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On the $ K $-Independence Number of Graphs On the k-independence number of graphs A. Abiada, G. Coutinhob, M. A. Fiolc aDepartment of Quantitative Economics Maastricht University, Maastricht, The Netherlands, [email protected] bDepartment of Computer Science Federal University of Minas Gerais, Belo Horizonte, Brazil [email protected] cDepartament de Matem`atiques Universitat Polit`ecnicade Catalunya, Barcelona, Catalonia Barcelona Graduate School of Mathematics [email protected] August 28, 2018 Abstract This paper generalizes and unifies the existing spectral bounds on the k-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than k. The previous bounds known in the literature follow as a corollary of the main results in this work. We show that for most cases our bounds outperform the previous known bounds. Some infinite graphs where the bounds are tight are also presented. Finally, as a byproduct, we derive some lower spectral bounds for the diameter of a graph. Keywords: Graph, k-independence number, Spectrum, Interlacing, Regular partition, An- tipodal distance-regular graph, Diameter. Mathematics Subject Classifications: 05C50, 05C69. arXiv:1803.07042v2 [math.CO] 26 Aug 2018 1 Introduction Given a graph G, let αk = αk(G) denote the size of the largest set of vertices such that any two vertices in the set are at distance larger than k. This choice of notation is no coincidence, since actually α1 is just the independence number of a graph. The parameter αk(G) therefore represents the largest number of vertices which can be k + 1 spread out in G. It is known that determining αk is NP-Hard in general [18]. 1 2 The k-independence number of a graph is directly related to other combinatorial pa- rameters such as the average distance [12], packing chromatic number [13], injective chro- matic number [17], and strong chromatic index [20]. Upper bounds on the k-independence number directly give lower bounds on the corresponding distance or packing chromatic number. Regarding it, Alon and Mohar [2] asked for the extremal value of the distance chromatic number for graphs of a given girth and degree. In this paper we generalize and improve the known spectral upper bounds for the k- independence number from [8] and [1]. For some cases, we also show that our bounds are sharp. As far as we are aware, there seems to be some conflict in the existing literature regarding the use of the term `k-independence number'. The following list contains the three conflicting definitions, which all, nonetheless, are a natural generalization of the concept of independence number. 1. Caro and Hansberg [6] use the term `k-independence number' to denote the max- imum size of a set of vertices in a graph whose induced subgraph has maximum degree k. Thus, α0 is the usual independence number. 2. Spacapanˇ [21] uses `k-independence number' to denote the size of the largest k- colourable subgraph of G. With this notation, α1 stands for the usual k-independence number of G. 3. Fiol [8] and Abiad, Tait, and Cioab˘a[1] use `k-independence number' to denote the size of the largest set of vertices such that any two vertices in the set are at distance larger than k. The latter definition is the one we use in this work. The first known spectral bound for the independence number α is due to Cvetkovi´c [7]. Theorem 1.1 (Cvetkovi´c[7]). Let G be a graph with eigenvalues λ1 ≥ · · · ≥ λn. Then, α ≤ minfjfi : λi ≥ 0gj and jfi : λi ≤ 0gjg: Another well-known result is the following bound due to Hoffman (unpublished; see for instance Haemers [15]). Theorem 1.2 (Hoffman [15]). If G is a regular graph on n vertices with eigenvalues λ1 ≥ · · · ≥ λn, then −λ α ≤ n n : λ1 − λn 3 Regarding the k-independence number, the following three results are known. The first is due to Fiol [8] and requires a preliminary definition. Let G be a graph with distinct eigenvalues θ0 > ··· > θd. Let Pk(x) be chosen among all polynomials p(x) 2 Rk(x), that is, polynomials of real coefficients and degree at most k, satisfying jp(θi)j ≤ 1 for all i = 1; :::; d, and such that Pk(θ0) is maximized. The polynomial Pk(x) defined above is called the k-alternating polynomial of G and was shown to be unique in [11], where it was used to study the relationship between the spectrum of a graph and its diameter. Theorem 1.3 (Fiol [8]). Let G be a d-regular graph on n vertices, with distinct eigenvalues θ0 > ··· > θd and let Pk(x) be its