This paper deals with vertex-magic total labellings of graphs. Earlier work by many authors has shown many infinite families of graphs to admit such labelings. The fact that many of these graphs are regular led MacDougall to conjecture that all non-trivial regular graphs are vertexmagic. Previously Gray and MacDougall showed that all odd-order rregular graphs (r ≥ 2) of order up to v = 19 are v...

In this paper we take a very special model of a random non-regular graph and study its non-backtracking spectrum. We study graphs consisting of a cycle with some random loops added; the graphs are not regular and their non-backtracking spectrum does not seem to be confined to some one-dimensional set in the complex plane. The nonbacktracking spectrum is required in some applications, and has no...

Distance regular graphs fall into three families: primitive, antipodal, and bipar-tite. Each antipodal distance regular graph is a covering graph of a smaller (usually primitive) distance regular graph; the antipodal distance graphs of diameter three are covers of the complete graph, and are the first non-trivial case. Many of the known examples are connected with geometric objects, such as pro...

In this paper, we generalize the construction of strongly regular graphs in Tan et al. (J. Comb. Theory, Ser. A 117:668–682, 2010) from ternary bent functions to p-ary bent functions, where p is an odd prime. We obtain strongly regular graphs with three types of parameters. Using certain non-quadratic p-ary bent functions, our constructions can give rise to new strongly regular graphs for small...

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of S+(U3), a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of orde...

We study the spectral radius of connected non-regular graphs. Let λ1(n,Δ) be the maximum spectral radius among all connected non-regular graphs with n vertices and maximum degree Δ. We prove that Δ− λ1(n,Δ)=Θ(Δ/n). This improves two recent results by Stevanović and Zhang, respectively. © 2007 Elsevier Inc. All rights reserved.

We construct distance-regular graphs with the same classical parameters as the Grassmann graphs on the e-dimensional subspaces of a (2e+1)-dimensional space over an arbitrary finite field. This provides the first known family of non-transitive distance-regular graphs with unbounded diameter.

A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Hoffman's bound). Such spreads give rise to colorings meeting Hoffman's lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (part...