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A complete characterization of spectra of the Randic matrix of level-wise regular trees

Punit Vadher, Devsi Bantva

TL;DR

The problem addressed is to completely characterize the eigenvalues of the Randić matrix $R(T^z)$ for level-wise regular trees. The authors develop a block-tridiagonal representation of $R(T^z)$ and introduce a three-term recurrence $\phi_j(\lambda)$ whose zeros determine the spectrum. The main result expresses $\sigma(R(T^1))$ and $\sigma(R(T^2))$ as unions of spectra of small symmetric matrices $P_j$ and $P^z_{h+1}$, yielding explicit eigenvalue multiplicities. This reduction enables efficient computation of the Randić energy and index for these trees and clarifies the spectral structure in terms of the height and degree sequence.

Abstract

Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1 \leq i \leq n$. The Randić matrix of $G$, denoted by $R(G) = [r_{i,j}]$, is the $n \times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j} = 1/\sqrt{d_id_j}$ if $v_i$ and $v_j$ are adjacent in $G$ and 0 otherwise. A tree is a connected acyclic graph. A level-wise regular tree is a tree rooted at one vertex $r$ or two (adjacent) vertices $r$ and $r'$ in which all vertices with the minimum distance $i$ from $r$ or $r'$ have the same degree $m_i$ for $0 \leq i \leq h$, where $h$ is the height of $T$. In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randić matrix of level-wise regular trees. We prove that the eigenvalues of the Randić matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence $(m_0,m_1,\ldots,m_{h-1})$ of level-wise regular trees.

A complete characterization of spectra of the Randic matrix of level-wise regular trees

TL;DR

The problem addressed is to completely characterize the eigenvalues of the Randić matrix for level-wise regular trees. The authors develop a block-tridiagonal representation of and introduce a three-term recurrence whose zeros determine the spectrum. The main result expresses and as unions of spectra of small symmetric matrices and , yielding explicit eigenvalue multiplicities. This reduction enables efficient computation of the Randić energy and index for these trees and clarifies the spectral structure in terms of the height and degree sequence.

Abstract

Let be a simple finite connected graph with vertex set . Denote the degree of vertex by for all . The Randić matrix of , denoted by , is the matrix whose -entry is if and are adjacent in and 0 otherwise. A tree is a connected acyclic graph. A level-wise regular tree is a tree rooted at one vertex or two (adjacent) vertices and in which all vertices with the minimum distance from or have the same degree for , where is the height of . In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randić matrix of level-wise regular trees. We prove that the eigenvalues of the Randić matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence of level-wise regular trees.
Paper Structure (3 sections, 9 theorems, 82 equations, 2 figures, 1 algorithm)

This paper contains 3 sections, 9 theorems, 82 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The spectrum of the Randić matrix of level-wise regular tree

Key Result

Theorem 2.4

(refer Cvetkovic1980) Suppose $A \in \mathbb{R}^{n \times n}$ is symmetric. Let $B \in \mathbb{R}^{m \times m}$ with $m < n$ be a principal submatrix (obtained by deleting both $i$-th row and $i$-th column for some value of $i$). Suppose $A$ has eigenvalues $\lambda_1 \leq \ldots \leq \lambda_n$ and And if $m=n-1$ then

Figures (2)

  • Figure 1: Tree $T^1_{4,4,3}$ and its vertex ordering.
  • Figure 2: Tree $T^2_{4,3,4}$ and its vertex ordering.

Theorems & Definitions (19)

  • Example 2.2
  • Example 2.3
  • Theorem 2.4: Eigenvalue Interlacing Theorem
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 9 more