On eigenvalues of Seidel matrices and Haemers' conjecture

For a graph G, let S(G) be the Seidel matrix of G and θ1(G), . . . , θn(G) be the eigenvalues of S(G). The Seidel energy of G is defined as |θ1(G)| + · · · + |θn(G)|. Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least 2n − 2, the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any α with 0 < α < 2, |θ1(G)| + · · ·+ |θn(G)| > (n − 1) + n − 1 if and only if |detS(G)| > n − 1. This, in particular, implies the Haemers’ conjecture for all graphs G with |detS(G)| > n− 1. A computation on the fraction of graphs with |detS(G)| < n−1 is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy |detS(G)| > n − 1. In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of Θ(n). Finally, we prove that self-complementary graphs G of order n ≡ 1 (mod 4) have detS(G) = 0. AMS Classification: 05C50