Related papers: Normality of one-matching semi-Cayley graphs over …

We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a…

A regular $t$-balanced Cayley map on a group $\Gamma$ is an embedding of a Cayley graph on $\Gamma$ into a surface with certain special symmetric properties. We completely classify regular $t$-balanced Cayley maps for a class of split…

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. In this paper, we classify all connected quintic Cayley graphs…

The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial.…

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd…

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_\Gamma)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_\Gamma)$ has at most order…

A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(\alpha,\beta)$-regular if $C$ induces an $\alpha$-regular subgraph and every vertex outside $C$ is adjacent to exactly $\beta$ vertices in $C$. In particular, if $C$ is an…

Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random…

A graph $\Gamma$ is called edge-regular whenever it is regular and for any two adjacent vertices, the number of their common neighbors is independent of the choice of vertices. A clique $C$ in $\Gamma$ is called regular whenever for any…

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are…

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $\Gamma_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include…

Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prism is the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and its complement $\bar{\Gamma}$ by adding a perfect matching, where each its edge connects two…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

We consider latin square graphs $\Gamma = \rm{LSG}(H)$ of the Cayley table of a given finite group $H$. We characterize all pairs $(\Gamma,G)$, where $G$ is a subgroup of autoparatopisms of the Cayley table of $H$ such that $G$ acts…

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The \textit{mixed Cayley graph} $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…