(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Given a group G with e as the identity element, the identity graph Γ_G having the vertex-set G and the edge-set E satisfies two conditions: (i) for every x,y G where x eq y, x and y are adjacent in Γ_G if and only if xy = e; (ii) for each x G, x and e are adjacent in Γ_G. The middle graph of G denoted by M(G) is the graph with vertex set V(G) 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 G and connecting the new obtained vertices if they are adjacent edges in 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.