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On the $D_α$ spectral radius of non-transmission regular graphs

Zengzhao Xu, Weige Xi, Ligong Wang

TL;DR

This paper studies the generalized distance spectral radius $\mu_\alpha(G)$ of connected non-transmission regular graphs via the generalized distance matrix $D_\alpha(G)$. It proves a sharp lower bound on the gap $Tr_{\max}(G)-\mu_\alpha(G)$ that depends on the parity of $n$ and on $\alpha\in[0,1)$, introducing the auxiliary quantity $(1-\alpha)\tau(G)$ with $\tau_n$ defined by a quadratic, and showing extremal graphs achieving equality. For odd $n$, equality occurs for the star-like graph $K_{1,2,\cdots,2}$; for even $n$, equality holds for any $(n-4)$-$DVDR$ graph. The results extend known bounds for distance-based spectral radii to the $D_\alpha$ setting, using equitable quotient matrices and Perron–Frobenius theory, and provide precise extremal characterizations that illuminate irregularity via distance spectra.

Abstract

Let $G$ be a connected graph with order $n$ and size $m$. Let $D(G)$ and $Tr(G)$ be the distance matrix and diagonal matrix with vertex transmissions of $G$, respectively. For any real $α\in[0,1]$, the generalized distance matrix $D_α(G)$ of $G$ is defined as $$D_α(G)=αTr(G)+(1-α)D(G).$$ The largest eigenvalue of $D_α(G)$ is called the $D_α$ spectral radius or generalized distance spectral radius of $G$, denoted by $μ_α(G)$. In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the $D_α$ spectral radius of non-transmission regular graphs, and we also characterize the extremal graphs attaining the bound.

On the $D_α$ spectral radius of non-transmission regular graphs

TL;DR

This paper studies the generalized distance spectral radius of connected non-transmission regular graphs via the generalized distance matrix . It proves a sharp lower bound on the gap that depends on the parity of and on , introducing the auxiliary quantity with defined by a quadratic, and showing extremal graphs achieving equality. For odd , equality occurs for the star-like graph ; for even , equality holds for any - graph. The results extend known bounds for distance-based spectral radii to the setting, using equitable quotient matrices and Perron–Frobenius theory, and provide precise extremal characterizations that illuminate irregularity via distance spectra.

Abstract

Let be a connected graph with order and size . Let and be the distance matrix and diagonal matrix with vertex transmissions of , respectively. For any real , the generalized distance matrix of is defined as The largest eigenvalue of is called the spectral radius or generalized distance spectral radius of , denoted by . In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the spectral radius of non-transmission regular graphs, and we also characterize the extremal graphs attaining the bound.
Paper Structure (3 sections, 6 theorems, 47 equations, 1 figure)

This paper contains 3 sections, 6 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['T2.6']}

Key Result

Theorem 1.1

Let $G$ be a connected non-transmission regular graph of order $n$. (1) If $n$ is odd and $\alpha\in[0,1)$, then equality holds if and only if $G\cong K_{1,2,2,\cdots,2}$ (the definition see Lemma L2.4). (2) If $n$ is even and $\alpha\in[0,1)$, then equality holds if and only if $G$ is isomorphic to any $(n-4)$-$DVDR$ graph (the definition see Page 4).

Figures (1)

  • Figure 1: Some examples of $DVDR$ graphs

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof