OOn a metric property of perfect colorings
A. A. Taranenko ∗ February 3, 2021
Abstract
Given a perfect coloring of a graph, we prove that the L distance between two rows ofthe adjacency matrix of the graph is not less than the L distance between the correspondingrows of the parameter matrix of the coloring. With the help of an algebraic approach, wededuce corollaries of this result for perfect 2-colorings, perfect colorings in distance- l graphsand in distance-regular graphs. We also provide examples when the obtained property rejectseveral putative parameter matrices of perfect colorings in infinite graphs. Keywords: perfect coloring, perfect structure, L distance, circulant graph, square grid,triangular grid. Following [8], we consider perfect colorings in a more general setting than perfect colorings ofsimple graphs or multigraphs. More specifically, we associate a graph G on n vertices with a real n × n -matrix M that is called its adjacency matrix . So under a graph we mean an oriented graphwith edges labelled by m u,v . We use V ( G ) to denote the vertex set of the graph G .Define a perfect k -coloring of a graph G with the parameter matrix S to be a partition of theset V ( G ) into disjoint classes J i , i = 1 , . . . , k , such that for all u ∈ J i it holds s i,j = (cid:80) v ∈ J j m u,v .This definition of a perfect coloring generelizes equitable partitions introduced by Delsarte [3].Note that classes J i can be considered as vertices of some graph H defined by the adjacencymatrix S . Then a perfect coloring can be defined as a map f : V ( G ) → V ( H ) that puts everyvertex v of the graph G to its color f ( v ), i.e., f ( v ) = i if and only if v ∈ J i .In many papers, a perfect coloring of simple graph G is said to be a partition of its vertex setinto color classes such that the colored neighborhood of each vertex is defined by a color of thevertex.At last, perfect colorings can be treated as special perfect structures [8]: a perfect coloringis a triple of matrices ( M, P, S ) connected by a relation
M P = P S , where M and S are squarematrices of orders n and k respectively, P is a (0 , n × k in which each rowcontains exactly one unity entry.In the present note, we bound the L distance between rows of the parameter matrix of aperfect coloring by the L distance between the corresponding rows of the adjacency matrix of agraph.Recall that the L distance between two n -tuples x = ( x , . . . , x n ) and y = ( y , . . . , y n ), x i , y i ∈ R , is d ( x, y ) = n (cid:88) i =1 | x i − y i | . Given a real matrix A of order n , we use [ A ] i to denote the i -th row of A . ∗ Sobolev Institute of Mathematics, Novosibirsk, Russia; [email protected] a r X i v : . [ m a t h . C O ] F e b heorem 1. Let f be a perfect k -coloring of a graph G with the parameter matrix S and let M be the adjacency matrix of G . Then for all u, v ∈ V ( G ) we have d ([ M ] u , [ M ] v ) ≥ d ([ S ] f ( u ) , [ S ] f ( v ) ) . Proof.
By the definition, d ([ S ] f ( u ) , [ S ] f ( v ) ) = k (cid:88) j =1 | s f ( u ) ,j − s f ( v ) ,j | . Since f is a perfect coloring, the set V ( G ) is partitioned into k disjoint subsets J , . . . , J k suchthat for every u ∈ V ( G ) we have s f ( u ) ,j = (cid:80) w ∈ J j m u,w . Consequently, k (cid:88) j =1 | s f ( u ) ,j − s f ( v ) ,j | = k (cid:88) j =1 | (cid:88) w ∈ J j ( m u,w − m v,w ) | . Using the inequality | a + b | ≤ | a | + | b | , we deduce k (cid:88) j =1 | (cid:88) w ∈ J j ( m u,w − m v,w ) | ≤ (cid:88) w ∈ V ( G ) | m u,w − m v,w | . It only remains to note that (cid:88) w ∈ V ( G ) | m u,w − m v,w | = d ([ M ] u , [ M ] v ) . Let us specialize Theorem 1 for some classes of graphs and colorings. In this section, we assumeeverywhere that G is an r -regular simple undirected graph with no loops. In other words, theadjacency matrix M of G is a symmetric (0 , r .Given a vertex v of a simple graph G , let N ( v ) denote the neighborhood of the vertex v thatis the set of all vertices u ∈ V ( G ) such that u and v are adjacent. Then for simple graphs G Theorem 1 takes the following form.
Theorem 2.
