Unified spectral bounds on the chromatic number

Abstract

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where µ 1 and µ n are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1 + µ 1 / − µ n . We recently generalised this bound to include all eigenvalues of the adjacency matrix.In this paper, we further generalize these results to include all eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. The various known bounds are also unified by considering the normalized adjacency matrix, and examples are cited for which the new bounds outperform known bounds.

Spectral Lower Bounds for the Orthogonal and Projective Ranks of a Graph

“…and (5), which states that x † v,i x w,i = 0 for all i ∈ [c] and all v ∼ w. Finally, condition (14) is proved similarly.…”

“…• The Clebsch graph on 16 vertices has spectrum (5 1 , 1 10 , −3 5 ) and χ f = 3.2 (see [7]). Therefore 1 + max n + n − , n − n + = 1 + n + n − = 3.2 = χ f ; but χ = ξ = 4.…”

More tales of Hoffman: bounds for the vector chromatic number of a graph

“…Note that for regular graphs the first three bounds are equal. Some of these bounds are further generalised in Elphick and Wocjan [4], which for reasons discussed in Section 5 we exclude here. Several of these bounds equal two for all bipartite graphs.…”

Distance Laplacian eigenvalues and chromatic number in graphs

“…For more results about interconnections between the chromatic number and the combinatorial Laplacian spectrum see [23,47]. A few results relating the spectrum of the normalized Laplacian (see e.g.…”

“…For more results about interconnections between the chromatic number and the signless Laplacian spectrum see [16,23,31] Concerning the interconnection between the distance spectrum and the chromatic number of a graph, only a few results are known. As an example, we recall the following lower bound from [38]:…”