Addressing Graph Products and Distance-Regular Graphs
Sebastian M. Cioabă, Randall J. Elzinga, Michelle Markiewitz, Kevin Vander Meulen, Trevor Vanderwoerd
aa r X i v : . [ m a t h . C O ] J a n Addressing Graph Products and Distance-Regular Graphs
Sebastian M. Cioab˘a ∗ , Randall J. Elzinga † ,Michelle Markiewitz ‡ , Kevin Vander Meulen § , and Trevor Vanderwoerd ¶ March 19, 2018
Abstract
Graham and Pollak showed that the vertices of any connected graph G can be assigned t -tuples withentries in { , a, b } , called addresses, such that the distance in G between any two vertices equals thenumber of positions in their addresses where one of the addresses equals a and the other equals b . Inthis paper, we are interested in determining the minimum value of such t for various families of graphs.We develop two ways to obtain this value for the Hamming graphs and present a lower bound for thetriangular graphs.Keywords: distance matrix, spectrum, triangular graphs, Hamming graphs, graph addressing. A t - address is a t -tuple with entries in { , a, b } . An addressing of length t for a graph G is an assignment of t -addresses to the vertices of G so that the distance between two vertices is equal to the number of locationsin the addresses at which one of the addresses equals a and the other address equals b . For example, we havea 3-addressing of a graph in Figure 1. Graham and Pollak [13] introduced such addressings, using symbols {∗ , , } instead { , a, b } , in the context of loop switching networks. ab b bbaaaa t such that G has an addressing of length t . We denote such amininum by N ( G ). Graham and Pollak [13, 14] showed that N ( G ) equals the biclique partition number ofthe distance multigraph of G . Specifically, the distance multigraph of G , D ( G ), is the multigraph with thesame vertex set as G where the multiplicity of any edge uv equals the distance in G between vertices u and v . The biclique partition number bp( H ) of a multigraph H is the minimum number of complete bipartitesubgraphs (bicliques) of H whose edges partition the edge-set of H . This parameter and its close coveringvariations have been studied by several researchers and appear in different contexts such as computationalcomplexity or geometry (see for example, [8, 13, 14, 15, 16, 19, 20, 25]). Graham and Pollak deduced that N ( G ) ≤ r ( n −
1) for any connected G of order n and diameter r and conjectured that N ( G ) ≤ n − G of order n . This conjecture, also known as the squashed cube conjecture , was proved byWinkler [24]. ∗ Department of Mathematical Sciences, University of Delaware, Newark, DE 19716-2553, USA; [email protected] . Researchpartially supported by NSF grant DMS-1600768. † Department of Mathematics, Royal Military College, Kingston, ON K7K 7B4, Canada; [email protected] . Currentaddress: Info-Tech Research Group, London, ON, N6B 1Y8, Canada. ‡ Department of Mathematical Sciences, University of Delaware, Newark, DE 19716-2553, USA; [email protected] . Researchsupported by the Summer Scholars Undergraduate Program at the University of Delaware. § Department of Mathematics, Redeemer University College, Ancaster, ON L9K 1J4, Canada; [email protected] . Re-search supported by NSERC Discovery Grant 203336. ¶ Department of Mathematics, Redeemer University College, Ancaster, ON L9K 1J4, Canada; [email protected]. N ( G ) below, Graham and Pollak used an eigenvalue argument on the adjacency matrix of D ( G ). Specifically, if M is a symmetric real matrix, let n + ( M ) , n − ( M ) , and n ( M ) denote the number ofeigenvalues of M (including multiplicity) that are positive, negative and zero, respectively. The inertia of M is the triple ( n + ( M ) , n ( M ) , n − ( M )). The adjacency matrix of D ( G ) will be denoted by D ( G ); we willalso refer to D ( G ) as the distance matrix of G . The inertia of distance matrices has been studied by variousauthors for many classes of graphs [3, 17, 18, 26]. Witsenhausen (cf. [13, 14]) showed that N ( G ) ≥ max( n + ( D ( G )) , n − ( D ( G ))) . (1)Letting J n denotes the all one n × n matrix and I n denotes the n × n identity matrix, and noting that n − ( D ( K n )) = n − ( J