Spectral Radius of Graphs with Size Constraints: Resolving a Conjecture of Guiduli

TL;DR

This paper proves a conjecture about the maximum spectral radius of graphs with size constraints on subgraphs, characterizes extremal graphs, and introduces a new potential function to analyze their structure.

Contribution

It resolves Guiduli's conjecture by establishing an upper bound on spectral radius for graphs with hereditary size properties and characterizes the extremal graphs achieving this bound.

Findings

Spectral radius of graphs with size constraints is bounded by a specific function.

Extremal graphs are characterized as join graphs with particular forest structures.

Introduction of a novel potential function $ ext{eta}(F)$ for structural analysis.

Abstract

We resolve a problem posed by Guiduli (1996) on the spectral radius of graphs satisfying the Hereditarily Bounded Property $P_{t,r}$, which requires that every subgraph $H$ with $|V(H)| \geq t$ satisfies $|E(H)| \leq t|V(H)| + r$. For an $n$-vertex graph $G$ satisfying $P_{t,r}$, where $t > 0$ and $r \geq -\binom{\lfloor t+1 \rfloor}{2}$, we prove that the spectral radius $\rho(G)$ is bounded above by $\rho(G) \leq c(s,t) + \sqrt{\lfloor t \rfloor n}$, where $s = \binom{\lfloor t \rfloor + 1}{2} + r$, thus affirmatively answering Guiduli's conjecture. Furthermore, we present a complete characterization of the extremal graphs that achieve this bound. These graphs are constructed as the join graph $K_{\lfloor t \rfloor} \nabla F$, where $F$ is either $K_3 \cup (n - \lfloor t \rfloor - 3)K_1$ or a forest consisting solely of star structures. The specific structure of such forests is meticulously characterized. Central to our analysis is the introduction of a novel potential function $\eta(F) = e(F) + (\lfloor t \rfloor - t)|V(F)|$, which quantifies the structural "positivity" of subgraphs. By combining edge-shifting operations with spectral radius maximization principles, we establish sharp bounds on $\eta^+(G)$, the cumulative positivity of $G$. Our results contribute to the understanding of spectral extremal problems under edge-density constraints and provide a framework for analyzing similar hereditary properties.