Kirchhoff index of a nested geometric graph with weighted multiple edges

TL;DR

This paper derives explicit formulas and asymptotic behavior for the Kirchhoff index of a sequence of nested weighted geometric graphs, revealing linear growth as the graph size increases.

Contribution

It introduces a recurrence relation for the Laplacian polynomial and provides an explicit formula for the Kirchhoff index of these graphs.

Findings

Kirchhoff index grows asymptotically linearly with graph size.

Derived a recurrence relation for the Laplacian characteristic polynomial.

Calculated the Kirchhoff index for a related 4-regular graph.

Abstract

Kirchhoff index, Kf(G), introduced by Klein and Randic in 1993, represents the total effective resistances between all pairs of vertices in a graph G, where each edge is regarded as a resistor. In this paper, the Kirchhoff indices of a particular sequence of nested geometric graphs with weighted multiple edges, denoted by Gn, are investigated. A recurrence relation for the characteristic polynomial of the Laplacian matrix L(Gn) is derived, and an explicit formula for Kf(Gn) is obtained. These facilitate the analysis of the variation of Kf(Gn) as as n goes to infinity. Consequently, Kf(Gn) is shown to grow asymptotically linearly, characterized by a specific asymptotic formula. In the course of this derivation, a recurrence relation for the determinant of a block tridiagonal matrix is established. The Kirchhoff index of a 4-regular graph constructed from Gn is also determined.