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arXiv · 2608.09879

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

Sebastian M. Cioabă·Abhay Jayarajan·M. Rajesh Kannan·Rahul Roy
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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 $n\geq 4$, 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 $n$.

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Sebastian M. Cioabă, Abhay Jayarajan, M. Rajesh Kannan, Rahul Roy. 2026-08-10. Maximizing the algebraic connectivity of graphs of given order and size: a proof of a conjecture of Kolokolnikov. https://arxiv.org/abs/2608.09879

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