Positive and Negative Square Energies of Graphs

Abstract

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^{+}(G),s^{-}(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if $G$ is a connected graph of order $n$, then $s^{+}(G)\ge n-1$ and $s^{-}(G)\ge n-1$. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.

References in corpus (1)

• New Eigenvalue Bound for the Fractional Chromatic Number

Cited by in corpus (2)

• Observations on Graph Invariants with the Lovász $\vartheta$-Function
• On the Positive and Negative $p$-Energies of Graphs under Edge Addition