On the sum of Laplacian eigenvalues of a signed graph.

For a signed graph Γ, let e ( Γ ) denote the number of edges and S k ( Γ ) denote the sum of the k largest eigenvalues of the Laplacian matrix of Γ. We conjecture that for any signed graph Γ with n vertices, S k ( Γ ) ≤ e ( Γ ) + ( k + 1 2 ) + 1 holds for k = 1 , … , n . We prove the conjecture for any signed graph when k = 2 , and prove that this conjecture is true for unicyclic and bicyclic signed graphs. [ABSTRACT FROM AUTHOR]