Extremal values on the Kirchhoff index of the line graph of trees

Abstract

The computation of resistance distance and the Kirchhoff index is a classical problem that has been extensively investigated by numerous mathematicians, physicists, and scientists. Consider a simple connected graph $G$ with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. If we replace each edge of $G$ with a resistor of 1 ohm resistance, we create an electrical network $N$; in that case, the distance between any two nodes between network $N$ is called resistance distance. The Kirchhoff index is a mathematical term that quantifies the complexity of a graph; it is defined as the total resistance distance among each pair of vertices in $G$. The line graph $L_G$ is constructed from $G$ by swapping out the edges for vertices; in $L_G$, if two vertices share endpoints in $G$, then they are connected in $L_G$. In this study, extremal values for the Kirchhoff index of the line graph of trees are calculated. Further, we will also calculate the Kirchhoff index for the line graph of some special trees and establish the relationship between them.