Extremal Graphs for the Energy-Independence Number Inequality

Abstract

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 $$\mathcal{E}(G)\ge 2(n-\alpha (G)).$$ 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 $K_{a,\dots ,a}$ and $K_{b,\dots ,b}$, with the same number $r\ge 3$ of parts, and then completely joining corresponding parts.