = { , } . Then ( T , T ) is a -perfect bitrade of size in H (3 , (see Figure 8). Figure 8: 1-perfect bitrade in H (3 , Example 2.
Let G = ( V, E ) be a graph. Suppose C and C be two distinct -perfect codes in G . Then ( C \ C , C \ C ) is a -perfect bitrade in G . Let ( T , T ) be a 1-perfect bitrade in a graph G = ( V, E ). We define thefunction f ( T ,T ) : V −→ {− , , } by the following rule: f ( T ,T ) ( x ) = , if x ∈ T ; − , if x ∈ T ;0 , otherwise.In what follows, in this subsection we will consider the following problem. Problem 4.
Let n ≥ and q ≥ . Find the minimum size of a -perfect bitradein H ( n, q ) . For q = 2 Problem 4 was essentially solved by Etzion and Vardy [38] andSolov’eva [87] (the results were formulated for more special cases of 1-perfectbitrades embedded into perfect binary codes, but both proofs work in the generalcase). In [71] Mogilnykh and Solov’eva for arbitrary q ≥ H ( q + 1 , q ) is 2 · q !.Now, using Theorem 1, we give a short solution of Problem 4 for q = 3 and q = 4. Firstly, we need the following result. Lemma 4 ([94], Lemma 6) . Let ( T , T ) be a -perfect bitrade in a graph G .Then f ( T ,T ) is a ( − -eigenfunction of G . Lemma 4 implies that we can consider Problem 4 only for n = qm + 1, where m ≥ − H ( n, q )).Suppose that ( T , T ) is a 1-perfect bitrade in H ( qm + 1 , q ). By Lemma 4we have that f ( T ,T ) is a ( − H ( qm + 1 , q ). We note that17 λ ( q − m +1 ( qm + 1 , q ). Applying Theorem 1 for n = qm + 1 and i = j =( q − m + 1, we obtain that | S ( f ( T ,T ) ) | ≥ ( q − m +1 · ( q − m for q ≥ | S ( f ( T ,T ) ) | ≥ m +1 · m for q = 3. Consequently, we have | T | + | T | = | S ( f ( T ,T ) ) | ≥ ( q − m +1 · ( q − m (3)for q ≥ | T | + | T | = | S ( f ( T ,T ) ) | ≥ m +1 · m (4)for q = 3. On the other hand, in [71] Mogilnykh and Solov’eva for arbitrary q ≥ H ( qm + 1 , q ) of size 2 · ( q !) m .Thus, the bounds (3) and (4) are sharp for q = 4 and q = 3 respectively, andwe obtain a solution of Problem 4 for q ∈ { , } . Finally, we note that Theorem1 implies that for q ≥ − H ( qm + 1 , q ) do not correspond to its 1-perfect bitrades (in this case we havea characterization of all optimal ( − q ≥
6. Doob graph
In this section, we give a survey of results on MS-problem for the Doobgraph. The
Shrikhande graph
Sh is the Cayley graph on the group Z with thegenerating set {± (0 , , ± (1 , , ± (1 , } . Figure 9: The Shrikhande graph.
The
Doob graph D ( m, n ), where m >
0, is the Cartesian product of m copiesof the Shrikhande graph and n copies of the complete graph K . In otherwords, we have D ( m, n ) = Sh m (cid:3) K n . This graph is a distance-regular graphwith the same parameters as the Hamming graph H (2 m + n, D ( m, n ) has 2 m + n + 1 distinct eigenvalues λ i ( m, n ) = 6 m + 3 n − i ,where 0 ≤ i ≤ m + n . In [8] Bespalov proved that the minimum cardinality18f the support of a λ ( m, n )-eigenfunction of D ( m, n ) is 6 · m + n − and ob-tained a characterization of optimal λ ( m, n )-eigenfunctions. He also showedthat the minimum cardinality of the support of a λ m + n ( m, n )-eigenfunctionof D ( m, n ) is 2 m + n and obtained a characterization of optimal λ m + n ( m, n )-eigenfunctions. In what follows, in this section we will consider the resultsobtained in [8].Now, we discuss constructions of optimal λ ( m, n )-eigenfunctions and λ m + n ( m, n )-eigenfunctions. It is interesting that as in the case of the Hamming graph suchfunctions are constructed as a tensor product of several elementary optimaleigenfunctions. Firstly, we define two sets of elementary optimal eigenfunctions.For a ∈ Z we define the function p a on the vertices of the Shrikhande graphby the following rule: p a ( x ) = , if x ∈ { a + (3 , , a + (3 , , a + (2 , } ; − , if x ∈ { a + (2 , , a + (1 , , a + (1 , } ;0 , otherwise.We note that the support of p a consists of two disjoint copies of the completegraph K . The function p (0 , is shown in Figure 10. Denote P = { p a | a ∈ Z } .For a ∈ Z and b ∈ { (0 , , (1 , , (1 , } we define the function r a,b on thevertices of the Shrikhande graph by the following rule: r a,b ( x ) = , if x ∈ { a, a + 2 b } ; − , if x ∈ { a + b, a + 3 b } ;0 , otherwise.We note that the vertices from the support of r a,b form a cycle of length4. The function r (0 , , (0 , is shown in Figure 10. Denote R = { r a,b | a ∈ Z , b ∈ { (0 , , (1 , , (1 , }} . We will also use the sets of functions A and C defined in Section 5. Figure 10: Functions p (0 , and r (0 , , (0 , . For m > I m,n the function that is defined on the vertices of D ( m, n ) and is identically equal to 1. For n ≥ I n the function thatis defined on the vertices of H ( n,
