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Gary R. W. Greaves

Publications and source records attributed to Gary R. W. Greaves.

22 records · Page 2Linked to original sources
arXiv2017

On a lower bound for the Laplacian eigenvalues of a graph

If $μ_m$ and $d_m$ denote, respectively, the $m$-th largest Laplacian eigenvalue and the $m$-th largest vertex degree of a graph, then $μ_m \geqslant d_m-m+2$. This inequality was conjectured by Guo in 2007 and proved by Brouwer and Haemers in 2008. Brouwer and Haemers gave several examples of graphs achieving equality, but a complete characterisation was not given. In this paper we consider the problem of characterising graphs satisfying $μ_m = d_m-m+2$. In particular we give a full classification of graphs with $μ_m = d_m-m+2 \leqslant 1$.

math.CO↗
arXiv2016

On the clique number of a strongly regular graph

We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs.

math.CO↗
arXiv2016

Biregular graphs with three eigenvalues

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.

math.CO↗