منابع مشابه
Chromatic number and spectral radius
Write (A) = 1 (A) min (A) for the eigenvalues of a Hermitian matrix A. Our main result is: let A be a Hermitian matrix partitioned into r r blocks so that all diagonal blocks are zero. Then for every real diagonal matrix B of the same size as A; (B A) B + 1 r 1 : Let G be a nonempty graph, (G) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality impli...
متن کاملChromatic number and the spectral radius
Let G be a graph, χ be its chromatic number, λ be the largest eigenvalue of its Laplacian, and µ be the largest eigenvalue of its adjacency matrix. Then, complementing a well-known result of Hoffman, we show that λ ≥ χ χ − 1 µ with equality holding for regular complete χ-partite graphs. We denote the eigenvalues of a Hermitian matrix A as µ (A) = µ 1 (A) ≥ · · · ≥ µ min (A). Given a graph G, we...
متن کاملSpectral Bounds on the Chromatic Number
The purpose of this paper is to discuss spectral bounds on the chromatic number of a graph. The classic result by Hoffman, where λ1 and λn are respectively the maximum and minimum eigenvalues of the adjacency matrix of a graph G, is χ(G) ≥ 1− λ1 λn . It is possible to discuss the coloring of Hermitian matrices in general. Nikiforov developed a spectral bound on the chromatic number of such matr...
متن کاملUnified spectral bounds on the chromatic number
One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenva...
متن کاملThe locating-chromatic number for Halin graphs
Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2007
ISSN: 0024-3795
DOI: 10.1016/j.laa.2007.06.005