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On the minimum spectral radius of connected graphs of given order and size

Sebastian M. Cioabă, Vishal Gupta, Celso Marques

TL;DR

This work addresses Hong's question on whether graphs minimizing the spectral radius among all connected graphs with fixed order $n$ and size $e$ are almost regular, delivering affirmative results in dense regimes and several sporadic cases. The authors leverage Perron–Frobenius theory, equitable partitions, eigenvalue interlacing, and Kelmans-type transformations to identify minimizer graphs and derive exact expressions for the minimal spectral radius $\rho_{\min}(n,e)$. They provide explicit minimizer structures for ranges $e\ge \binom{n-2}{2}$ and $e=\binom{n-1}{2}-2$, among others, and exhibit several cases where the minimizers are joins of large regular cores with highly regular subgraphs (e.g., cocktail party graphs). The results yield precise formulas and constructive descriptions (e.g., $K_{n-2p}\vee CP_{2p}$, $K_{n/2,n/2}-uv$, and three-part joins), sharpening our understanding of how spectral radius constraints enforce near-regularity and revealing both the structure and limitations of such minimizers in dense graphs. The paper thus advances a positive resolution of Hong's question in broad dense regimes and lays out conjectures and counterexamples guiding future work in spectral extremal graph theory.

Abstract

In this paper, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices $n$ and number of edges $e$, the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this paper, we answer Hong's question positively for various values of $n$ and $e$ and in several cases, we determined the graphs with minimum spectral radius.

On the minimum spectral radius of connected graphs of given order and size

TL;DR

This work addresses Hong's question on whether graphs minimizing the spectral radius among all connected graphs with fixed order and size are almost regular, delivering affirmative results in dense regimes and several sporadic cases. The authors leverage Perron–Frobenius theory, equitable partitions, eigenvalue interlacing, and Kelmans-type transformations to identify minimizer graphs and derive exact expressions for the minimal spectral radius . They provide explicit minimizer structures for ranges and , among others, and exhibit several cases where the minimizers are joins of large regular cores with highly regular subgraphs (e.g., cocktail party graphs). The results yield precise formulas and constructive descriptions (e.g., , , and three-part joins), sharpening our understanding of how spectral radius constraints enforce near-regularity and revealing both the structure and limitations of such minimizers in dense graphs. The paper thus advances a positive resolution of Hong's question in broad dense regimes and lays out conjectures and counterexamples guiding future work in spectral extremal graph theory.

Abstract

In this paper, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices and number of edges , the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this paper, we answer Hong's question positively for various values of and and in several cases, we determined the graphs with minimum spectral radius.
Paper Structure (11 sections, 31 theorems, 91 equations, 6 figures)

This paper contains 11 sections, 31 theorems, 91 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dense graphs
  4. The case $e\geq \frac{n(n-2)}{2}$
  5. The case ${n-1\choose 2} \leq e \leq \frac{n(n-2)}{2}$
  6. The case $e={n-1\choose 2}-2$
  7. Sporadic cases
  8. The case $e=\left(\frac{n-1}{2}\right)\left(\frac{n+1}{2}\right)$
  9. The case $e = \frac{n^2}{4} -1$
  10. The case $e = \frac{n^2}{3} -1$
  11. Final Remarks

Key Result

Lemma 2.1

The $p$-mean $d^{(p)}$, considered as a function of $p$, is continuous and strictly increasing if and only if there exists $i,j\in [n]$ such that $d_i\not=d_j.$

Figures (6)

  • Figure 1: $B(p,q,r)$ (left) and $P(p,q,r)$ (right).
  • Figure 2: Minimizers on 6 vertices and 8 edges
  • Figure 3: All $n=8,e=15$ non-isomorphic minimizers.
  • Figure 4: The minimizer for $n=8, e=17.$
  • Figure 5: Bicyclic minimizers on $7$ vertices.
  • ...and 1 more figures

Theorems & Definitions (55)

  • Lemma 2.1: Hof
  • Theorem 2.2: Hof
  • Theorem 2.3: Hof
  • Theorem 2.4: Theorem 8.1.3, CRS
  • Theorem 2.5: Theorem 8.1.10, CRS
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Proposition 3.1
  • proof
  • ...and 45 more