On the Identity-Order Sum Graph of finite groups: a new Graph Constructed from Identity and Order Sum Graphs of

Abstract

This paper introduces a novel graph structure known as the identity-order sum graph of the finite cyclic group , denoted . This graph combines properties of both the identity graph and the order sum graph, providing a unified structure to explore element relationships in the group. Specifically, two distinct vertices  and  in  are adjacent if and only if , or , or , and the sum of their orders satisfies . The paper investigates several graph-theoretic properties of this construction, including bipartite, girth, planarity, and diameter. The results show that  has diameter , girth , not bipartite and is planar for all primes , with a single isolated vertex. 

Keywords: identity graph, order sum graph, cyclic group, girth, diameter, planarity.