Author

Rahul Roy

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Preprint Aug 2026

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

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 $\alp...

S. Cioabă, Abhay Jayarajan, M. Kannan et al. · 1 citation

Preprint Aug 2026

A Vertex-Localized Positive Square-Energy Strengthening of Tur\'an's Theorem

Let $G$ be a graph of order $n$ with the adjacency eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[...

Abhay Jayarajan, M. Kannan, Shivaramakrishna Pragada et al. · 0 citations

Preprint Aug 2026

The Equality Case in the Positive Square-Energy Strengthening of Tur\'an's Theorem

Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G)$, and let $s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2.$ Recently Liu, Tang, and Zhang proved the positive square-energy strengthening of Tur\'an's theorem \[\sqrt{s_+(G)}\leq \left(1-\frac1r\right)n.\] where $r=\omega(G)$ is the...

Abhay Jayarajan, M. Kannan, Shivaramakrishna Pragada et al. · 0 citations