Elphick–Liu–Ning conjecture on the square energies of connected graphs

Let $G$ be a connected graph with $n$ vertices and adjacency-matrix eigenvalues $\mu_1\geq\cdots\geq\mu_n$. Define

$$s^+(G)=\sum_{\mu_i>0}\mu_i^2,\qquad s^-(G)=\sum_{\mu_i<0}\mu_i^2.$$

Elphick–Liu–Ning conjecture. For any connected graph,

$$\min\{s^+(G),s^-(G)\}\geq n-1.$$

The conjecture is a central proposed lower bound for positive and negative square energies in spectral graph theory. The source describes it as an unsolved problem and notes evidence for the importance of connectedness; it is known for several graph families, including odd cycles, but remains open in general.