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For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…

For a graph $\Gamma=(V(\Gamma),E(\Gamma))$, a subset $C$ of $V(\Gamma)$ is called an $(\alpha,\beta)$-regular set in $\Gamma$, if every vertex of $C$ is adjacent to exactly $\alpha$ vertices of $C$ and every vertex of $V(\Gamma)\setminus C$…

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$…

A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and…

A walk $W$ in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in $W$ is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity…

In this paper, we begin by partitioning the edges (or arcs) of a circulant (di)graph according to which generator in the connection set leads to each edge. We then further refine the partition by subdividing any part that corresponds to an…

We introduce the concept of alternate-edge-colourings for maps, and study highly symmetric examples of such maps. Edge-biregular maps of type $(k,l)$ occur as smooth normal quotients of a particular index two subgroup of $T_{k,l}$, the full…

A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_4\times C_p^2$, where $p$ is a prime, is a…

We study finite four-valent graphs Gamma admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on Gamma, and such that at least one G-normal quotient is a cycle (a quotient modulo the…

Let $G$ be an $n$-vertex connected graph. A cyclic base ordering of $G$ is a cyclic ordering of all edges such that every cyclically consecutive $n-1$ edges induce a spanning tree of $G$. In this project, we study cyclic base ordering of…

A Cayley Graph for a group $G$ is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the Holomorph of $G$ (the normaliser of a regular copy of $G$ in $\operatorname{Sym}(G)$). We complete the…

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition…

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…

As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…

In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is…

Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct…

We characterize connected tetravalent graphs $\Gamma$ which admit groups $M<H$ of automorphisms such that $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive. Examples for each case are constructed, including a counter-example to a…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$…

This paper investigates the enumeration of Cayley digraphs, focusing on counting Cayley digraphs on dihedral groups up to CI-isomorphism. By leveraging the Cauchy-Frobenius Lemma and properties of automorphisms, we derive an explicit…

We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order $p^2q^2$, where $p$ and $q$ are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main…