The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{p^k·q^r}) power graph

Abstract

The undirected $P(Z_n)$ power graph of a finite group of $Z_n$ is a connected graph, the set of vertices of which is $Z_n$. Here $\langle u, v\rangle \in P(Z_n)$ are two diverse adjacent vertices if and only if $u \ne v$ and $\langle v \rangle \subseteq \langle u \rangle$ or $\langle u \rangle \subseteq \langle v \rangle$. We will shortly name the undirected $P(Z_n)$ power graph as the power graph $P(Z_n)$. The Wiener, hyper-Wiener, Harary and SK indices of the $P(Z_n)$ power graph are in order as follows $$\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right), \ \frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right)+\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,{{d}^{2}}\left(u,v \right),$$ $$\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,\frac{1}{d\left(u,v \right)} \mbox{ and } \frac{1}{2}\underset{uv\in E\left( G \right)}{\mathop \sum }\,\left( {{d}_{u}}+{{d}_{v}} \right).$$ In this article we focus more on the indices of $P(Z_n)$ power graph by Wiener, hyper-Wiener, Harary and SK the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and SK indices of the power graph $P(Z_n)$ corresponding to the vertex $n = p^k \cdot q^r$. These are as follows: $p, q$ are distinct primes and $k, r$ are nonnegative integers.