On eigenvalues of Seidel matrices and Haemers' conjecture

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and ${\text{te}}_1(G),\mathrm{.}\mathrm{.}\mathrm{.},{\text{te}}_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $\mathrm{\mid }{\text{te}}_1(G)\mathrm{\mid }+\mathrm{.}\mathrm{.}\mathrm{.}+\mathrm{\mid }{\text{te}}_n(G)\mathrm{\mid }$. 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 $\text{al}$ with $0<\text{al}<2$, $\mathrm{\mid }{\text{te}}_1(G){\mathrm{\mid }}^{\text{al}}+\mathrm{.}\mathrm{.}\mathrm{.}+\mathrm{\mid }{\text{te}}_n(G){\mathrm{\mid }}^{\text{al}}\text{g}(n-1{)}^{\text{al}}+n-1$ if and only if $|{\rm det} S(G)|\g n-1$. This, in particular, implies the Haemers’ conjecture for all graphs $G$ with $\mathrm{\mid }\mathrm{d}\mathrm{e}\mathrm{t}S(G)\mathrm{\mid }\text{g}n-1$.