4) and is identically equal to 1.Now, we define two classes of optimal λ ( m, n )-eigenfunctions and one classof optimal λ m + n ( m, n )-eigenfunctions.19e say that a function f defined on the vertices of D ( m, n ) belongs to theclass G ( m, n ) if f = c · g . . . g m · I n , where c is a real non-zero constant, g i ∈ P for some i ∈ { , . . . , m } and g j = I , for any j ∈ { , . . . , m } \ i .Let n ≥
2. We say that a function f defined on the vertices of D ( m, n )belongs to the class G ( m, n ) if f = c · I m, · h . . . h n − , where c is a realnon-zero constant, h i ∈ A for some i ∈ { , . . . , n − } and h j = I for any j ∈ { , . . . , n − } \ i .We say that a function f defined on the vertices of D ( m, n ) belongs to theclass G ( m, n ) if f = c · g . . . g m · h . . . h n , where c is a real non-zero constant, g i ∈ R for any i ∈ { , . . . , m } and h j ∈ C for any j ∈ { , . . . , n } .The main results proved in [8] are the following. Theorem 2 ([8], Theorem 1) . Let f be a λ ( m, n ) -eigenfunction of D ( m, n ) ,where m > . Then | S ( f ) | ≥ · m + n − . Moreover, if | S ( f ) | = 6 · m + n − ,then the following statements hold: If n ≥ , then f ∈ G ( m, n ) or f ∈ G ( m, n ) . If n ∈ { , } , then f ∈ G ( m, n ) . Theorem 3 ([8], Theorem 2) . Let f be a λ m + n ( m, n ) -eigenfunction of D ( m, n ) ,where m > . Then | S ( f ) | ≥ m + n . Moreover, if | S ( f ) | = 2 m + n , then f ∈ G ( m, n ) . Remark 1.
We note that the bound proved in Theorem 3 can also be obtainedby applying the weight distribution bound for the smallest eigenvalue of the Doobgraph.
7. Johnson graph
In this section, we give a survey of results on MS-problem for the Johnsongraph. The
Johnson graph J ( n, ω ) is defined as follows. The vertices of J ( n, ω )are the binary vectors of length n with ω ones; and two vertices are adjacent ifthey have exactly ω − J ( n, ω ) has ω + 1distinct eigenvalues λ i ( n, ω ) = ( ω − i )( n − ω − i ) − i , where 0 ≤ i ≤ ω . In [99]Vorob’ev et al. showed that for a fixed ω and sufficiently large n the minimumcardinality of the support of a λ i ( n, ω )-eigenfunction of J ( n, ω ) is 2 i · (cid:0) n − iω − i (cid:1) and obtained a characterization of optimal λ i ( n, ω )-eigenfunctions. Thus, MS-problem for the Johnson graph is asymptotically solved for all eigenvalues.Now we discuss the main results obtained in [99]. Firstly, we define thefunction f i,ω,n on the vertices of the Johnson graph J ( n, ω ) by the followingrule: f i,ω,n ( x , . . . , x n ) = , if x k − + x k = 1 for any 1 ≤ k ≤ i and x + x + . . . + x i − is even; − , if x k − + x k = 1 for any 1 ≤ k ≤ i and x + x + . . . + x i − is odd;0 , otherwise.So, the support of f i,ω,n consists of binary vectors ( x , . . . , x n ) of weight ω suchthat the product ( x − x ) · . . . · ( x i − − x i ) is not equal to zero. In [99,20roposition 1] it was shown that f i,ω,n is a λ i ( n, ω )-eigenfunction of J ( n, ω )and | S ( f i,ω,n ) | = 2 i · (cid:0) n − iω − i (cid:1) . The main result proved in [99] is the following. Theorem 4 ([99], Theorem 4) . Let i and ω be positive integers, ω ≥ i . Thereis n ( i, ω ) such that for all n ≥ n ( i, ω ) and any λ i ( n, ω ) -eigenfunction f of J ( n, ω ) the following holds: | S ( f ) | ≥ i · (cid:18) n − iω − i (cid:19) , (5) and any function that attains the bound (5) is equivalent to f i,ω,n up to a per-mutation of coordinate positions and the multiplication by a scalar. Remark 2.
We note that the bound (5) for i = ω and arbitrary n can also beobtained by applying the weight distribution bound for the smallest eigenvalue ofthe Johnson graph. Now, we discuss the main ideas of the proof of Theorem 4.Let f be a real-valued function defined on the vertices of the Johnson graph J ( n, ω ) and let j , j ∈ { , , . . . , n } , j < j . We define a function f j ,j on thevertices of J ( n − , ω −
1) as follows: for any vertex y = ( y , y , . . . , y j − , y j +1 , . . . , y j − , y j +1 , . . . , y n )of J ( n − , ω − f j ,j ( y ) = f ( y , y , . . . , y j − , , y j +1 , . . . , y j − , , y j +1 , . . . , y n ) −− f ( y , y , . . . , y j − , , y j +1 , . . . , y j − , , y j +1 , . . . , y n ) . One of the important ingredients in the proof of Theorem 4 is the following.
