Abstract
The undirected power graph P(Zn) of a finite group Zn is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if u ≠ v and or . The Wiener index W(P(Zn)) of an undirected power graph P(Zn) is defined to be sum of distances between all unordered pair of vertices in P(Zn). Similarly, the edge-Wiener index We(P(Zn)) of P(Zn) is defined to be the sum of distances between all unordered pairs of edges in P(Zn). In this paper, we concentrate on the wiener index of a power graph , P(Zpq) and P(Zp). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph P(Zn), using m,n and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of Zn.
Highlights
We define an undirected power graph P (G) for a group G as follows
The undirected power graph P ( Zn ) of a finite group Zn is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if u ≠ v and u ⊆ v or v ⊆ v
Our aim is to give our main results on the Wiener index and the edge-Wiener index of an undirected power graph P ( Zn ) for n = pk, or n = pq, where p and q are distinct prime numbers and k is a nonnegative integer
Summary
We define an undirected power graph P (G) for a group G as follows. Let us de-{ } note the cylic subgroup genarated by u ∈ G by u , that is= , u um | m ∈ , where denotes the set of naturel numbers. We obtain new results on the wiener index and edge-wiener index of power graph P (Zn ) , using m, n and Euler φ function. We obtain an equivalence between the edge-wiener index and wiener index of a power graph of Zn .
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