n − I n ), Graham and Pollak [13, 14] used the bound (1) to conclude that N ( K n ) = n − . (2)Graham and Pollak [13, 14] also determined N ( K n,m ) for many values of n and m . The determinationof N ( K n,m ) for all values of n and m was completed by Fujii and Sawa [11]. A more general addressingscheme, allowing the addresses to contain more than two different nonzero symbols, was recently studied byWatanabe, Ishii and Sawa [23]. The parameter N ( G ) has been determined when G is a tree or a cycle [14],as well as one particular triangular graph T [25], described in Section 5. For the Petersen graph P , Elzinga,Gregory and Vander Meulen [10] showed that N ( P ) = 6. To the best of our knowledge, these are the onlygraphs G for which addressings of length N ( G ) have been determined. We will say a t -addressing of G is optimal if t = N ( G ). An addressing is eigensharp [19] if equality is obtained in (1).In this paper, we study optimal addressings of Cartesian graph products and the distance-regular graphsknown as triangular graphs. Let H ( n, q ) is the Hamming graph whose vertices are the n -tuples over analphabet with q letters with two n -tuples being adjacent if and only if their Hamming distance is 1. Wegive two different proofs showing that N ( H ( n, q )) = n ( q − H (1 , q ) = K q . We determine that the triangular graphs are not eigensharp. Suppose that G i , i = 1 , . . . , k are graphs and that each graph G i has vertex set V ( G i ) and order n i = | V ( G i ) | .The Cartesian product G (cid:3) G (cid:3) · · · (cid:3) G k of G , G , . . . , G k is the graph with vertex set V ( G ) × V ( G ) ×· · · × V ( G k ), order n = n n · · · n k , and with vertices x = ( x , . . . , x k ) and y = ( y , . . . , y k ) adjacent if forsome index j , x j is adjacent to y j in G j while x i = y i for all remaining indices i = j . Thus, if d and d i denote distances in G and G i , respectively, then d ( x , y ) = k X i =1 d i ( x i , y i ) (3)It follows that if each G i , i = 1 , . . . , k is given an addressing, then each vertex x of G may be addressed byconcatenating the addresses of its components x i . Therefore, the parameter N is subadditive on Cartesianproducts; that is, if G = G (cid:3) · · · (cid:3) G k (4)then N ( G ) ≤ N ( G ) + · · · + N ( G k ) (5)Note that N ( G ) + · · · + N ( G k ) ≤ (cid:16)P ki =1 n i (cid:17) − k ≤ (cid:16)Q ki =1 n i (cid:17) − n −
1. Thus (5) can improve onWinkler’s upper bound of n − G is a Cartesian product. Question 2.1.
Must equality holds in (5) for all choices of G i ? Remark 3.4 might provide a possiblecounterexample. If v , . . . , v n denote the vertices of a connected graph G , the distance matrix D ( G ) of G is the n × n matrixwith entries D ( G ) ij = d ( v i , v j ). Because G is connected, its adjacency matrix A ( G ) and its distance matrix D ( G ) are irreducible symmetric nonnegative integer matrices and by the Perron-Frobenius Theorem (see [5,Proposition 3.1.1] or [12, Theorem 8.8.1]), the largest eigenvalue of each of these matrices has multiplicity1. We call this largest eigenvalue the Perron value of the matrix and often denote it by ρ .To obtain a formula for the distance matrix of a Cartesian product of graphs, we will use an additiveanalogue of the Kronecker product of matrices. Recall first that if A is an n × n matrix and B an m × m matrix, with x ∈ R n , y ∈ R m , then the Kronecker products A ⊗ B and x ⊗ y are defined by A ⊗ B = a B a B · · · a m Ba B a B · · · a m B ... ... ... a m B a m B · · · a mm B and x ⊗ y = x yx y ... x n y (6)For the additive analogue, we use the symbol ⋄ and define A ⋄ B and x ⋄ y by A ⋄ B = a + B a + B · · · a m + Ba + B a + B · · · a m + B ... ... ... a m + B a m + B · · · a mm + B and x ⋄ y = x + yx + y ... x n + y (7)If G = G (cid:3) G (cid:3) · · · (cid:3) G k , then the additive property (3) implies that D ( G ) = D ( G ) ⋄ D ( G ) ⋄ · · · ⋄ D ( G k ) (8)Observe that A ⋄ B = A ⊗ J m + J n ⊗ B and x ⋄ y = x ⊗ m + n ⊗ y (9)where n ∈ R n denotes the column vector whose entries are all one. Let n ∈ R n denote the column vectorwith all zero entries. The following two lemmas are due to D.A. Gregory. Lemma 3.1.
