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A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of…
One version of the polycirculant conjecture states that every vertex-transitive graph has a semiregular automorphism. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open case.
For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive…
A graph $\Gamma$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(\Gamma)$ is transitive on the set of vertices of $\Gamma$ and the set of $s$-arcs of $\Gamma$, where for an integer $s \ge 1$ an $s$-arc of $\Gamma$ is a sequence of…
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of…
We construct two families of distance-regular graphs, namely the subgraph of the dual polar graph of type B_3(q) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type D_4(q) induced on the vertices…
More than $30$ years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive $2$-design, with blocks of size $k$, could leave invariant a nontrivial point-partition, but only if the number of points was bounded…
In this work, we define an orthogonal graph on the set of equivalence classes of $(2\nu + \delta)-$tuples over $\mathbb{Z}_{2^n}$ where $n$ and $\nu$ are positive integers and $\delta = 0, 1$ or $2$. We classify our graph if it is strongly…
We show that the categories of directed and undirected reflexive graphs carry exactly two (up to isomorphism) biclosed monoidal structures.
A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…
We give a non-abelian analogue of Whitney's 2-isomorphism theorem for graphs. Whitney's theorem states that the cycle space determines a graph up to 2-isomorphism. Instead of considering the cycle space of a graph which is an abelian…
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…
In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…
Recently, Ehrenborg and Van Willenburg defined a class of bipartite graphs that correspond naturally to Ferrers diagrams, and proved several results about them. We give bijective proofs for the (already known) expressions for the number of…
We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard…
In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information…
Let $p$ be an odd prime, and $D_{2p}=\langle \tau,\sigma\mid \tau^p=\sigma^2=e,\sigma\tau\sigma=\tau^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we provide the number of (connected) Cayley (di-)graphs on $D_{2p}$ up to…
This study is the first part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. Circulant graphs $C_n(R)$ and $C_n(S)$ are said to be \emph{Adam's isomorphic} if there exist some $a\in…
The 2-switch-degree of $G$ is the number of distinct 2-switches acting on a graph $G$. In this work we study structural properties of the 2-switch-degree, with a focus on split graphs. Our approach is motivated by the Tyshkevich…
Let $S$ be a subset of the cyclic group $\Z_n$. The cyclic Haar graph $H(\Z_n,S)$ is the bipartite graph with color classes $\Z_n^+$ and $\Z_n^-,$ and edges $\{x^+,y^-\},$ where $x,y \in \Z_n$ and $y - x \in S$. In this paper we give…