Spectral Bounds on the Chromatic Number

Transcription

1 Spectral Bounds on the Chromatic Number Naneh Apkarian March 14, 2013 Abstract The purpose of this paper is to discuss spectral bounds on the chromatic number of a graph. The classic result by Hoffman, where λ 1 and λ n are respectively the maximum and minimum eigenvalues of the adjacency matrix of a graph G, is χ(g) 1 λ1 λ n. It is possible to discuss the coloring of Hermitian matrices in general. Nikiforov developed a spectral bound on the chromatic number of such matrices, which enables the formulation of a chromatic bound based on eigenvalues of tweaked adjacency matrices - specifically the normalized Laplacian. 1 Introduction We begin by generalizing the concept of graph coloring to the coloring of a Hermitian matrix. Let V = {1,..., n}. A Hermitian matrix A C n n can be colored with c colors if there exists a partition of V into disjoint subsets V 1,..., V c such that [A] ij = 0 for all i, j V k, over each partition V k. Clearly when A is the standard adjacency matrix, this is the regular notion of coloring. Recall that the chromatic number χ is the minimum c such that a graph (or now, matrix) can be colored with c colors. Our proofs begin with the following theorem [4]: Theorem 1.1 (Nikiforov 2007). Let A C n n be an arbitrary Hermitian matrix which is colorable by c colors. Then, for any real diagonal matrix B C n n, we have λ max (B A) λ max (B + 1 c 1 A) 1

2 This theorem immediately implies many of the known bounds on the chromatic number. Reformulating the theorem as the following corollary make this more obvious. Corollary 1.2. For the hypotheses of Theorem 1.1, we have and consequently λ max (B A) λ max (B + A) c 2 c 1 λ max(a) c 1 + λ max (B + A) + λ max (B A) Proof sketch. In the result of Theorem 1.1, rewrite B + 1 c 1 A as B + A c 2 c 1A, and use the fact that, for arbitrary Hermitian matrices X and Y, λ max (X Y ) λ max (X) λ max (Y ). 2 Normalization Note that declaring A to be the adjacency matrix of a graph G with maximum eigenvalue λ 1 and minimum eigenvalue λ n, and B = 0 in Corollary 1.2 gives Hoffman s bound [3]: χ(g) = χ(a) = χ(a + 0) λ max ( A) λ 1 = 1 + λ 1 λ 1 λ n = 1 λ 1 λ n (1) Moving towards our goal, we rediscover some definitions. Let G be a graph on n labeled vertices, none of which are isolated. Let D denote the diagonal matrix whose entries [D] ii = d i, the degree of vertex i. Then the normalized adjacency matrix [5] is A = D 1/2 AD 1/2, and the normalized Laplacian [1] is L = I A. 2

3 Following the methods of Elphick and Wocjan [2], we take Corollary 1.2 and set A = A, B = I. Thus we see that χ(g) = χ(a) 1 + λ max (I + A) + λ max (I A) = 1 + (1 + ) + λ max (L) 1 = 1 + λ max (L) 1 (2) Giving us a bound on the chromatic number based on the spectrum of the normalized Laplacian. 3 Performance The usefulness of a new bounds on χ is of course dependent on it improving a previous bound. In 2013, Elphick and Wocjan [2] used Wolfram to analyze the relative performance of several bounds on the chromatic number of different types of graph. The inbuilt function ChromaticNumber[g] can be used to find the true chromatic number, and comparing the true value to the bounds can be informative. Elphick and Wocjan surveyed named graphs on 16, 18, 25, and 28 vertices. They found that for all regular graphs, the bound (2) equals the Hoffman bound (1). For irregular graphs, however, there are more varied results. Wolfram s NoPerfectMatchingGraph on 16 vertices has χ = 4, while bound (1) gives 2.5 and bound (2) gives 2.4. The new bound also underperforms on Sun(8), giving 3.0 while Hoffman s bound gives 4.1. However, the situation is different for Windmill[3,6]. Here, Hoffman s bound gives 3.7, while (2) gives 6, the true chromatic number. 4 Proof of Theorem 1.1 Here we present a version of Nikiforov s proof, found in [4]. Let A C n n be a c-colorable Hermitian matrix, and B R n n a diagonal matrix. Let {N i } r be the partition of the index set [n], and let b 1,..., b n R be the 3

4 diagonal entries of B. To simplify the notation to follow, let L = B A and K = (c 1)B + A. Let µ denote the largest eigenvalue of a matrix, and select a unit eigenvector x = (x 1,..., x n ) corresponding to µ(k). We will proceed by constructing n-vectors y 1,..., y c to show that c(c 1)µ(L) µ(l) y i 2 Ly i, i i c Kx, x = cµ(k). For i = 1 [c], we define y i = (y i1,..., y in ) as { (c 1)xj if j N y ij = i x j if j / N i Note that, by construction, y i 2 = x j 2 + (c 1) 2 x j 2 j / N i j N i = 1 x j 2 + (c 1) 2 x j 2 j N i j N i = 1 + c(c 2) j N i x j 2 So that r y i 2 = c(c 1), and the Rayleigh principle implies that µ(l) y i 2 Ly i, y i. Further note that Ly i, y i = = n b j y ij 2 a jk y ik y ij j=1 j,k [n] n b j xj 2 + c(c 2) j=1 j,k [n] j N i b j x j 2 a jk y ik y ij 4

5 We rewrite the final term, combine our equations, and with slight algebraic manipulation, we conclude that c(c 1)µ(L) µ(l) y i 2 Ly i, i i c Kx, x = cµ(k). Recalling our definitions of L and K, this is Which, as desired, implies that (c 1)µ(B A) µ((c 1)B + A) µ(b A) µ(b + 1 c 1 A). 5 Conclusions We have discussed a normalized bound (2) for the chromatic number χ, which in some irregular graphs improves upon Hoffman s classical bound. At present there does not seem to be a way of predicting which bound will be better, but using the pair, we are able to improve the bound overall. To summarize, for a graph G with chromatic number χ, adjacency matrix A, and normalized Laplacian L, { χ max 1 λ } max(a) λ min (A), λ max (L) 1 References [1] Fan R. K. Chung, Spectral graph theory, CBMS, no. 92, [2] C. Elphick and P. Wocjan, New spectral bounds on the chromatic number, Cornell University Library arxiv.org, arxiv: [math.co], January [3] A. J. Hoffman, On eigenvalues and colourings of graphs, Graph Theory and its Applications, pp , Academic Press, New York, [4] V. Nikiforov, Chromatic number and spectral radius, Linear Algebra Applications 426 (2007),

6 [5] V. S. Shigehalli and V.M. Shettar, Spectral techniques using normalized adjacency matrices for graph matching, International Journal of Computational Science and Mathematics 3 (2011), no. 4,