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Shivaramakrishna Pragada

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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 clique number of $G$. We characterize the families of graphs for which the above inequality is sharp. Precisely, we prove that, for $r\geq 2$, equality holds if and only if $r\mid n$ and $G$ is the complete regular $r$-partite graph $K_{n/r,\ldots,n/r}$.

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

Preprint Aug 2026

Extremal graphs for the $k$-th eigenvalue

For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.

Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al. · 0 citations