k-alternating polynomial. Then, 2n αk ≤ : Pk(θ0) + 1 The second and third bounds are due to Abiad, Cioab˘a,and Tait [1]. The first is a Cvetkovi´c-like approach, whereas the second resembles Hoffman’s. Theorem 1.4 (Abiad, Cioab˘a,Tait [1]). Let G be a graph on n vertices with adjacency matrix A, with eigenvalues λ1 ≥ · · · ≥ λn. Let wk and Wk be respectively the smallest and the largest diagonal entries of Ak. Then, k k αk ≤ jfi : λi ≥ wk(G)gj and αk ≤ jfi : λi ≤ Wk(G)gj: Theorem 1.5 (Abiad, Cioab˘a,Tait [1]). Let G be a δ-regular graph on n vertices with adjacency matrix A, whose distinct eigenvalues are θ0(= δ) > ··· > θd. Let Wfk be the 2 k largest diagonal entry of A + A + ··· + A . Let θ = maxfjθ1j; jθdjg. Then, Pk j Wfk + j=1 θ αk ≤ n : Pk j Pk j j=1 δ + j=1 θ 2 Preliminaries For basic notation and results see [3, 14]. Let G = (V; E) be a graph with n = jV j vertices, m1 md m = jEj edges, and adjacency matrix A with spectrum sp G = fθ0 > θ1 > ··· > θd g. When the eigenvalues are presented with possible repetitions, we shall indicate them by λ1 ≥ λ2 ≥ · · · ≥ λn. Let us consider the scalar product in Rd[x]: d 1 1 X hf; gi = tr(f(A)g(A)) = m f(θ )g(θ ): G n n i i i i=0 The so-called predistance polynomials p0(= 1); p1; : : : ; pd are a sequence of orthogonal polynomials with respect to the above product, with dgr pi = i, and are normalized in 2 such a way that kpikG = pi(θ0) (this makes sense since it is known that pi(θ0) > 0) for 4 i = 0; : : : ; d. Therefore they are uniquely determined, for instance, following the Gram- Schmidt process. They were introduced by Fiol and Garriga in [10] to prove the so-called `spectral excess theorem' for distance-regular graphs. We also use the sum polynomials qi = p0 +···+pi, for i = 0; : : : ; d−1, which are also a sequence of orthogonal polynomials, now with respect to the scalar product d−1 1 X hf; gi = m (θ − θ )f(θ )g(θ ); [G] n i 0 i i i i=0 and satisfy 1 = q0(θ0) < q1(θ0) < ··· < qd−1(θ0) < n. See [5] for further details and applications. Eigenvalue interlacing is a powerful and old technique that has found countless appli- cations in combinatorics and other fields. This technique will be used in several of our proofs. For more details, historical remarks and other applications see Fiol and Haemers [9, 15]. Given square matrices A and B with respective eigenvalues λ1 ≥ · · · ≥ λn and µ1 ≥ · · · ≥ µm, with m < n, we say that the second sequence interlaces the first if, for all i = 1; : : : ; m, it follows that λi ≥ µi ≥ λn−m+i: Theorem 2.1 (Interlacing [9, 15]). Let S be a real n × m matrix such that ST S = I, and T let A be a n × n matrix with eigenvalues λ1 ≥ · · · ≥ λn. Define B = S AS, and call its eigenvalues µ1 ≥ · · · ≥ µm. Then, (i) The eigenvalues of B interlace those of A. (ii) If µi = λi or µi = λn−m+i, then there is an eigenvector v of B for µi such that Sv is eigenvector of A for µi. (iii) If there is an integer k 2 f0; : : : ; mg such that λi = µi for 1 ≤ i ≤ k, and µi = λn−m+i for k + 1 ≤ i ≤ m (tight interlacing), then SB = AS. Two interesting particular cases where interlacing occurs (obtained by choosing ap- propriately the matrix S) are the following. Let A be the adjacency matrix of a graph G = (V; E). First, if B is a principal submatrix of A, then B corresponds to the adjacency matrix of an induced subgraph G0 of G. Second, when, for a given partition of the vertices of Γ, say V = U1 [···[ Um, B is the so-called quotient matrix of A, with elements bij, i; j = 1; : : : ; m, being the average row sums of the corresponding block Aij of A. Actually, the quotient matrix B does not need to be symmetric or equal to S>AS, but in this case B is similar to|and therefore has the same spectrum as| S>AS. In the second case, if the interlacing is tight, Theorem 2.1(iii) reflects that S corresponds to a regular (or equitable) partition of A, that is, each block of the partition has constant row and column sums. Then the bipartite induced subgraphs Gij, with adjacency matrices Aij, for i 6= i, are biregular, and the subgraphs Gii are regular. 5 We finally recall that the Kronecker product of two matrices A = (aij) and B, denoted by A ⊗ B, is obtained by replacing each entry aij with the matrix aijB, for all i and j. Then, if u and v are eigenvectors of A and B, with corresponding eigenvalues λ and µ, respectively, then u ⊗ v (seeing u and v as matrices) is an eigenvector of A ⊗ B, with eigenvalue λµ.
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