A Linear Lower Bound for the Square Energy of Graphs

Key Points

The result proves that for every connected graph of order n, the square energy is at least 3n/4.
It derives bounds for square energy based on the positive and negative eigenvalues of disjoint induced subgraphs.
The conjectured bound of n-1 for square energy remains unestablished for all graph types.
This work improves the understanding of eigenvalue characteristics in connected graphs.

Abstract

Let G be a graph of order n with eigenvalues ₁ ₙ. Let ^+ (G) =㶁>₀ ᵢ², s^- (G) =㶁<₀ ᵢ². \ The smaller value, s (G) =\s^+ (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 ^+ (G) ₈=₁^k s^+ (Hᵢ) and s^- (G) ₈=₁^k s^- (Hᵢ), \ and then use this result to prove that s (G) 3n4 for every connected graph G of order n 4.

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