Sombor Index of GCD Graph for Ring of Integers Modulo Power of Prime

Abstract

The GCD graph of the group or ring is a simple undirected graph and connected, has its vertex set as all the elements of the corresponding algebraic structure. Two distinct vertices are adjacent if and only if the greatest common divisor (gcd) of the orders of the two vertices equals the order of their product. The Sombor index of the graph is the summation of the square root of the sum of the squares of the degrees of two distinct adjacent vertices. This article examines certain properties of the GCD graph for the ring of integers modulo a power of a prime, including the degree of every vertex, the number of vertices, the number of edges, and the observation of subgraphs. We also derive the formula of the Sombor index and investigate its upper and lower bounds.