Prime coprime graph of a finite group

In this paper, a new graph structure called the prime coprime graph of a finite group G denoted by Θ(G) has been introduced. The coprime graph of a finite group introduced by Ma, Wei and Yang [The coprime graph of a group. International Journal of Group Theory, 3(3), pp.13-23.] is a subgraph of the prime coprime graph introduced in this paper. The vertex set of Θ(G) is G, and any two vertices x, y in Θ(G) are adjacent if and only if gcd(o(x), o(y)) is equal to 1 or a prime number. We study how the graph properties of Θ(G) and group properties of G are related among themselves. We provide a necessary and sufficient condition for Θ(G) to be Eulerian for any finite group G. We also study Θ(G) for certain finite groups like Zn and Dn and derive conditions when it is connected, complete, planar and Hamiltonian for various n ∈ N. We also study the vertex connectivity of Θ(Zn) for various n ∈ N. Finally we have computed the signless Laplacian spectrum of Θ(G) when G = Zn and G = Dn for n ∈ {pq, p m} where p, q are distinct primes and m ∈ N.