On the Graph Theoretic Properties of the Inverse-Oder Sum Graph of finite groups: a new Graph Constructed from Inverse and Order Sum

Abstract

This paper introduces the inverse-order sum graph of the group , , a new graph structure that combines the properties of the inverse graph and the order sum graph. We define the inverse-order sum graph of  as a simple graph whose vertices are the elements of , and two distinct vertices are adjacent if either the sum of their orders is greater than or equal to the order of  or they are inverses of each other. By combining these two properties, we create a graph that provides a visual representation of the intricate relationships between elements of .We investigate the graph theoretic properties of the inverse-order sum graph of the cyclic group , obtaining results on connectivity, completeness, regularity, vertex degree, and graph size. Findings indicated that for  being an odd prime,  is connected, not complete, and not regular. Specifically, the vertex degree and the graph size were obtained.