Co-prime order graphs of finite Abelian groups and dihedral groups

The co-prime order graph Theta(G) of a given finite group is a simple undirected graph whose vertex set is the group G itself, and any two vertexes x,y in Theta(G) are adjacent if and only if gcd(o(x), o(y)) = 1 or prime. In this paper, we derive a precise formula to count the vertex's degree in the co-prime order graph of a finite Abelian group or dihedral group.We also investigate the Laplacian spectrum of the co-prime order graph Theta(G) when G is a finite Abelian p-group, Z(p)(t) x Z(q)(s )or a dihedral group D-p(n).