Let A and B be n × n and m × m real matrices respectively. If A x = λ x and x ⊤ n = P x i = 0 ,then ( A ⋄ B )( x ⋄ m ) = mλ ( x ⋄ m ) . Also, if 1 Tm y = 0 and By = µy , then ( A ⋄ B )( n ⋄ y ) = nµ ( n ⋄ y ) .Proof . We use properties of Kronecker products:( A ⋄ B )( x ⋄ ) = ( A ⊗ J m + J n ⊗ B )( x ⊗ m + n ⊗ )= Ax ⊗ J m m + J n x ⊗ B m + A n ⊗ J m J n ⊗ B = Ax ⊗ J m m = λm ( x ⊗ m )= λm ( x ⊗ m + n ⊗ ) = λm ( x ⋄ )A similar argument work the vector ( n ⋄ y ).Throughout we will say a square matrix is k - regular if it has constant row sum k . Lemma 3.2. If A is ρ A -regular and B is ρ B -regular then A ⋄ B is ( mρ A + nρ B )-regular.Proof . Using properties of Kronecker products,( A ⋄ B )( n ⊗ m ) = ( A ⊗ J m + J n ⊗ B )( n ⊗ m )= A n ⊗ J m m + J n n ⊗ B m )= ρ A m ( n ⊗ m ) + nρ B ( n ⊗ m )= ( ρ A m + nρ B )( n ⊗ m ) . Thus ( A ⋄ B ) = ( ρ A m + nρ B ) . This approach was suggested by the late David A. Gregory.
Lemma 3.3. If G = G (cid:3) G (cid:3) · · · (cid:3) G k and each D ( G i ) , i = 1 , . . . , k is regular then(a) n − ( D ( G )) ≥ P i n − ( D ( G i )) , and(b) n + ( D ( G )) ≥ P i ( n + ( D ( G i )) − .Proof . Because D ( G i ) regular, D ( G i ) n i = ρ i n i where ρ i is the Perron value of D ( G i ). Then n i isa ρ i -eigenvector of D ( G i ) and R n i has an orthogonal basis of eigenvectors of D ( G i ) that includes n i as amember. Thus, Lemma 3.1 with A = D ( G i ) and B = D ( (cid:3) j = i G j ) implies that the n i − D ( G i ) in the basis other than n i contribute n i − D ( G ).An eigenvector of D ( G i ) with eigenvalue λ = ρ i contributes an eigenvector of D ( G ) with eigenvalue λ ( n n · · · n k ) /n i = λn/n i . This eigenvalue has the same sign as λ if λ = 0. Also, if i = j , then each of the n i − D ( G ) by D ( G i ) is orthogonal to each of the analogous n j − D ( G ) by D ( G j ). Thus, the inequality (a) claimed for n − follows. Also, by Lemma 3.1, n isan eigenvector of D ( G ) with a positive eigenvalue ρ , so the inequality (b) for n + follows. Remark 3.4. (Observed by D.A. Gregory) The inequality in Lemma 3.3 need not hold if the regularityassumption is dropped. For example, suppose G = G (cid:3) G where G is the graph on 6 vertices obtained K , by inserted an edge incident to the two vertices in the part of size 2. Then n − ( D ( G )) = 5 but n − ( D ( G )) = 9 < N ( G ) = 5, so 9 ≤ N ( G ) ≤
10 by (1) and (5). An affirmative answer toQuestion 2.1 would imply N ( G ) = 10.If each D ( G i ) in (8) is regular, then Lemma 3.3 accounts for 1 + P i (rank D ( G i ) −
1) = k − P i rank D ( G i ) of the rank D ( G ) nonzero eigenvalues of D ( G ). The following results imply that if each D ( G i ) is regular then all of the remaining eigenvalues must be equal to zero. Equivalently, the results willimply that if each D ( G i ) in (8) is regular, then equality must hold in Lemma 3.3(a) and (b).The next result (proved by D.A. Gregory) is obtained by exhibiting an orthogonal basis of R mn consistingof eigenvectors of A ⋄ B when A and B are symmetric and regular. Theorem 3.5.
