Diameters of Graphs with Spectral Radius at most $3/2\sqrt{2}$

TL;DR

This paper investigates the diameters of graphs with spectral radius at most $3/2\sqrt{2}$, establishing tight bounds for open and closed quipus, and characterizes minimal spectral radius graphs for certain diameters, settling a conjecture.

Contribution

It provides tight bounds on diameters of specific graph classes with spectral radius constraints and characterizes minimal spectral radius graphs, confirming a conjecture for a range of diameters.

Findings

Diameter bounds for open quipus are tight and depend on the number of vertices.

Diameter bounds for closed quipus are asymptotically tight, with exact upper bounds.

The minimal spectral radius graphs for certain diameters are characterized as attaching paths to antipodal vertices of an even cycle.

Abstract

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph $G$ with $\rho(G)\leq 3/2{\sqrt{2}}$ is either a dagger, an open quipu, or a closed quipu. The reverse statement is not true. Many open quipus and closed quipus have spectral radius greater than $3/2{\sqrt{2}}$. In this paper we proved the following results. For any open quipu $G$ on $n$ vertices ($n\geq 6$) with spectral radius less than $3/2{\sqrt{2}}$, its diameter $D(G)$ satisfies $D(G)\geq (2n-4)/3$. This bound is tight. For any closed quipu $G$ on $n$ vertices ($n\geq 13$) with spectral radius less than $3/2{\sqrt{2}}$, its diameter $D(G)$ satisfies $\frac{n}{3}< D(G)\leq \frac{2n-2}{3}$. The upper bound is tight while the lower bound is asymptotically tight. Let $G^{min}_{n,D}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $D$. We applied the results and found $G^{min}_{n,D}$ for some range of $D$. For $n\geq 13$ and $D\in [\frac{n}{2}, \frac{2n-7}{3}]$, we proved that $G^{min}_{n,D}$ is the graph obtained by attaching two paths of length $D-\lfloor\frac{n}{2}\rfloor$ and $D-\lceil\frac{n}{2}\rceil$ to a pair of antipodal vertices of the even cycle $C_{2(n-D)}$. Thus we settled a conjecture of Cioab-van Dam-Koolen-Lee, who previously proved a special case $D=\frac{n+e}{2}$ for $e=1,2,3,4$.