Lemma 5 ([99], Lemma 1) . Let f be a λ i ( n, ω ) -eigenfunction of J ( n, ω ) , where j , j ∈ { , , . . . , n } and j < j . Then f j ,j is a λ i − ( n − , ω − -eigenfunctionof J ( n − , ω − or the all-zero function. Lemma 5 is a very useful tool for studying of eigenfunctions of the Johnsongraph. In particular, this lemma allows to apply induction on n , ω and i (we canuse the induction assumption for the function f j ,j ). Moreover, we suppose thatLemma 5 can be useful not only for the MS-problem but also for other problems.For example, recently in [97] Vorob’ev applied Lemma 5 for characterizationof equitable 2-partitions of the Johnson graph with the eigenvalue λ ( n, ω ).Finally, we note that Lemma 5 is an analogue of Lemma 3 (see Subsection 5.2).Let v = ( v , . . . , v n ) be a real non-zero vector such that v + . . . + v n = 0.We define the function f v on the vertices of the Johnson graph J ( n, ω ) by thefollowing rule: f v ( x , . . . , x n ) = (cid:88) ≤ i ≤ n : x i =1 v i . For k ∈ { , . . . , n − } denote v k = (1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) k , − kn − k , . . . , − kn − k (cid:124) (cid:123)(cid:122) (cid:125) n − k ) .
21n [73] Mogilnykh et al. proved the following improvement of Theorem 4 for λ ( n, ω )-eigenfunctions of J ( n, ω ). Theorem 5 ([73], Theorem 1) . Let f be an optimal λ ( n, ω ) -eigenfunction of J ( n, ω ) , where n ≥ ω and ω ≥ . Then f is f ,ω,n or f v k for some k ∈{ , . . . , n − } such that kωn ∈ N up to a permutation of coordinate positions andthe multiplication by a scalar.
8. Grassmann Graph
In this section, we give a survey of results on MS-problem for the Grassmanngraph. The Grassmann graph J q ( N, m ) is a distance-regular graph with the ver-tex set consisting of all m -dimensional subspaces of a vector space of dimension N over a finite field F q . Two vertices are adjacent whenever the correspondingsubspaces intersect in a ( m − t -designs of the lattices of subspaces over a finite field. In [52] G. D. Jamesmade a conjecture about the minimum support size of non-zero null t -designsof the lattices of subspaces over a finite field. S. Cho confirms the conjecturein [26] and in [27] characterizes all the null t -designs with minimum supports interms of maximal isotropic spaces of some bilinear form.Coming back to the Grassmann graph, we obtain the following theorem thatgives us the characterization of optimal λ D -eigenfunctions for the Grassmanngraph (compare with Theorems 1,2 from [27] and Theorem 5 from [66]). Formore details about null t -designs and totally isotropic spaces the reader is re-ferred to [27] and Chapter 18 of [52]. Theorem 6.
Suppose f is an optimal λ D -eigenfunction of the Grassmann graph J q ( N, m ) , where N ≥ m and λ D is its minimum eigenvalue. Then the cardi-nality of its support is D (cid:80) i =0 (cid:20) mi (cid:21) q · q i ( i − / which is also equal to the value of theweight distribution bound and the non-zeros of the function f correspond to themaximal totally isotropic subspaces of a m -dimensional space, equipped witha bilinear form B with a Gram matrix (cid:18) E m E m (cid:19) up to the equivalence (or,equivalently, with respect to a non-degenerate quadratic form Q ). Thus for the minimum eigenvalue of the Grassmann graph J q ( N, m ) MS-problem is solved and the weight distribution bound is achieved.
9. Bilinear Forms Graph
In this section, we give a survey of results on MS-problem for bilinear formsgraph. More details can be found [91]. The bilinear forms graph Bil q ( n, m ) is22 distance-regular graph with the vertex set V consisting of all n × m matricesover a finite field F q and two vertices being adjacent when their matrix differencehas a rank 1. For the sake of convenience, we will further suppose that m ≤ n .Thus the diameter D of the bilinear forms graph Bil q ( n, m ) is equal to m .Here as well as in the previous section we consider MS-problem only for thecase of minimum eigenvalue λ D . In this case we have the following lower boundfor the minimum support cardinality: m (cid:88) i =0 (cid:20) mi (cid:21) q · q i ( i − / It is interesting that the weight distribution for bilinear forms graph coincideswith that of the Grassmann graph. Later we will see the importance of thisconnection.The key idea here is that bilinear forms graph belongs to a family of so-called Delsarte cliques graphs (each edge lies in a constant number of Delsartecliques). Recall that a clique in a distance-regular graph of degree k is calledDelsarte clique if it consists of exactly 1 − k/λ D vertices. For more details aboutDelsarte cliques graphs, the reader is referred to [7].This property leads to the following observations: • Theorem 2 from [66] implies that for a Delsarte cliques graph G a function f is a λ D -eigenfunction of G if and only if for every Delsarte clique C itholds (cid:80) v ∈ C f ( v ) = 0. • Theorem 3 from [66] tells us that for a Delsarte clique graph G in caseof D = 2 if a weight distribution bound is achieved then non-zeros ofoptimal λ D -eigenfunction induce a complete bipartite graph. Note thatfor bilinear forms graph Bil q (2 ,