Let A be a regular symmetric real n × n matrix with A n = ρ A n with ρ A > and let B bea regular symmetric matrix of order m with B m = ρ B m with ρ B > . Then(a) n − ( A ⋄ B ) = n − ( A ) + n − ( B ) ,(b) n + ( A ⋄ B ) = n + ( A ) + n + ( B ) − , and(c) n o ( A ⋄ B ) = nm − n − m + 1 + n o ( A ) + n o ( B ) .Proof . As in Lemma 3.3, Lemma 3.1 can be used to provide eigenvectors that imply that n − ( A ⋄ B ) ≥ n − ( A ) + n − ( B ) and n + ( A ⋄ B ) ≥ n + ( A ) + n + ( B ) −
1. It remains to exhibit an adequate number of linearlyindependent eigenvectors of A ⋄ B for the eigenvalue 0.If ⊤ n x = 0 and ⊤ m y = 0, then( A ⋄ B )( x ⊗ y ) = ( A ⊗ J m + J n ⊗ B )( x ⊗ y ) = Ax ⊗ m + 0 n ⊗ By = 0 nm This accounts for at least ( n − m −
1) = nm − n − m + 1 orthogonal eigenvectors of A ⋄ B with eigenvalue0. Moreover, if Au = 0 then ⊤ n u = 0 and hence, by Lemma 3.1, ( A ⋄ B )( u ⋄ m ) = 0 nm . Likewise, if Bv = 0then ⊤ m v = 0 and by Lemma 3.1, ( A ⋄ B )(0 n ⋄ v ) = 0 nm . If each set of vectors x , each set of vectors y , eachset of vectors u and each set of vectors v that occur above are chosen to be orthogonal, then the resultingvectors x ⊗ y, u ⊗ m , n ⊗ v will be orthogonal. Thus, n o ( A ⋄ B ) ≥ nm − n − m + 1 + n o ( A ) + n o ( B ). Addingthe three inequalities obtained above, we get nm = n − ( A ⋄ B ) + n + ( A ⋄ B ) + n o ( A ⋄ B ) ≥ n − ( A ) + n − ( B ) + n + ( A ) + n + ( B ) − nm − n − m + 1 + n o ( A ) + n o ( B )= nm. Thus equality holds in each of the three inequalities.
Corollary 3.6. If G = G (cid:3) G (cid:3) · · · (cid:3) G k and each D ( G i ) , i = 1 , . . . , k is regular then(a) n − ( D ( G )) = P i n − ( D ( G i )) , and (b) n + ( D ( G )) = 1 + P i ( n + ( D ( G i )) − . Remark 3.7.
In the proof of Theorem 3.5, whether or not A and B are symmetric and regular, we alwayshave ( A ⋄ B )( x ⊗ y ) = 0 nm whenever ⊤ n x = 0 and ⊤ m y = 0. Thus,Nul( A ⋄ B ) ≥ ( n − m − A and B of orders n and m , respectively.In order to apply Lemma 3.3 to the Cartesian product (4), it would be helpful to have conditions onthe graphs G i that imply that the distance matrices D ( G i ) are regular. The following remark gives a fewexamples of graphs whose distance matrix has constant row sums. Remark 3.8. (Regular distance matrices)
1. If G is either distance regular or vertex transitive, then D ( G ) is ρ -regular where ρ is equal to the sumof all the distances from a particular vertex to each of the others.2. If G is a regular graph of order n and the diameter of G is either one or two, then D ( G ) is ρ -regularwith ρ = 2( n − − ρ A where ρ A is the Perron value of the adjacency matrix A of G . For if A is the adjacencymatrix of G , then D ( G ) = A + 2( J n − I n − A ) = 2( J n − I n ) − A . This holds, for example, when G is thePetersen graph or G = K n (the complete graph on n vertices) or when G = K m,m (the complete balancedbipartite graph on n = 2 m vertices). Question 3.9.
What are other conditions on a graph that imply that its distance matrix is regular?
Theorem 3.10.
Let G = G (cid:3) G (cid:3) · · · (cid:3) G k . If D ( G i ) is regular and N ( G i ) = n − ( D ( G i )) for i = 1 , . . . , k, then N ( G ) = P ki =1 N ( G i ) .Proof . By the lower bound (1) and the subadditivity property (5), P i N ( G i ) ≥ N ( G ) ≥ n − D ( G ), whereby Lemma 3.3(a), n − ( D ( G )) ≥ P i n − ( D ( G i )) = P i N ( G i ). Example 3.11.
