Lower Bound on the Chromatic Number by Spectra of Weighted Adjacency Matrices

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Spectral 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...

Lower Bounds of Copson Type for Hausdorff Matrices on Weighted Sequence Spaces

Let = be a non-negative matrix. Denote by the supremum of those , satisfying the following inequality: where , , and also is increasing, non-negative sequence of real numbers. If we used instead of The purpose of this paper is to establish a Hardy type formula for , where is Hausdorff matrix and A similar result is also established for where In particular, we apply o...

Unified 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...

New Spectral Bounds on the Chromatic Number Encompassing all Eigenvalues of the Adjacency Matrix

The purpose of this article is to improve existing lower bounds on the chromatic number χ. Let μ1, . . . , μn be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound χ > 1 + maxm{ ∑m i=1 μi/ − ∑m i=1 μn−i+1} for m = 1, . . . , n − 1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case...