2) we have λ D = − q −
1, thus non-zerosof optimal λ D -eigenfunction achieving weight distribution bound induce acomplete bipartite graph K q +1 ,q +1 if such a function exists.It appears that in case of strongly regular bilinear forms graphs Bil p (2 , D = 2) the weight distribution bound can be achieved. An explicitconstruction of an optimal λ D -eigenfunction can be found in [91]. Below arethe statements that summarize this construction, but first let us introduce addi-tional notation. Suppose a is a generating element of the multiplicative group F ∗ p . Denote a = 0 , a = a , . . . , a p − = a p − , a p − = a p − = 1 e ∗ = [0 , , e = [1 , , e = [1 , a ] , . . . , e p − = [1 , a p − ] Theorem 7 ([91], Theorem 3) . Let
Bil p (2 , be a bilinear forms graph overa prime field F p . For any ν ∈ F p , such that ν (cid:54) = − ξ for all ξ ∈ F p , and b i = a i ν +1 the independent set N = (cid:110)(cid:20) (cid:21) e ∗ , b (cid:20) a ν (cid:21) e , . . . , b p − (cid:20) a p − ν (cid:21) e p − (cid:111) ogether with P = (cid:110)(cid:20) (cid:21) e ∗ ; b (cid:20) a ν (cid:21) e + b (cid:20) − a (cid:21) e ∗ ; . . . ; b p − (cid:20) a p − ν (cid:21) e + b p − (cid:20) − a p − (cid:21) e ∗ (cid:111) form non-zeros of λ D -eigenfunction f as two parts of a complete bipartite graph K p +1 , p +1 and f ( v ) = c, for v ∈ P , − c, for v ∈ N , , elsefor some constant c (cid:54) = 0 . Let us illustrate this theorem with some small example. Consider a bilinearforms graph Bil (2 , N = (cid:110)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21)(cid:111) , P = (cid:110)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21)(cid:111) . Here under the notation above a = 0, a = 2, a = 1, e ∗ = [0 , e = [1 , e = [1 , e = [1 , ν = 1, b = 1, b = 2, b = 2.Thus we proved that there exists a family of optimal λ D -eigenfunctions ofthe bilinear forms graph Bil p (2 ,
2) over a prime field F p that achieve the lowerbound. However the construction described above does not provide the fullcharacterization of all optimal λ D -eigenfunctions.What happens if we look at bilinear forms graphs of larger diameter? Itappears that the weight distribution bound cannot be achieved. And for provingthis the connection between bilinear graphs and the Grasssmann graphs comesin handy. The bilinear forms graph Bil q ( n, m ) with m ≤ n can be consideredas a subgraph of the Grassman graph J q ( n + m, m ) as follows: given a fixedsubspace W of dimension n , all m -spaces U such that U ∩ W = 0 are the verticesof Bil q ( n, m ). This embedding leads to the following result about the Delsartecliques of these graphs (see Lemma 8 from [91]): Lemma 6.
Delsarte cliques of bilinear forms graph
Bil q ( n, m ) are embedded inDelsarte cliques of a Grassmann graph J q ( n + m, m ) in the sense that for anyDelsarte cliques C and (cid:98) C of a bilinear forms graph and the Grassmann graphcorrespondingly, either C ⊂ (cid:98) C or C ∩ (cid:98) C = ∅ . Since for any λ D -eigenfunction the sum of its values over a Delsarte clique iszero, from the previous Lemma we immediately obtain the following Corollarywhich simply tells us that we can extend eigenfunctions of bilinear forms graphto those of the Grassmann graph: 24 orollary 1. Suppose f is a λ D -eigenfunction of a bilinear forms graph Bil q ( n, m ) .Then (cid:98) f is an eigenfunction of the Grassmann graph J q ( n + m, m ) , where (cid:98) f ( M ) = (cid:40) f ( M ) , if M ∈ V (Bil q ( n, m ))0 , else This corollary is crucial for the final result:
Theorem 8 ([91], Theorem 7) . Let
Bil q ( n, m ) be a bilinear forms graph ofdiameter D ≥ . Then the minimum support of an eigenfunction correspondingto the minimum eigenvalue does not achieve the weight distribution bound. The main idea behind the proof of this theorem can be described as follows.Suppose the opposite holds and f is an optimal λ D -eigenfunction that achievesthe weight distribution bound. Under the notation of Corollary 1, (cid:98) f is anoptimal λ D -eigenfunction of the Grassmann graph J q ( n + m, m ). According tothe Theorem 6 characterizing optimal eigenfunctions of the Grassmann graphs,the non-zeros of (cid:98) f correspond to the maximal totally isotropic spaces of a non-degenerate quadratic form Q . Now we recall the graphs embedding construction:there exists a subspace W of dimension n that trivially intersects with all themaximal totally isotropic subspaces. A well-known corollary from the Chevalleytheorem states that any non-degenerate quadratic form is isotropic on a vectorspace of dimension not less that 3 over the finite field F q (here the diameter ofa graph plays its role). Thus there exists a non-zero vector w ∈ W such that Q ( w ) = 0, therefore
10. Paley graph
