On co-prime order graphs of finite abelian p-groups

For a finite group G, the co-prime order graph Θ(G) of G is defined as the graph with vertex set G, the group itself, and two distinct vertices u, v in Θ(G) are adjacent if and only if gcd(o(u),o(v)) = 1 or a prime number. In this paper, some properties and some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the co-prime order graph of finite abelian p-group are studied. We also figure out the metric dimension and resolving polynomial of the co-prime order graph of finite abelian p-group.