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- W1853207795 abstract "For a graph G, let S(G) be the Seidel matrix of G and $${theta }_1(G),ldots ,{theta }_n(G)$$ be the eigenvalues of S(G). The Seidel energy of G is defined as $$|{theta }_1(G)|+cdots +|{theta }_n(G)|$$ . 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 $$alpha $$ with $$0<alpha <2,|{theta }_1(G)|^alpha +cdots +|{theta }_n(G)|^alpha geqslant (n-1)^alpha +n-1$$ if and only if $$|hbox {det},S(G)|geqslant n-1$$ . This, in particular, implies the Haemers’ conjecture for all graphs G with $$|hbox {det},S(G)|geqslant n-1$$ . A computation on the fraction of graphs with $$|hbox {det},S(G)|<n-1$$ is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy $$|hbox {det},S(G)|geqslant n-1$$ . In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of order $$Theta (n^{3/2})$$ . Finally, we prove that self-complementary graphs G of order $$nequiv 1pmod 4$$ have $$det S(G)=0$$ ." @default.
- W1853207795 created "2016-06-24" @default.
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- W1853207795 date "2016-07-16" @default.
- W1853207795 modified "2023-10-17" @default.
- W1853207795 title "On eigenvalues of Seidel matrices and Haemers’ conjecture" @default.
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- W1853207795 doi "https://doi.org/10.1007/s10623-016-0248-x" @default.
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