الملخص

In this paper, we study the extended graph theory in the sum group via Zpnq  which is   Zpnq  by two distinct orders, the sum is greatest than the order of the group Zpnq where p,q are prime numbers. We have some results that the group sum graph of Zpnq  are connected, cyclic, etc., if they satisfy some properties of the graph theory and compute all the degrees of graphs. Furthermore, we shall calculate the famous topological indices via generalized it.