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S. A. Mojallal

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Preprint Aug 2026

Extremal Graphs for the Energy-Independence Number Inequality

For a graph $G$ of order $n$, let $\mathcal E(G)$ denote its adjacency energy and let $\alpha(G)$ denote its independence number. A recent theorem of Kumar and Pragada states that $$\mathcal E(G)\ge 2\bigl(n-\alpha(G)\bigr).$$ We determine all graphs attaining equality. More precisely, equality holds if and only if every connected component of $G$ is an isolated vertex, a balanced complete multipartite graph, or a graph obtained by taking the disjoint union of $K_{a,\ldots,a}$ and $K_{b,\ldots,b}$, with the same number $r\ge3$ of parts, and then completely joining corresponding parts.

S. A. Mojallal · 1 citation

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