The Cartesian product of complete graphs, G = K n (cid:3) K n (cid:3) · · · (cid:3) K n k is also known as aHamming graph. By (2) and Theorem 3.10, it follows that N ( G ) = P ki =1 ( n i − Let n ≥ q ≥ H ( n, q ) can be described as thewords of length n over the alphabet { , . . . , q } . Two vertices ( x , . . . , x n ) and ( y , . . . , y n ) are adjacent ifandonly if their Hamming distance is 1. If n = 1, H (1 , q ) is the complete graph K q . The following result, canbe derived from Example 3.11, but we provide another interesting and constructive argument. Theorem 4.1. If n ≥ and q ≥ , then N ( H ( n, q )) = n ( q − .Proof . We first prove that the length of any addressing of H ( n, q ) is at least n ( q − ≤ k ≤ n ,let A k denote the distance k adjacency matrix of H ( n, q ). The adjacency matrix of the distance multigraphof H ( n, q ) is D ( H ( n, q )) = P nk =1 kA k . The graph H ( n, q ) is distance-regular and therefore, A , . . . , A n aresimultaneously diagonalizable. The eigenvalues of the matrices A , . . . , A n were determined by Delsarte inhis thesis [9] (see also [22, Theorem 30.1]). Proposition 4.2.
Let k ∈ { , . . . , n } . The eigenvalues of A k are given by the Krawtchouk polynomials: λ k,x = k X i =0 ( − q ) i ( q − k − i (cid:18) n − ik − i (cid:19)(cid:18) xi (cid:19) (10) with multiplicity (cid:0) nx (cid:1) ( q − x for x ∈ { , , . . . , n } . The Perron value of A k equals (cid:0) nk (cid:1) ( q − k . Thus, the Perron value of D ( H ( n, q )) equals P nk =1 (cid:0) nk (cid:1) k ( q − k = nq n − ( q −
1) and has multiplicity one. The other eigenvalues of D ( H ( n, q )) are µ x = n X k =1 kλ k,x = n X k =1 k k X i =0 ( − q ) i ( q − k − i (cid:18) n − ik − i (cid:19)(cid:18) xi (cid:19) = n X i =0 ( − q ) i (cid:18) xi (cid:19) n X k = i k ( q − k − i (cid:18) n − ik − i (cid:19) = n X i =0 ( − q ) i (cid:18) xi (cid:19) n − i X t =0 ( i + t )( q − t (cid:18) n − it (cid:19) = n X i =0 ( − q ) i (cid:18) xi (cid:19) (cid:0) nq n − i − ( n − i ) q n − i − (cid:1) = q n − n X i =0 (cid:18) xi (cid:19) ( − i i = ( − q n − if x = 10 if x ≥ . with multiplicity (cid:0) nx (cid:1) ( q − x for 1 ≤ x ≤ n . Thus, the spectrum of D ( H ( n, q )), with multiplicities, is (cid:18) nq n − ( q − − q n − n ( q − q n − − q ( n − (cid:19) . (11)where the first row contains the distinct eigenvalues of D ( H ( n, q )) and the second row contains their multi-plicities. Thus, max( n − ( D ( H ( n, q ))) , n + ( D ( H ( n, q )))) = n ( q −
1) and Witsenhausen’s inequality (1) impliesthat N ( H ( n, q )) ≥ n ( q − n ( q −
1) is the optimal length of an addressing of H ( n, q ), we describe a partition of the edge-setof the distance multigraph of H ( n, q ) into exactly n ( q −
1) bicliques. For 1 ≤ i ≤ n and 1 ≤ t ≤ q −
1, definethe biclique B i,t whose color classes are { ( x , . . . , x n ) : x i = t } and { ( x , . . . , x n ) : x i ≥ t + 1 } . One can check easily that if u and v are two distinct vertices in H ( n, q ), there exactly d H ( u, v ) bicliques B i,t containing the edge uv . Thus, the n ( q −
1) bicliques B i,t partition the edge set of the distance multigraphof H ( n, q ) and N ( H ( n, q )) ≤ n ( q − H ( n, q ) was also computed by Indulal [17]. The triangular graph T n is the line graph of the complete graph K n on n vertices. When n ≥
4, the triangulargraph T n is a strongly regular graph with parameters (cid:0)(cid:0) n (cid:1) , n − , n − , (cid:1) . The adjacency matrix of T n has spectrum (cid:18) n − n − − n − (cid:0) n (cid:1) − n (cid:19) (12)and therefore, the distance matrix D ( T n ) has spectrum (cid:18) ( n − n −
