Maximum degree and spectral radius of graphs in terms of size

TL;DR

This paper investigates the relationship between the spectral radius and signless Laplacian spectral radius of large graphs and their structural properties, establishing conditions that guarantee the presence of specific subgraphs.

Contribution

It extends existing results by linking spectral bounds to the existence of certain subgraphs like cycles and stars in large graphs.

Findings

If rho(G)  \u2265 sqrt{m-k}, then G contains either a 4-cycle or a star K_{1,m-k}.

If q(G)    m-k, then G contains a star K_{1,m-k}.

Results apply to large graphs and extend previous spectral graph theory findings.

Abstract

Research on the relationship of the (signless Laplacian) spectral radius of a graph with its structure properties is an important research project in spectral graph theory. Denote by $\rho(G)$ and $q(G)$ the spectral radius and the signless Laplacian spectral radius of a graph $G$, respectively. Let $k\ge 0$ be a fixed integer and $G$ be a graph of size $m$ which is large enough. We show that if $\rho(G)\ge\sqrt{m-k}$, then $C_4\subseteq G$ or $K_{1,m-k}\subseteq G$. Furthermore, we prove that if $q(G)\ge m-k$, then $K_{1,m-k}\subseteq G$. Both these two results extend some known results.