Let G be a simple r -regular graph and f be a perfect coloring of G with the parametermatrix S . Assume that there are vertices u, v ∈ V ( G ) of colors f ( u ) = i , f ( v ) = j such that |N ( u ) ∩ N ( v ) | = h . Then d ([ S ] i , [ S ] j ) ≤ r − h ) . If the inequality becomes an equality, then distributions of colors in sets N ( u ) ∩N ( v ) , N ( u ) \N ( v ) ,and N ( v ) \ N ( u ) are determined by the parameter matrix S and do not depend on the coloring f .Proof. The theorem follows from Theorem 1 and the fact that M ( u, v ) = 2( r − |N ( u ) ∩ N ( v ) | ).For perfect colorings of simple graphs the equality d ([ M ] u , [ M ] v ) = d ([ S ] f ( u ) , [ S ] f ( v ) ) meansthat the symmetric difference N ( u )∆ N ( v ) contains exactly | s f ( u ) ,j − s f ( v ) ,j | vertices of each color j . Since [ S ] f ( u ) and [ S ] f ( v ) are distributions of colors in sets N ( u ) and N ( v ) respectively, weknow color distributions for sets N ( u ) ∩ N ( v ), N ( u ) \ N ( v ), and N ( v ) \ N ( u ).2 .1 Perfect -colorings The parameter matrix of perfect colorings in 2 colors is usually written as S = (cid:18) a bc d (cid:19) . It is easy to see that if G is an r -regular graph, then S has two different eigenvalues: the trivialeigenvalue λ = r and the second eigenvalue λ = r − ( b + c ) = a − c . Since a + b = c + d = r ,the parameters b and c uniquely define the matrix S . Thus we will say that a perfect coloring incolors with the parameter matrix is a ( b, c )-coloring. Lemma 1.
Let S = (cid:18) a bc d (cid:19) be the parameter matrix of a perfect ( b, c ) -coloring of an r -regular graph G . Then d ([ S ] , [ S ] ) = 2 | λ | = 2 | r − ( b + c ) | . Proof.
By equalities a + b = c + d = r and λ = r − ( b + c ), we have d ([ S ] , [ S ] ) = | a − c | + | b − d | = 2 | r − ( b + c ) | = 2 | λ | . For further applications, we state Theorem 2 for perfect ( b, c )-colorings.
Theorem 3.
Let G be an r -regular graph and f be a perfect ( b, c ) -coloring of G . Assume thatthere are vertices u, v ∈ V ( G ) of different colors such that |N ( u ) ∩ N ( v ) | = h . Then h ≤ b + c ≤ r − h. If the left inequality attains an equality, then all vertices from N ( u ) \ N ( v ) (and N ( v ) \ N ( u ) )have the same color as the vertex u (vertex v ). If the right inequality is achieved, then all verticesfrom N ( u ) \ N ( v ) (or N ( v ) \ N ( u ) ) have the same color as the vertex v (vertex u ).At last, if the vertices u and v are adjacent, then h + 2 ≤ b + c .Proof. By Theorem 2, we have d ([ S ] f ( u ) , [ S ] f ( v ) ) ≤ r − h ) . Since vertices u and v have different colors, Lemma 1 gives d ([ S ] f ( u ) , [ S ] f ( v ) ) = d ([ S ] , [ S ] ) = 2 | r − ( b + c ) | ≤ r − h ) , that is equivalent to the required inequalities.Equality 2( r − ( b + c )) = 2( r − h ) means that the sum b + c attains the minimal possible value.Then the set N ( u ) (and N ( v )) contains the minimal possible number of vertices whose color isdifferent from f ( u ) ( f ( v )). Similarly, if 2( r − ( b + c )) = − r − h ), then the sum b + c attains themaximal possible value, and N ( u ) ( N ( v )) contains the maximal possible number of vertices withcolors different from f ( u ) ( f ( v )). Thus sets N ( u ) \ N ( v ) and N ( v ) \ N ( u ) are monochromatic.If vertices u and v are adjacent, then we can slightly improve the inequalities. Indeed, in thiscase the sets N ( u ) \ N ( v ) and N ( v ) \ N ( u ) contain at least one vertex of color N ( v ) and N ( u ),respectively. It means that 2( r − ( b + c )) ≤ r − h − .2 Colorings in distance graphs As before, we use M for the adjacency matrix of a simple graph G . It is well known, that powersof M count the number of paths from one vertex of G to another: the number of paths of length l in G from a vertex u to a vertex v is equal to the ( u, v )-entry of the matrix M l . We will saythat M l is the adjacency matrix of the distance- l graph G l . Note that in most cases G l is not asimple graph but a multigraph.In [8] it was proved the following. Proposition 1.
If a triple of matrices ( M, P, S ) is a perfect coloring, then for every polynomial p ( x ) ∈ R [ x ] the triple ( p ( M ) , P, p ( S )) is also a perfect coloring. So we can specialize Theorem 1 for distance- l graphs. Theorem 4.
Let G be a simple graph and f be a perfect coloring of G with the parameter matrix S . Then for all l ∈ N and for all vertices u, v ∈ V ( G ) it holds d ([ M l ] u , [ M l ] v ) ≥ d ([ S l ] f ( u ) , [ S l ] f ( v ) ) . Proof.
