Energy of signed graphs and the vertex cover number

Abstract

The energy $\mathcal{E}(Γ)$ of a signed graph $Γ=(G,σ)$ is the sum of the absolute values of the eigenvalues of its adjacency matrix $A(Γ)$. In this article, we prove that $\mathcal{E}(Γ)\geq 2τ(G)$, where $τ(G)$ is the vertex cover number of $G$. The case of unsigned graphs, a 1980s conjecture proved only recently, is a special case of this result. Moreover, we completely characterize the equality case: equality holds if and only if every connected component of $Γ$ is switching equivalent to an \emph{additive rook graph} $\Rook_N(a,b)$. This graph is constructed on an $N\times N$ board with $a_i+b_j$ vertices on square $(i,j)$. Two vertices are joined by a positive edge if they share only a row, and by a negative edge if they share only a column. As consequences, we obtain a balanced/antibalanced trichotomy for the extremal signed graphs and deduce the characterization of the extremal unsigned graphs. Finally, an example shows that the bound does not extend to the Hermitian adjacency matrices of mixed graphs or to complex unit gain graphs.