Nonregular graphs with a given maximum degree attaining maximum spectral radius

TL;DR

This paper investigates the structure of connected nonregular graphs with maximum degree 5 that maximize spectral radius, confirming conjectures and characterizing graphs for specific degree and order conditions.

Contribution

It confirms Liu and Li4s conjecture for 5=3,4, and characterizes the structure of graphs for 5=n-2 and n-3 under certain conditions.

Findings

Confirmed conjecture for 5=3,4.

Fully characterized graphs for 5=n-2 with n55.

Fully characterized graphs for 5=n-3 with n55.

Abstract

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with $\delta<\Delta$. For $\Delta=3$ and $\Delta=4$, Liu (2024) confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed $\Delta$ and sufficiently large $n$, stating that the above $\delta=\Delta-1$ if $\Delta$ and $n$ are both odd, $\delta=1$ if $\Delta$ is odd and $n$ is even, and $\delta=\Delta-2$ if $\Delta$ is even. For the cases where $\Delta=n-2$ with $n\ge 5$, and $\Delta=n-3$ with $n\ge 59$, we fully characterize the structure of $G$.