Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs

TL;DR

This paper investigates the relationship between the spectral radius of graphs and the presence of specific subgraphs, establishing thresholds that guarantee certain cycles or bipartite structures, and extends classical spectral graph theory results.

Contribution

It introduces new spectral bounds that determine the existence of cycles and bipartite subgraphs in graphs with fixed size, generalizing previous inequalities and conjectures.

Findings

Graphs with spectral radius at least (m)\u00a0contain specific subgraphs like cycles or bipartite graphs.

Thresholds for spectral radius guarantee the presence of cycles of various lengths.

Certain extremal graphs, like stars or books, are characterized as exceptions.

Abstract

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $\rho(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if $\rho(G)>\sqrt{m}$. In this paper, we follow this direction on determining which subgraphs will be contained in $G$ if $\rho(G)> f(m)$, where $f(m)\sim\sqrt{m}$ as $m\rightarrow \infty$. We first show that if $\rho(G)\geq \sqrt{m}$, then $G$ contains $K_{2,r+1}$ unless $G$ is a star; and $G$ contains either $C_3^+$ or $C_4^+$ unless $G$ is a complete bipartite graph, where $C_t^+$ denotes the graph obtained from $C_t$ and $C_3$ by identifying an edge. Secondly, we prove that if $\rho(G)\geq{\frac12+\sqrt{m-\frac34}}$, then $G$ contains pentagon and hexagon unless $G$ is a book; and if $\rho(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$, then $G$ contains $C_t$ for every $t\leq 2k+2$. In the end, some related conjectures are provided for further research.