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

Abstract

Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where $n⩾3$. 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 ${\mathcal{G}}_n^r$ be the set of subgraphs obtained from $G_n$ by deleting any $r$ vertical edges of $G_n$, where $0⩽r⩽n$. Explicit formulas for the Kirchhoff index and the number of spanning trees for any graph $G_n^r\in {\mathcal{G}}_n^r$ are completely established, respectively. Finally, it is interesting to see that the Kirchhoff index of $G_n^r$ is almost one-sixth of its Wiener index.

References in corpus (1)

• Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed polyomino chains