A graph energy conjecture through the lenses of semidefinite programming

Let $G$ be a graph on $n$ vertices with independence number $\alpha(G)$. Let $\mathcal{E}(G)$ be the energy of a graph, defined as the sum of the absolute values of the adjacency eigenvalues of $G$. Using Graffiti, Fajtlowicz conjectured in the 1980s that $$\frac{1}{2}\mathcal{E}(G) \geq n - \alpha(G).$$ In this paper we derive a semidefinite program formulation of the graph energy, and we use it to obtain several results that constitute a first step towards proving this conjecture. In particular, we show that $$\frac{1}{2}\mathcal{E}(G) \geq n - \chi_f(\overline{G}) \quad \text{ and } \quad \frac{1}{2}\mathcal{E}(G) \geq n - H(G),$$ where $\chi_f(G)$ is the fractional chromatic number and $H(G)$ is Hoffman's ratio number. As a byproduct of the SDP formulation we obtain several lower bounds for the graph energy that improve and refine previous results by Hoffman (1970) and Nikiforov (2007). The later author showed that the conjecture holds for almost all graphs. However, the graph families known to attain the conjecture with equality are highly structured and do not represent typical graphs. Motivated by this, we prove the following bound in support of the conjecture for the class of highly regular graphs $$\frac{1}{2}\mathcal{E}(G) \geq n - \vartheta^-(G),$$ where $\vartheta^-$ is Schrijver's theta number.