Hosoya-Polynomial Of Sum Graph Of The Group Z2τπ

In this paper, we search the application of graph theory to the addition group using Z_2τπ, where τ and π are prime numbers. We explore the notion of two different orders, where the sum is constantly greater than and equal to the order of the group Z_2τπ. We display that the group sum graph of Z_(2τπ )is, among other properties, connected and cyclic when certain graph-theoretic conditions are satisfied. In addition, we determined all the degrees of the graphs and calculated The Hosoya polynomial and Wiener index.