Energy and independence number

Abstract

For a graph $G$ of order $n$, with adjacency eigenvalues $\hat{I}{»}_1(G)\ge \cdots \ge \hat{I}{»}_n(G)$, the \emph{energy} of $G$ is defined to be \[\mathcal{E}(G)=\sum_{i=1}^{n} |λ_i(G)|.\] A well-known conjecture from the 1980s by Fajtlowicz states that for any graph $G$, \[\mathcal{E}(G) \ge 2\left(n-α(G)\right),\] where $\hat{I}\pm (G)$ denotes the independence number. We prove this conjecture.

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