The result follows from Theorem 1 and Proposition 1.In special graphs, it is possible to express some subsets of vertices by the means of a polynomialon the adjacent matrix. One of the most famous examples of such sets and graphs are balls andspheres in distance-regular graphs.For vertices u, v of a simple graphs G , let ρ ( u, v ) denote the distance between u and v (thelength of the shortest path between them). The ball B r ( u ) of a radius r and with center u is a set { v : ρ ( u, v ) ≤ r } , and the sphere W r ( u ) of a radius r and with center u is a set { v : ρ ( u, v ) = r } .It is well known (see, e.g. [2]) that in a distance-regular graph G for every r ≤ diam ( G ) thereare polynomials p Br and p Wr such that for each u ∈ V ( G ) rows [ p Br ( M )] u and [ p Wr ( M )] u are theindicator functions of a ball B r ( u ) and a sphere W r ( u ), respectively. Thus, for distance-regulargraphs we have the following theorem. Theorem 5.
Let G be a simple distance-regular graph with polynomials p Br ( x ) and p Wr ( x ) ex-pressing balls and spheres of radius r in G , respectively. Suppose that f is a perfect coloring of G with the parameter matrix S . Then for all vertices u, v ∈ V ( G ) | B r ( u )∆ B r ( v ) | ≥ d ([ p Br ( S )] f ( u ) , [ p Br ( S )] f ( v ) ); | W r ( u )∆ W r ( v ) | ≥ d ([ p Wr ( S )] f ( u ) , [ p Wr ( S )] f ( v ) ) . The above results can be applied to reject some putative parameter matrices of perfect coloringsfor a given graph G . For perfect 2-colorings, our method is more useful if the second eigenvalueof the parameter matrix has a large absolute value. It is especially interesting for infinite graphs(i.e., graphs with an infinite number of vertices) because the standard spectral condition on theexistence of perfect colorings is not applicable for them. Our first example is a simple proof that there are no perfect (4 , square grid is an infinite 4-regular graph with the vertex set Z and edges (( x, y ) , ( x + 1 , y ))and (( x, y ) , ( x, y + 1)) for x, y ∈ Z . 4uppose that f is a perfect (4 , u = ( x, y ) and v = ( x + 1 , y + 1) in the square grid of different colors in the coloring f because h = |N ( u ) ∩ N ( v ) | = 2 and 7 = b + c > r − h = 6. So for a given vertex ( x, y ) all vertices( x + t, y + t ), t ∈ Z have the same color in f as the vertex ( x, y ). Then every vertex is adjacentto an even number of vertices of each color that contradicts to the parameters of f .An approach similar to the presented one was used in a characterization of 3-colorings in thesquare grid [6] and in studying multiple coverings of the square grid with balls of a constantradius [1].We can also apply this technique to perfect 2-colorings in the triangular grid that is an infinite6-regular graph with the following local structure:For every adjacent vertices u and v in the triangular grid we have h = |N ( u ) ∩ N ( v ) | = 2.By Theorem 3, for every perfect ( b, c )-coloring of the triangular it holds 4 ≤ b + c ≤
10. Inparticular, there are no perfect (1 , , , , , , , b + c achieves one of the possible equalities). Indeed, Theorem 3 allows us to determinecolors of vertices in sets N ( u ) \ ( N ( v ) ∪ { v } ) and N ( v ) \ ( N ( u ) ∪ { u } ) that gives a contradictionto the parameters of the coloring at the vertex w :The similar reasoning implies that there is a unique (up to transformations of the plane)perfect (2 , .2 Circulant graphs Our results can be widely used for perfect colorings in circulant graphs. Perfect colorings in someclasses of such graphs were previously studied in [4, 5].Given a (multi)set D = { d , . . . , d m } , d i ∈ N , a circulant (multi)graph C ( d , . . . , d m ) is a2 m -regular (multi)graph with the vertex set Z and edges ( x, y ), where | x − y | ∈ D . It is easyto see that every perfect coloring f of C ( d , . . . , d m ) is periodic: there is some T ∈ N such that f ( x + T ) = f ( x ) for all x ∈ Z . Theorem 6.
Assume that for a (multi)set D = { d , . . . , d m } and for t ∈ N we have |{± d , . . . , ± d m } ∩ { t ± d , . . . , t ± d m }| = h. If b + c > m − h or b + c < h (or b + c < h + 2 if t ∈ D ), then the period T of every perfect ( b, c ) -coloring of the circulant (multi)graph C ( d , . . . , d m ) divides t .Proof. Let f be a perfect ( b, c )-coloring of the (multi)graph C ( d , . . . , d m ) with period T . If T is not a divisor of t , then there are vertices x and x + t in C ( d , . . . , d m ) that are colored withdifferent colors by f . By the condition of the theorem, |N ( x ) ∩ N ( x + t ) | = h . It only remainsto note that the demanded inequalities on b + c contradict to Theorem 3.For example, consider a circulant graph C (1 , , t = 3 we have h = |{± , ± , ± } ∩ { ± , ± , ± }| = 4 . So all ( b, c )-colorings of C (1 , ,
4) with b + c < b + c > T such that T divides3, and so T = 3. Searching all colorings of C (1 , ,
4) of period 3, it is easy to see that there areno (1 , , , , , , , C (1 , , Acknowledgements
This work was carried out within the framework of the state contract of the Sobolev Institute ofMathematics (project no. 0314-2019-0016).
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