Maxima of spectral radius of irregular graphs with given maximum degree

TL;DR

This paper investigates the maximum spectral radius of irregular graphs with given maximum degree, confirming conjectures about asymptotic behavior and providing bounds for specific graph classes.

Contribution

It determines the maximum spectral radius for connected subcubic bipartite graphs and establishes new bounds for irregular and regular graphs' spectral radii.

Findings

Identified the unique connected subcubic bipartite graph with maximum spectral radius.

Provided lower bounds for the spectral radius gap in irregular and regular graphs.

Confirmed the asymptotic conjecture relating spectral radius and maximum degree.

Abstract

Let $\lambda^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $\Delta$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(\Delta-\lambda^{*}))/(\Delta-1)=\pi^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $\lambda_{1}(G)$, we present a lower bound for $\Delta-\lambda_{1}(G)$. Moreover, if $H$ is a proper subgraph of a $k$-connected $\Delta$-regular graph, a lower bound for $\Delta-\lambda_{1}(H)$ is also obtained. These bounds improve some previous results.