Maximizing the algebraic connectivity of graphs of given order and size: a proof of a conjecture of Kolokolnikov

Abstract

The algebraic connectivity of a graph $G$ is a well-studied graph invariant that is related to other properties of the graph such as connectivity and expansion. Given $n$ and $m$, $\alpha (n,m)$ is the maximum algebraic connectivity of a graph with $n$ vertices and $m$ edges. In 2015, Kolokolnikov conjectured that $\alpha (n,2n-4)=2$ for $n\ge 4$, and verified this claim computationally for $n\le 12$. In this paper, we prove Kolokolnikov's conjecture. We also show that $\alpha (n,3(n-3))=3$ is false in general. %Combined with the computational verification for $n\le 12$, this yields $\alpha (n,2n-4)=2$ for all admissible values of $n$.

Corrected minor typos and included the appendix