The (multiplicative degree-)Kirchhoff index of graphs derived from the Catersian product of $S_n$ and $K_2$

TL;DR

This paper derives explicit formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index, and spanning trees of graphs formed from the Cartesian product of a star graph and a complete graph, solving a recent open problem.

Contribution

It provides the first complete solutions with closed-form formulas for these indices of graphs from the Cartesian product of $S_n$ and $K_2$, advancing spectral graph theory.

Findings

Explicit formulas for Kirchhoff index and multiplicative degree-Kirchhoff index

Closed-form expression for the number of spanning trees

Complete solution to the problem posed by Li et al.

Abstract

Recently, Li et al. [Appl. Math. Comput. 382 (2020) 125335] proposed the problem of determining the Kirchhoff index and multiplicative degree-Kirchhoff index of graphs derived from $S_n \times K_2$, the Catersian product of the star $S_n$ and the complete graph $K_2$. In the present paper, we completely solve this problem. That is, the explicit closed-form formulae of Kirchhoff index, multiplicative degree-Kirchhoff index, and number of spanning trees are obtained for some graphs derived from $S_n \times K_2$.