2) 2 − n n − (cid:0) n (cid:1) − n (cid:19) (13)Witsenhausen’s inequality (1) implies that N ( T n ) = bp ( D ( T n )) ≥ n − n ≥ T is equivalent to determining the biclique partition number of the multigraphobtained from K by adding one perfect matching. This formulation of the problem was studied by Zaks [25]and Hoffman [16] (see also Section 6). Zaks proved that N ( T ) = 4 and hence T is not eigensharp. We willreprove the lower bound of Zaks [25] in Lemma 5.2 using a technique of [10]. The argument of Lemma 5.2will then be used to show that T n is not eigensharp for any n ≥ addressing matrix of a t -addressing is the n -by- t matrix M ( a, b ) where the i -th row of M ( a, b ) is theaddress of vertex i . M ( a, b ) can be written as a function of a and b : M ( a, b ) = aX + bY, where X and Y are matrices with entries in { , } . Elzinga et al. [10] use the addressing matrix, along withresults from Brandenburg et al. [4] and Gregory et al. [15], to create the following theorem: Theorem 5.1. [10] Let M ( a, b ) be the address matrix of an eigensharp addressing of a graph G . Then forall real scalars a, b , each column of M ( a, b ) is orthogonal to the null space of D ( G ) . Also, the columns of M (1 , are linearly independent, as are the columns of M (0 , . In [10, Theorem 3], Elzinga et al. use Theorem 5.1 to show that the Petersen graph does not have aneigensharp addressing. We will use a similar approach on triangular graphs.
Lemma 5.2.
The triangular graph T is not eigensharp, that is, N ( T ) ≥ .Proof . Suppose T is eigensharp. Let D = D ( T ). By Theorem 5.1, all vectors in the null space of D areorthogonal to the columns of a 6 × M ( a, b ).We can construct null vectors of D in the following manner, referring to the entries of the null vector as labels : choose any two non-adjacent vertices, and label them with zeroes. The remaining four vertices forma 4-cycle, which will be alternatingly labelled with 1 and −
1, as in Figure 2.1 − − T with a D ( T ) null-vector labellingLet w ( a, b ) be any column of M ( a, b ). We claim that w ( a, b ) has at least three a -entries, and at leastthree b -entries. For convenience, we’ll refer to vertices corresponding to the a -entries of w ( a, b ) as a -vertices .If there are no a -vertices, then since M ( a, b ) = aX + bY , one of the columns of X is the zero vector. Itwould follow that the columns of M (1 ,
0) are linearly dependent, contradicting Theorem 5.1. Thus w ( a, b )has at least one a and at least one b entry.Suppose w ( a, b ) has at most 2 a -entries. There are three cases we will consider: there are two adjacent a -vertices, there are two non-adjacent a -vertices, or there is exactly one a -vertex. In each case, we willconstruct a null vector x of D which is not orthogonal to u = w (1 , w ( a, b ) has two adjacent a -vertices. Label one of the a -vertices with a zero and the adjacent a -vertex with 1. Then x T u = 1 = 0. Suppose w ( a, b ) has two non-adjacent a -vertices. Label the two a -vertices with 1. Then x T u = 2 = 0. Suppose there is only one a -vertex in w . Label the a -vertex with 1.Then x T u = 1 = 0.Therefore, at least three positions of w ( a, b ) have the value a . Similarly, at least three positions of w ( a, b )must have value b .Since each column of M ( a, b ) has at least three a and three b -entries, there are at least nine a, b pairingscorresponding to each column. Since M ( a, b ) has three columns, there are 27 a, b column-wise pairs in total.However the number of column-wise a, b pairs in the addressing matrix M ( a, b ) is simply the number ofedges in D ( T ), namely 18. This contradiction implies that T is not eigensharp. Theorem 5.3.
