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Kirchhoff index of a nested geometric graph with weighted multiple edges

Da-yeon Huh

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.

Kirchhoff index of a nested geometric graph with weighted multiple edges

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.
Paper Structure (5 sections, 10 theorems, 77 equations, 8 figures, 1 table)

This paper contains 5 sections, 10 theorems, 77 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Determinant of a block tridiagonal matrix
  3. Kirchhoff index of $\mathbb{G}_n$
  4. Kirchhoff index of a 4-regular graph
  5. Discussion

Key Result

Lemma 2.1

horn2012matrix Suppose $A$ is a nonsingular square submatrix of a block matrix $M=\in \mathbf{R}^{n\times n}$. Then In particular, when $D=d\in \mathbf{R}$, $B=\boldsymbol{x}$ and $C=\boldsymbol{y}^T$, where $\boldsymbol{x}, \boldsymbol{y}\in \mathbf{R}^{n-1}$, Moreover,

Figures (8)

  • Figure 1: Simple circuit with three resistors
  • Figure 2: Sequence of nested geometric graphs
  • Figure 3: Sequence of nested goemetric graphs with resistance
  • Figure 4: Directed graph of $\mathbb{G}_2$ used to derive $Q$
  • Figure 5: Plot of $K\!f(\mathbb{G}_n)$ and $Y_{asym}$ for $1\leq n \leq 30$
  • ...and 3 more figures

Theorems & Definitions (16)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • proof
  • Theorem 3.1
  • ...and 6 more