Resistance distances in corona and neighborhood corona graphs with Laplacian generalized inverse approach

TL;DR

This paper derives formulas for resistance distances in corona and neighborhood corona graphs using the Laplacian generalized inverse, providing efficient computational methods and practical examples.

Contribution

It introduces a novel approach to compute resistance distances in corona and neighborhood corona graphs via Laplacian generalized inverse.

Findings

Derived explicit formulas for resistance distances in both graph types.

Validated the method with practical examples demonstrating accuracy.

Showed the efficiency of the proposed approach in computations.

Abstract

Let $G_1$ and $G_2$ be two graphs on disjoint sets of $n_1$ and $n_2$ vertices, respectively. The corona of graphs $G_1$ and $G_2$, denoted by $G_1\circ G_2$, is the graph formed from one copy of $G_1$ and $n_1$ copies of $G_2$ where the $i$-th vertex of $G_1$ is adjacent to every vertex in the $i$-th copy of $G_2$. The neighborhood corona of $G_1$ and $G_2$, denoted by $G_1\diamond G_2$, is the graph obtained by taking one copy of $G_1$ and $n_1$ copies of $G_2$ and joining every neighbor of the $i$-th vertex of $G_1$ to every vertex in the $i$-th copy of $G_2$ by a new edge. In this paper, the Laplacian generalized inverse for the graphs $G_1\circ G_2$ and $G_1\diamond G_2$ are investigated, based on which the resistance distances of any two vertices in $G_1\circ G_2$ and $G_1\diamond G_2$ can be obtained. Moreover, some examples as applications are presented, which illustrate the correction and efficiency of the proposed method.