Spectral Radius of Graphs with Size Constraints: Resolving a Conjecture of Guiduli
Rui Li, Anyao Wang, Mingqing Zhai
TL;DR
This work resolves Guiduli's conjecture by proving that every $n$-vertex graph $G$ with the Hereditarily Bounded Property $P_{t,r}$ satisfies $\rho(G) \le c(s,t) + \sqrt{\lfloor t\rfloor n}$, with $s = \binom{\lfloor t\rfloor+1}{2} + r$, for $t>0$ and $r \ge -\binom{\lfloor t+1 \rfloor}{2}$. A stability theorem shows that large spectral radius compels a tight $\lfloor t\rfloor$-clique-like core with high degree, enabling a precise structural reduction to join-graphs $K_{\lfloor t\rfloor} \nabla F$. The extremal graphs are completely characterized: they are joins $K_{\lfloor t\rfloor} \nabla F$ where $F$ is either $K_3 \cup (n-\lfloor t\rfloor-3)K_1$ or a carefully constrained forest of stars, with a detailed parameterization of the star composition. The analysis introduces a potential function $\eta(F)$ and uses edge-shifting and walk-based comparisons to obtain sharp bounds, providing a robust framework for spectral extremal problems under hereditary edge-density constraints.
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 $ρ(G)$ is bounded above by $ρ(G) \leq c(s,t) + \sqrt{\lfloor t \rfloor n}$, 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 $η(F) = e(F) + (\lfloor t \rfloor - t)|V(F)|$, which quantifies the structural "positivity" of subgraphs. By combining edge-shifting operations with spectral radius maximization principles, we establish sharp bounds on $η^+(G)$, 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.