In this section, we give a survey of results on MS-problem for the Paleygraph. Let q be an odd prime power, where q ≡ P ( q )is the Cayley graph on the additive group F + q of the finite field F q with thegenerating set of all squares in the multiplicative group F ∗ q . This graph is astrongly regular with parameters ( q, q − , q − , q − ). The eigenvalues of P ( q ) are λ = q − , λ = − √ q and λ = − −√ q . In [45] Goryainov et al. for i ∈ { , } proved that the minimum cardinality of the support of a λ i -eigenfunction of P ( q ), where q is an odd prime power, is q + 1. In what follows, in this sectionwe will discuss the results obtained in [45].Let q be an odd prime power and let β be a primitive element of the finitefield F q . Denote ω = β q − , Q = (cid:104) ω (cid:105) and Q = ω (cid:104) ω (cid:105) . We define the function25 β on the vertices of the Paley graph P ( q ) by the following rule: f β ( x ) = , if x ∈ Q ; − , if x ∈ Q ;0 , otherwise.One of the main results proved in [45] is the following. Theorem 9 ([45], Theorem 2) . Let q be an odd prime power and let β be aprimitive element of the finite field F q . Then the following statements hold: If q ≡ , then f β is a λ -eigenfunction of P ( q ) and | S ( f β ) | = q + 1 . If q ≡ , then f β is a λ -eigenfunction of P ( q ) and | S ( f β ) | = q + 1 . Since the Paley graph P ( q ) is self-complementary, Theorem 9 implies thatfor any i ∈ { , } P ( q ) has λ i -eigenfunction f such that | S ( f ) | = q + 1. On theother hand, by the weight distribution bound we obtain that a λ -eigenfunctionof P ( q ) has at least q + 1 non-zero values. Since P ( q ) is self-complementary,the same bound holds for a λ -eigenfunction of P ( q ). Thus, the minimumcardinality of the support of a λ i -eigenfunction of P ( q ), where i ∈ { , } , is q + 1.Now we discuss one interesting connection between the sets Q and Q andmaximal cliques of the Paley graph P ( q ). The maximum possible size of aclique of P ( q ) is q (all cliques of such size are Delsarte cliques). Blokhuis [20]determined all cliques and all cocliques of size q in P ( q ) and showed that theyare affine images of the subfield F q . Baker et al. [6] found maximal cliques oforder q +12 and q +32 for q ≡ q ≡ q +12 (from q +32 , respectively) to q .Kiermaier and Kurz [58] studied maximal integral point sets in affine planesover finite fields and found maximal cliques of size q +32 in P ( q ) for q ≡ Q and Q defined above, Goryainov et al. [45] constructed newmaximal cliques of size q +12 and q +32 for q ≡ q ≡ P ( q ). Theorem 10 ([45], Theorem 1) . Let q be an odd prime power and let β be aprimitive element of the finite field F q . Then the following statements hold: If q ≡ , then Q and Q are maximal cocliques of size q +12 in the graph P ( q ) . If q ≡ , then Q ∪ { } and Q ∪ { } are maximal cliques of size q +32 in the graph P ( q ) .
11. The Star graph
In this section, we give a survey of results on MS-problem for the Star graph.The
Star graph S n , n ≥
3, is the Cayley graph on the symmetric group Sym n { (1 i ) | i ∈ { , . . . , n }} . This graph is not distance-regular. The spectrum of the Star graph is integral [25, 62]. For n ≥
4, theeigenvalues of S n are ± ( n − k ), where 1 ≤ k ≤ n ; and the eigenvalues of S are {− , − , , } . The multiplicities of eigenvalues of the Star graph were studiedin [3, 54, 55]. In particular, explicit formulas for calculating multiplicities ofeigenvalues ± ( n − k ), where 2 ≤ k ≤
12, were found. In [53] Kabanov et al.found the minimum cardinality of the support of an ( n − S n and obtained a characterization of optimal ( n − n ≥ n = 3. In what follows, in this section we will consider the results obtained in[53].Now, we discuss one construction of optimal ( n − i ∈ { , . . . , n } and j, k ∈ { , . . . , n } , where j (cid:54) = k . We definethe function f j,ki on the vertices of the Star graph S n by the following rule: f j,ki ( π ) = , if π ( j ) = i ; − , if π ( k ) = i ;0 , otherwise.In [53, Lemma 2] it was shown that f j,ki is an ( n − S n and | S ( f j,ki ) | = 2( n − F = { f j,ki | i ∈ { , . . . , n } , j, k ∈ { , . . . , n } , j (cid:54) = k } . The main result proved in [53] is the following.
Theorem 11 ([53], Theorem 20) . Let f be an ( n − -eigenfunction of S n ,where n ≥ or n = 3 . Then | S ( f ) | ≥ n − . Moreover, | S ( f ) | = 2( n − if and only if f = c · ˜ f , where c is a real non-zero constant and ˜ f ∈ F . Now, we discuss the main ideas of the proof of Theorem 11. Firstly, we needsome definitions.Let M = ( m i,j ) be a real n × n matrix. We say that M is special if M isnon-zero and the following conditions hold:1. m i, = 0 for any i ∈ { , . . . , n } .2. m ,j = 0 for any j ∈ { , . . . , n } .3. (cid:80) nj =1 m i,j = 0 for any i ∈ { , . . . , n } .For a real n × n matrix M = ( m i,j ) denote g M ( n ) = |{ π ∈ Sym n | n (cid:88) i=1 m i ,π (i) (cid:54) = 0 }| . The key point of the proof of Theorem 11 is the following. For an arbitrary ( n − f of S n we can construct some special n × n matrix M ( f ) andmatch the permutations from Sym n with diagonals of M ( f ) in such a way that27he value of f on a permutation π is the sum of elements of the correspondingdiagonal of M ( f ). In other words, we have the equality | S ( f ) | = g M ( f ) ( n ) (6)for any ( n − f of S n . This observation allows us to reduce MS-problem for the Star graph S n and its eigenvalue n − n × n matrices. Problem 5.
