A Linear Lower Bound for the Square Energy of Graphs

Let G be a graph of order n with eigenvalues λ₁ ≥ ⋯ ≥λₙ. Let \[s^+(G)=∑λ_i>0 λ_i^2, s^-(G)=∑λ_i<0 λ_i^2.\] The smaller value, s(G)=min⁺(G), s⁻(G)\ is called the square energy of G. In 2016, Elphick, Farber, Goldberg, and Wocjan conjectured that for every connected graph G of order n, s(G)≥ n-1. No linear bound for $s(G)$ in terms of n is known. Let H₁, …, Hₖ be disjoint induced subgraphs of G. In this note, we prove that \[s^+(G)≥∑ᵢ₌₁ᵏ s^+(H_i) and s^-(G)≥∑ᵢ₌₁ᵏ s^-(H_i),\] and then use this result to prove that s(G)≥ 3n/4 for every connected graph G of order n≥ 4.