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arXiv · 1901.02047

Proof of a conjecture on the algebraic connectivity of a graph and its complement

Mostafa Einollahzadeh·Mohammad Mahdi Karkhaneei
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Abstract

For a graph $G$, let $λ_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that $λ_2(G) + λ_2(\overline{G}) \geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove this conjecture in the general case. Also, we will show that $\max\{λ_2(G), λ_2(\overline{G})\} \geq 1 - O(n^{-\frac 13})$, where $n$ is the number of vertices of $G$.

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Mostafa Einollahzadeh, Mohammad Mahdi Karkhaneei. 2021-06-24. Proof of a conjecture on the algebraic connectivity of a graph and its complement. https://arxiv.org/abs/1901.02047

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