On the structure of graphs with given odd girth and large algebraic connectivity

Abstract

A classical result of Andrásfai, Erdős, and Sós states that every $n$-vertex graph with odd girth at least $2k+1$ and minimum degree larger than $\frac{2n}{2k+1}$ is bipartite. Rather than imposing a minimum-degree condition, in this paper we investigate conditions on algebraic connectivity that force graphs of given odd girth to have a simple structure. The algebraic connectivity of a graph $G$, denoted by ${\mu }_2(G)$, is the second smallest eigenvalue of its Laplacian matrix. Our main results are as follows. 1. Every $n$-vertex triangle-free graph $G$ with ${\mu }_2(G)\ge \frac{n}{3}$ is bipartite. Moreover, the constant $\frac{1}{3}$ is asymptotically best possible. 2. For $k\ge 3$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with ${\mu }_2(G)>\frac{4n}{6k-1}$ is bipartite. 3. For $k\ge 22$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with ${\mu }_2(G)>\frac{3456n}{k^3}$ is bipartite. Moreover, the term $k^{-3}$ is asymptotically best possible.