Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extremal distance spectral radius of graphs with fixed size

Hongying Lin, Bo Zhou

TL;DR

This work addresses the problem of determining, among connected graphs with fixed size $m$, which graphs minimize the distance spectral radius $\rho(G)$, the largest eigenvalue of the distance matrix $D(G)$. By linking the problem to the structure of complements of forests and introducing the parameters $n=\left\lceil\frac{1+\sqrt{8m}}{2}\right\rceil$ and $s=m-{n-1\choose 2}$, the authors prove that for a broad range ${n-1\choose 2}+\max\{\frac{n-6}{2},1\} \le m \le {n\choose 2}$ the unique minimizer is $P_{n,s+1}$, and they provide a detailed degree-structure analysis across subranges of $s$. They also characterize the extremal graphs achieving maximum distance spectral radius among complements of forests and establish auxiliary results for trees using Perron-Frobenius theory and Rayleigh quotients. The paper closes with a conjecture for the remaining small-$s$ regime and discusses potential extensions via forest-complement configurations, contributing to a deeper understanding of distance-based extremal graph theory.

Abstract

Let $m$ be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size $m$. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size $m$. Given $m$, there is exactly one positive integer $n$ such that ${n-1\choose 2} <m\leq {n\choose 2}$. We establish some structural properties of the extremal graphs for all $m$ and solve the problem for ${n-1\choose 2}+\max\{\frac{n-6}{2},1\}\le m\leq {n\choose 2}$. We give a conjecture for the remaining case. To prove the main results, we also determine the the complements of forests of fixed order with large and small distance spectral radius.

Extremal distance spectral radius of graphs with fixed size

TL;DR

This work addresses the problem of determining, among connected graphs with fixed size , which graphs minimize the distance spectral radius , the largest eigenvalue of the distance matrix . By linking the problem to the structure of complements of forests and introducing the parameters and , the authors prove that for a broad range the unique minimizer is , and they provide a detailed degree-structure analysis across subranges of . They also characterize the extremal graphs achieving maximum distance spectral radius among complements of forests and establish auxiliary results for trees using Perron-Frobenius theory and Rayleigh quotients. The paper closes with a conjecture for the remaining small- regime and discusses potential extensions via forest-complement configurations, contributing to a deeper understanding of distance-based extremal graph theory.

Abstract

Let be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size . After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size . Given , there is exactly one positive integer such that . We establish some structural properties of the extremal graphs for all and solve the problem for . We give a conjecture for the remaining case. To prove the main results, we also determine the the complements of forests of fixed order with large and small distance spectral radius.

Paper Structure

This paper contains 7 sections, 11 theorems, 47 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs which are the complements of forests with large and small distance spectral radius
  4. Proof of Theorem \ref{['max=1']}
  5. Proof of Theorems \ref{['size2023-result1=1']}
  6. Proofs of Theorem \ref{['size2025-result3']}
  7. Concluding remarks

Key Result

Theorem 1.1

Let $G\in \mathbb{G}(m)\setminus \{P_{m+1}, A_{m+1}\}$, where $m\ge 5$. Then $\rho(G)\leq \rho(B_{m+1})<\rho(A_{m+1})<\rho(P_{m+1})$ with equality in the first inequality if and only if $G\cong B_{m+1}$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 25 more