The triangular graph T n is not eigensharp for any n ≥ , that is, N ( T n ) ≥ n for all n ≥ .Proof . Note that T is an induced subgraph of T n since K is an induced subgraph of K n . Let T be aninduced subgraph of T n isomorphic to T .Suppose T n is eigensharp. Let M ( a, b ) be an eigensharp addressing matrix of T n . By Theorem 5.1, thecolumns of M ( a, b ) are orthogonal to any null vector of D ( T n ). Let w be one of the columns of M ( a, b ). Wecan construct a null vector y of D ( T n ) by labelling the vertices corresponding to T as described in Figure 2and labelling the remaining vertices of T n with zeroes. In [10], it is described that the columns of anaddressing matrix correspond to bicliques that partition the edgeset of D ( T n ). Every biclique decompositionof D ( T n ) induces a decomposition of D ( T ), an induced subgraph of D ( T n ). Lemma 5.2 tells us that at least4 bicliques are needed to decompose D ( T ). Therefore, there must be at least four columns of M ( a, b ) whose6 entries corresponding to T have at least one a and one b . The proof of Lemma 5.2 guarantees that each ofthese 4 vectors, restricted to the vertices of T , has at least three a entries and three b entries. Since D ( T ) isan induced subgraph of D ( T n ), there are the same number of edges between the corresponding vertices inthe two graphs. However, a contradiction occurs: the eigensharp addressing implies that there are at least36 edges in D ( T ), but there are in fact 18. Therefore T n is not eigensharp.For the triangular graph T (the complement of the Petersen graph), the following six bicliques partitionthe edge set of D ( T ): { , , , } ∪ { , , , , , }{ , } ∪ { , , , , }{ , } ∪ { , , , , }{ , , } ∪ { , , }{ } ∪ { , , , }{ } ∪ { , , } . Thus, by Theorem 5.3, we know that 5 ≤ N ( T ) ≤ K ,..., K ,..., with m colorclasses of size 2 is a highly non-trivial open problem. It is equivalent to finding the biclique partition numberof the multigraph obtained from the complete graph K m by adding a perfect matching. Motivated byquestions in geometry involving nearly-neighborly families of tetrahedra, this problem was studied by Zaks[25] and Hoffman [16]. The best current results for N ( K ,..., ) = bp ( D ( K ,..., )) are due to these authors(the lower bound is due to Hoffman [16] and the upper bound is due to Zaks [25]): m + ⌊√ m ⌋ − ≤ N ( K ,..., ) ≤ ( m/ − m is even(3 m − / m is odd . (14) We conclude this paper with some open problems.1. Must equality hold in (5) for all choices of G i ?2. It is known that determining bp ( G ) for a graph G is an NP-hard problem (see [19]). This problem isNP-hard even when restricted to graphs G with maximum degree ∆( G ) ≤ N ( G ) for general graphs G ? How about graphs with ∆( G ) >
3, or other familiesof graphs ?3. What is N ( T n ) for n ≥ T n is a special case of a Johnson graph. For n ≥ m ≥
2, the Johnson graph J ( n, m ) has as its vertex set the m -subsets of [ n ] with two m -subsets being adjacent if and only iftheir intersection has size m −
1. The Johnson graph is distance-regular and its eigenvalues weredetermined by Delsarte in his thesis [9] (see also [22, Theorem 30.1]). Atik and Panigrahi [3] computedthe spectrum of the distance matrix D ( J ( n, m )): (cid:18) s − sn − (cid:0) nm (cid:1) − n n − (cid:19) (15)where s = P mj =1 j (cid:0) mj (cid:1)(cid:0) n − mj (cid:1) . Inequality (1) implies that N ( J ( n, m )) ≥ n −
1. What is N ( J ( n, m )) ?5. The Clebsch graph is the strongly regular graph with parameters (16 , , ,
2) that is obtained fromthe 5-dimensional cube by identifying antipodal vertices. The eigenvalue bound gives N ≥
11 and theconnection with the 5-dimensional cube might be useful to find a good biclique decomposition of thedistance multigraph of this graph.6. What is N ( G ) if G is a random graph ? Winkler’s work [24], Witsenhausen inequality 1 and the Wignersemicircle law imply that n − ≥ N ( G ) ≥ n/ − c √ n for some positive constant c . Recently, Chungand Peng [6] (see also [1, 2]) have shown for a random graph G ∈ G n,p with p ≤ / p = Ω(1),almost surely n − o ((log b n ) ǫ ) ≤ bp( G ) ≤ n − b n (16)for b = 1 /p and any positive constant ǫ . Here G n,p is the Erd˝os-R´enyi random graph model. Acknowledgement.
Some of the initial threads of this project, in particular Section 2 and 3, started inconversation with the late D.A. Gregory. We are grateful for his discussion and his leadership over the years.
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