Graphs with Diameter $n-e$ Minimizing the Spectral Radius

TL;DR

This paper proves conjectures about the structure of graphs with minimal spectral radius for fixed diameter and determines the minimal spectral radius graphs for specific diameter values, advancing understanding of spectral graph properties.

Contribution

The paper confirms three conjectures on the structure of minimal spectral radius graphs for diameters 6 and 7, and determines the minimal spectral radius graph for diameter 8.

Findings

Confirmed conjectures for diameters 6 and 7.

Determined the minimal spectral radius graph for diameter 8.

Characterized the structure of extremal graphs in the specified family.

Abstract

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \sim 1 \sim 2 \sim ... \sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\in\{1,2,...,t\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioab\v{a}-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed $e\geq 6$, $G^{min}_{n,n-e}$ is in the family ${\cal P}_{n,e}=\{P_{2,1,...1,2,n-e+1}^{2,m_2,...,m_{e-4},n-e-2}\mid 2<m_2<...<m_{e-4}<n-e-2\}$. For $e=6,7$, they conjectured $G^{min}_{n,n-6}=P^{2,\lceil\frac{D-1}{2}\rceil,D-2}_{2,1,2,n-5}$ and $G^{min}_{n,n-7}=P^{2,\lfloor\frac{D+2}{3}\rfloor,D- \lfloor\frac{D+2}{3}\rfloor, D-2}_{2,1,1,2,n-6}$. In this paper, we settle their three conjectures positively. We also determine $G^{min}_{n,n-8}$ in this paper.