Resistance distance-based graph invariants and spanning trees of graphs derived from the strong product of $P_2$ and $C_n$

TL;DR

This paper derives explicit formulas for resistance-based graph invariants and spanning trees of graphs formed by the strong product of a path and a cycle, revealing proportional relationships with classical indices.

Contribution

It provides new explicit expressions for Kirchhoff and related indices, and counts of spanning trees for these specific product graphs and their subgraphs.

Findings

Kirchhoff index is approximately one-sixth of Wiener index for the graphs.

Explicit formulas for the number of spanning trees are established.

Resistance indices relate closely to classical graph invariants.

Abstract

Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where $n\geqslant3$. In this paper, explicit expressions for the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ are determined, respectively. It is surprising to find that the Kirchhoff (resp. multiplicative degree-Kirchhoff) index of $G_n$ is almost one-sixth of its Wiener (resp. Gutman) index. Moreover, let $\mathcal{G}^r_n$ be the set of subgraphs obtained from $G_n$ by deleting any $r$ vertical edges of $G_n$, where $0\leqslant r\leqslant n$. Explicit formulas for the Kirchhoff index and the number of spanning trees for any graph $G^r_n\in \mathcal{G}^r_{n}$ are completely established, respectively. Finally, it is interesting to see that the Kirchhoff index of $G^r_n$ is almost one-sixth of its Wiener index.