Clive Elphick

New measures of graph irregularity

In this paper, we define and compare four new measures of graph irregularity. We use these measures to prove upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for irregular graphs and investigate to what extent Turans theorem can be similarly strengthened for generalized r-partite graphs. We conclude by relating these new measures to the Randic index and using the measures to devise new normalised indices of network heterogeneity.

Conjectured bounds for the sum of squares of positive eigenvalues of a graph

A well known upper bound for the spectral radius of a graph, due to Hong, is that µ 1 2 ź 2 m - n + 1 if ź ź 1 . It is conjectured that for connected graphs n - 1 ź s + ź 2 m - n + 1 , where s + denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q -partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.

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 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 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.