Middle Graph of the Identity Graph of Finite Cyclic and Dihedral Groups

Abstract

 Given a group $\mathbb G$ with $e$ as the identity element, the identity graph $\Gamma_\mathbb G$ having the vertex-set $\mathbb G$ and the edge-set $E$ satisfies two conditions: (i) for every $x,y \in \mathbb G$ where $x \neq y$, $x$ and $y$ are adjacent in $\Gamma_\mathbb G$ if and only if $xy = e$; (ii) for each $x \in \mathbb G$, $x$ and $e$ are adjacent in $\Gamma_\mathbb G$. The middle graph of $G$ denoted by $M(G)$ is the graph with vertex set $V(G) \cup E(G)$ where two vertices will be adjacent if and only if they are either adjacent edges of $G$ or one is a vertex and the other is an edge incident to it. It can be obtained by inserting a new vertex into every edge of $\mathbb G$ and connecting the new obtained vertices if they are adjacent edges in $\mathbb G$. In this paper, we constructed the middle graph of the identity graph particular for finite cyclic and dihedral groups.  Some parameters of a graph such as the size, order, graph measurements, independence number, domination number, vertex chromatic number and edge chromatic number were also investigated.