Related papers: Strongly regular graphs with the same parameters a…
Finite connected cubic symmetric graphs of girth 6 have been classified by K. Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an abelian automorphism group with two orbits on the vertex set. In this paper all cubic…
It is known that the automorphism group of the elementary abelian $2$-group $Z_2^n$ is isomorphic to the general linear group $GL(n,F_2)$ of degree $n$ over $F_2$. Let $W$ be the collection of permutation matrices of order $n$. It is clear…
In this paper we construct distance-regular graphs admitting a transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$. From the code spanned by…
We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a…
Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
The theory of voltage graphs has become a standard tool in the study graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting…
We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from…
We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost…
We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show…
In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, \lambda, \mu)$ satisfying $\mu = \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the…
The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral…
A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley…
In this article we construct a series of new infinite families of strongly regular graphs with the same parameters as the point-graphs of non-singular quadrics in PG(n,2).
In this paper, we give a new lifting construction of "hyperbolic" type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of…
A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian…
We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of…
Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters, yet the existence of many of them is still under the question. In this paper, we continue the study of the famuly of strongly regular…
We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least $3$ over a finite field of order $q$. We show that these graphs are non-isomorphic to the…
{\em Honeycomb toroidal graphs} are a family of cubic graphs determined by a set of three parameters, that have been studied over the last three decades both by mathematicians and computer scientists. They can all be embedded on a torus and…