A best bound for ${\lambda }_2(G)$ to guarantee $\kappa (G)\ge 2$

Abstract

Let $G$ be a connected $d$-regular graph with a given order and the second largest eigenvalue ${\lambda }_2(G)$. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for ${\lambda }_2(G)$ which guarantees that $\kappa (G)\ge t+1$, where $1\le t\le d-1$ and $\kappa (G)$ is the vertex-connectivity of $G$, which was also mentioned by Cioabă. As a starting point, we solve this problem in the case $t=1$, and characterize all families of extremal graphs.

10 pages

References in corpus (1)

• Edge-connectivity in regular multigraphs from eigenvalues