Proof of a Conjecture on the Seidel Energy of Graphs

Let $G$ be a graph with the vertex set $\{v_1,\dots ,v_n\}$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $v_i$ and $v_j$ are adjacent and otherwise is $1$. The Seidel energy of $G$ is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of $G$. Haemers conjectured that the Seidel energy of any graph of order $n$ is at least $2n-2$ and, up to Seidel equivalence, the equality holds for $K_n$. We establish the validity of Haemers’ Conjecture in general.