Given a positive integer n , to find the minimum value of g M ( n ) for the class of special n × n matrices M . In [53, Theorem 19] for n ≥ n = 3 it was proved that g M ( n ) ≥ n − n × n matrix M . Moreover, in [53, Theorem 19] a classificationof special matrices in the equality case was obtained. Using these results andthe equality (6), we can finish the proof of Theorem 11.
12. Some remarks on optimal eigenfunctions of graphs
In this section, we give some observations on optimal eigenfunctions ofgraphs.Recall that MS-problem is formulated for arbitrary real-valued functionsfrom the corresponding eigenspace. Surprisingly, in many cases optimal eigen-functions take only three distinct values (for example, see Theorems 1, 2, 3, 4,11). But, in general case it is not true. For example, there are optimal ( − Figure 11: Optimal ( − There is an interesting connection between optimal eigenfunctions corre-sponding to the second largest eigenvalue of a given graph and completely regu-lar codes in this graph. In particular, an arbitrary optimal λ ( n, q )-eigenfunction( λ ( n, ω )-eigenfunction) of the Hamming graph H ( n, q ) (the Johnson graph J ( n, ω )) is the difference of the characteristic functions of two completely regu-lar codes of covering radius 1 (see [93, Theorem 3] and [99, Theorem 4]). The28tar graph S n does not have completely regular codes of covering radius 1 withthe eigenvalue n −
2. However, an arbitrary optimal ( n − S n is the difference of the characteristic functions of two completely regular codesof covering radius 2 (see [53, Lemma 22]).
13. Open problems
In this section, we briefly recall the main results on MS-problem and formu-late several open problems.Recall that Problem 3 is completely solved for all n ≥ q ≥
2. Inparticular, MS-problem for the Hamming graph H ( n, q ) is solved for all eigen-values. Moreover, a characterization of functions that are optimal in the space U [ i,j ] ( n, q ) was obtained for q ≥ i + j ≤ n and q ≥ i = j , i > n . Taking intoaccount these results, we formulate the following two problems for the Hamminggraph. Problem 6.
Characterize functions that are optimal in the space U [ i,j ] ( n, q ) forthe cases q = 2 and q ≥ , i + j > n (in this problem we assume that i < j ). Problem 7.
Characterize optimal λ i ( n, q ) -eigenfunctions of the Hamming graph H ( n, q ) for q ∈ { , } and i > n . MS-problem for the Doob graph D ( m, n ) is solved for the second largesteigenvalue λ ( m, n ) and the smallest eigenvalue λ m + n ( m, n ). So, it seems veryinteresting to consider the following question. Problem 8.
Solve MS-problem for the third largest eigenvalue λ ( m, n ) of theDoob graph D ( m, n ) . MS-problem for the bilinear forms graph Bil q ( n, m ) is solved for the smallesteigenvalue λ D in case n = m = 2 and q is prime. For bilinear forms graphs oflarger diameters over the arbitrary field it is proved that the weight distributionbound cannot be attained. This leads to the following interesting questions: Problem 9.
For the bilinear forms graph
Bil q ( n, m ) of diameter D : • Characterize optimal λ D -eigenfunctions in case of D = 2 for a prime q (including the case of n (cid:54) = m ). • Solve MS-problem for the smallest eigenvalue λ D in case of D = 2 andarbitrary q . • Solve MS-problem for the smallest eigenvalue λ D in case of D ≥ andarbitrary q . MS-problem for the Grassmann graph J q ( N, m ) is solved for the smallesteigenvalue λ D . Since the Grassmann graph can be considered as a q -analogueof the Johnson graph it may be interesting to consider the following question:29 roblem 10. Solve MS-problem for the second largest eigenvalue of the Grass-mann graph J q ( N, m ) . MS-problem for the Paley graph P ( q ) is solved for both non-principal eigen-values. We formulate the following problem for optimal eigenfunctions. Problem 11.
Characterize optimal λ -eigenfunctions and λ -eigenfunctions ofthe Paley graph P ( q ) . MS-problem for the Star graph S n is solved only for the second largesteigenvalue. So, the following question is very natural. Problem 12.
Solve MS-problem for the third largest eigenvalue of the Stargraph S n . At the end of this section we also would like to bring the attention of thereader to the following problems:
Problem 13.
For distance-regular graphs find the conditions for the weightdistribution bound to be achieved.
Problem 14.
For distance-regular graphs find a sharper lower bound on thecardinality of a graph eigenfunction support than the weight distribution bound.
Problem 15.
Find a lower bound on the cardinality of a graph eigenfunctionsupport for the Cayley graphs.
14. Acknowledgements
The authors are grateful to Evgeny Bespalov, Denis Krotov, Vladimir Potapovand Konstantin Vorob’ev for helpful discussions.
