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arXiv · 2608.09822

On a conjecture of Kolokolnikov on algebraic connectivity

Cheng Chi·Junjie Wang·Jiaxin Zheng
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Abstract

For a graph $G$, let $\alpha(G)$ be the second smallest eigenvalue of the Laplacian matrix of $G$, also known as the algebraic connectivity. Algebraic connectivity plays an important role in characterizing the connectivity of graphs and convergence properties of networks. Kolokolnikov conjectured that among all graphs on $n$ vertices with exactly $2n-4$ edges, $\alpha(G)\leq 2$ and one of the maximizers is the complete bipartite graph whose two parts have sizes two and $n-2$, respectively. In this paper, we completely resolve this conjecture.

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Cheng Chi, Junjie Wang, Jiaxin Zheng. 2026-08-10. On a conjecture of Kolokolnikov on algebraic connectivity. https://arxiv.org/abs/2608.09822

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