On Sombor Index and Sombor polynomial of Co-prime Graphs of Finite Groups

Abstract

The Sombor polynomial is a polynomial whose power of the indeterminate is the sum of the squares of the degrees of adjacent vertices in a graph. In this paper the graph considered is the co-prime graph of a finite group $G$; the graph has all elements of $G$ as vertices with two distinct elements adjacent in the graph if and only if their orders are relatively prime. Formulae for obtaining the Sombor index and Sombor polynomial for the graphs of $p$—groups and groups of order the product of two distinct primes. Two variations of the graph are also considered; the co-prime order graph and the co-prime power order graph, and it is shown that their Sombor index and Sombor polynomial are the same for $p$—groups and groups of order the product of two primes.