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Let $O_\tau(\Gamma)$ be a family of algebras \textit{quantizing} the coordinate ring of $\mathbb{C}^2 / \Gamma$, where $\Gamma$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$, and let $G_{\Gamma}$ be the automorphism group of $O_\tau$.…

We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of…

Let $G$ be a permutation group on a set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted…

For a group $G,$ the generating graph of $G,$ denoted by $\Gamma(G).$ We define $Q_n=\langle x,y: x^{2n}=y^4=1, x^n=y^2,y^{-1}xy=x^{-1}\rangle,$ the dicyclic group of order $4n.$ This paper primarily delves into exploring the graph…

Let $R$ be a commutative ring with identity. We define a graph $\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in \aut$ such that…

Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial…

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set of vertices $D\subseteq V(G)$ is called a total outer-independent dominating set of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and…

We give necessary and sufficient conditions on the parameters of a regular graph $\Gamma$ (with or without loops) such that $E(\Gamma)=E(\overline \Gamma)$. We study complementary equienergetic cubic graphs obtaining classifications up to…

For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive alliance if for each vertex $v$ in $V-S$ at least half the vertices from the…

Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal G}$" is the group with generators in one-to-one correspondence with the vertices of ${\cal G}$ and with relations stating two generators commute if their associated…

Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an…

Let $G$ be 2-generated group. The generating graph $\Gamma(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a…

Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G…

A graph is almost bipartite if it contains exactly one odd cycle, and it is Konig-Egervary if the sum of the independence number and the matching number equals the order of the graph. We introduce the class of Bipartite-Almost Bipartite…

A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $\Gamma$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $\Gamma$, and we…

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $\Gamma$. If $G$ acts primitively on the points of $\Gamma$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group…

Let $G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base…

Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2\nu,j)$, of connected $2\nu$-valent labeled graphs with $j$ vertices…

Let $\Gamma$ be a finite connected graph and $G$ a vertex-transitive group of its automorphisms. The pair $(\Gamma, G)$ is said to be locally-$L$ if the permutation group induced by the action of the vertex-stabiliser $G_v$ on the set of…