References [1] R. Ahlswede, H. K. Aydinian, L. H. Khachatrian, On perfect codes andrelated concepts, Designs, Codes and Cryptography 22(3) (2001) 221–237.[2] N. Alon, H. Huang, B. Sudakov, Nonnegative k -sums, fractional covers, andprobability of small deviations, Journal of Combinatorial Theory, Series B102(3) (2012) 784–796.[3] S. V. Avgustinovich, E. N. Khomyakova, E. V. Konstantinova, Multiplic-ities of eigenvalues of the Star graph, Siberian Electronic MathematicalReports 13 (2016) 1258–1270.[4] B. Aspvall, J. Gilbert, Graph Coloring Using Eigenvalue Decomposition,SIAM Journal on Algebraic Discrete Methods 5(4) (1984) 526–538.[5] R. B. Bapat, T. E. S. Raghavan, Nonnegative Matrices and Applications,Encyclopedia of Mathematics and its Applications, Cambridge UniversityPress, Cambridge, 1997. 306] R. D. Baker, G. L. Ebert, J. Hemmeter, A. J. Woldar, Maximal cliquesin the Paley graph of square order, Journal of Statistical Planning andInference 56(1) (1996) 33–38.[7] S. Bang, A. Hiraki, J. H. Koolen, Delsarte clique graphs, European Journalof Combinatorics 28(2) (2007) 501–516.[8] E. A. Bespalov, On the minimum supports of some eigenfunctions in theDoob graphs, Siberian Electronic Mathematical Reports 15 (2018) 258–266.[9] E. Bespalov, D. Krotov, A. Matiushev, A. Taranenko, K. Vorob’ev, Perfect2-colorings of Hamming graphs, arXiv:1911.13151, November 2019.[10] T. Bier, A distribution invariant for association schemes and strongly reg-ular graphs, Linear Algebra and its Applications 57 (1984) 105–113.[11] T. Bier, Some distribution numbers of the triangular association scheme,European Journal of Combinatorics 9(1) (1988) 19–22.[12] T. Bier, P. Delsarte, Some bounds for the distribution numbers of an asso-ciation scheme, European Journal of Combinatorics 9(1) (1988) 1–5.[13] T. Bier, P. Delsarte, Some distribution numbers of the hypercubic associ-ation scheme, European Journal of Combinatorics 9(1) (1988) 7–17.[14] T. Bier, N. Manickam, The first distribution invariant of the Johnsonscheme, SEAMS Bull. Math. 11 (1987) 61–68.[15] E. J. Billington, Combinatorial trades: A survey of recent results, in: De-signs, 2002, Further computational and constructive design theory (W. D.Wallis, ed.), 47–67, Math. Appl. 563, Kluwer Acad. Publ., Boston, 2003.[16] T. Biyiko˘glu, A discrete nodal domain theorem for trees, Linear Algebraand its Applications 360 (2003) 197–205.[17] T. Biyiko˘glu, W. Hordijk, J. Leydold, T. Pisanski, P. F. Stadler, GraphLaplacians, nodal domain and hyperplane arrangements, Linear Algebraand its Applications 390 (2004) 155–174.[18] T. Biyiko˘glu, J. Leydold, P. F. Stadler, Nodal domain theorems and bipar-tite subgraphs, Electronic Journal of Linear Algebra 13 (2005) 344–351.[19] T. Biyiko˘glu, J. Leydold, P. F. Stadler, Laplacian Eigenvectors of Graphs -Perron-Frobenius and Faber-Krahn type theorems, Lecture Notes in Math-ematics 1915, Springer, 2007.[20] A. Blokhuis, On subsets of GF ( q ) with square differences, IndagationesMathematicae 87(4) (1984) 369–372.[21] J. Borges, J. Rif`a, V. A. Zinoviev, On completely regular codes, Problemsof Information Transmission 55(1) (2019) 1–45.3122] A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-Regular Graphs,Berlin Heidelberg: Springer-Verlag, 1989.[23] A. E. Brouwer, W. H. Haemers, Spectra of graphs, Springer, New York,2012.[24] N. J. Cavenagh, The theory and application of Latin bitrades: a survey,Mathematica Slovaca 58(6) (2008) 691–718.[25] G. Chapuy, V. Feray, A note on a Cayley graph of Sym n arXiv:1202.4976v2(2012) 1–5.[26] S. Cho, On the support size of null designs of finite ranked posets, Combi-natorica, 19(4) (1999) 589–595.[27] S. Cho, Minimal null designs of subspace lattice over finite fields, Linearalgebra and its applications 282 (1998) 199–220.[28] A. Chowdhury, G. Sarkis, S. Shahriari, The Manickam-Mikl´os-Singhi con-jectures for sets and vector spaces, Journal of Combinatorial Theory, SeriesA 128 (2014) 84–103.[29] Y. Colin de Verdi`ere, Multiplicit`es des valeurs propres Laplaciens discretset laplaciens continus, Rendiconti di Matematica 13 (1993) 433–460 (inFrench).[30] D. Cvetkovi´c, M. Doob and H. Sachs, Spectra of graphs, 3rd edition, JohannAmbrosius Barth Verlag, Heidelberg, 1995.[31] D. Cvetkovi´c, P. Rowlinson and S. Simi´c, Eigenspaces of graphs, Encyclo-pedia of Mathematics and its Applications 66, Cambridge University Press,Cambridge, 1997.[32] D. Cvetkovi´c, P. Rowlinson and S. Simi´c, An introduction to the theory ofgraph spectra, London Mathematical Society Student Texts 75, CambridgeUniversity Press, Cambridge, 2010.[33] F. R. K. Chung, Spectral Graph Theory, American Mathematical Society,1997.[34] K. Ch. Das, The Laplacian spectrum of a graph, Computers & Mathematicswith Applications 48(5) (2004) 715–724.[35] E. R. van Dam, J. H. Koolen, H. Tanaka, Distance-regular graphs, TheElectronic Journal of Combinatorics (2016) t -design, EuropeanJournal of Combinatorics 4(1) (1983) 21–23.[42] G. Frobenius, ¨Uber Matrizen aus nicht negativen Elementen, Preussis-che Akademie der Wissenschaften Berlin: Sitzungsberichte der PreußischenAkademie der Wissenschaften zu Berlin, 1912.[43] J. Friedman, Some geometric aspects of graphs and their eigenfunctions,Duke Mathematical Journal 69(3) (1993) 487–525.[44] C. D. Godsil, Algebraic Combinatorics, Chapman and Hall MathematicsSeries, Chapman Hall, New York, 1993.[45] S. Goryainov, V. Kabanov, L. Shalaginov, A. Valyuzhenich, On eigenfunc-tions and maximal cliques of Paley graphs of square order, Finite Fieldsand Their Applications 52 (2018) 361–369.[46] A. S. Hedayat, G. B. Khosrovshahi, Trades, in: Handbook of CombinatorialDesigns (C. J. Colbourn and J. H. Dinitz, eds.), Second Edition, 644–648,Chapman Hall/CRC Press, Boca Raton, 2007.[47] O. Heden, A survey of perfect codes, Advances in Mathematics of Commu-nications 2(2) (2008) 223–247.[48] H. Huang, B. Sudakov, The minimum number of nonnegative edges inhypergraphs, The Electronic Journal of Combinatorics 21(3) (2014) v, k, t ) trades, Journal of StatisticalPlanning and Inference 13 (1986) 179–191.[50] F. Ihringer, A note on the Manickam-Mikl´os-Singhi conjecture for vectorspaces, European Journal of Combinatorics 52(A) (2016) 27–39.[51] F. Ihringer, K. Meagher, Mikl´os-Manickam-Singhi conjectures on partialgeometries, Designs, Codes and Cryptography 86 (2018) 1311–1327.[52] G. D. James, Representations of general linear groups, LMS Lecture NoteSeries 94, Cambridge University Press, 1984.3353] V. Kabanov, E. V. Konstantinova, L. Shalaginov, A. Valyuzhenich, Mini-mum supports of eigenfunctions with the second largest eigenvalue of theStar graph, The Electronic Journal of Combinatorics 27(2) (2020) q -arySteiner and other-type trades, Discrete Mathematics 339(3) (2016) 1150–1157[67] C. R. MacCluer, The Many Proofs and Applications of Perron’s Theorem,SIAM Review 42(3) (2000) 487–498.3468] N. Manickam, D. Mikl´os, On the number of non-negative partial sums of anon-negative sum, Colloq. Math. Soc. Janos Bolyai 52 (1988) 385–392.[69] N. Manickam, N. M. Singhi, First distribution invariants and EKR theo-rems, Journal of Combinatorial Theory, Series A 48(1) (1988) 91–103.[70] R. Merris, Laplacian graph eigenvectors, Linear Algebra and its Applica-tions 278 (1998) 221–236.[71] I. Yu. Mogilnykh, F. I. Solov’eva, On existence of perfect bitrades in Ham-ming graphs, Discrete Mathematics 343(12) (2020) 112128.[72] I. Mogilnykh, A. Valyuzhenich, Equitable 2-partitions of the Hamminggraphs with the second eigenvalue, Discrete Mathematics 343(11) (2020)112039.[73] I. Mogilnykh, K. Vorob’ev, A. Valyuzhenich, MMS-type problems for John-son scheme, Siberian Electronic Mathematical Reports 15 (2018) 1663–1670.[74] A.Y. Ng, Michael Jordan, and Y Weiss, On Spectral Clustering: Analysisand an Algorithm, Proceedings of the 14th International Conference onNeural Information Processing Systems: Natural and Synthetic (2001) 849–856.[75] L. Page, S. Brin, R. Motwani, T. Winograd, The PageRank Citation Rank-ing: Bringing Order to the Web, Technical Report, Stanford InfoLab, 1999.[76] O. Perron, Zur theorie der matrices, Mathematische Annalen 64 (1907)248–263.[77] A. Pokrovskiy, A linear bound on the Manickam-Mikl´os-Singhi conjecture,Journal of Combinatorial Theory, Series A 133 (2015) 280–306.[78] V. N. Potapov, On perfect colorings of Boolean n -cube and correlationimmune functions with small density, Siberian Electronic MathematicalReports 7 (2010) 372–382 (in Russian).[79] V. N. Potapov, Cardinality spectra of components of correlation immunefunctions, bent functions, perfect colorings, and codes, Problems of Infor-mation Transmission 48(1) (2012) 47–55.[80] V. N. Potapov, On perfect 2-colorings of the q -ary n -cube, Discrete Math-ematics 312(6) (2012) 1269–1272.[81] A. Pothen, H. D. Simon, K.-P. Liou, Partitioning Sparse Matrices withEigenvectors of Graphs, SIAM Journal on Matrix Analysis and Applica-tions 11(3) (1990) 430–452.[82] D. L. Powers, Graph partitioning by eigenvectors, Linear Algebra and itsApplications 101 (1988) 121–133. 3583] N. Saito, Laplacian eigenfunctions resource page,