Some Bounds on the Energy of Signed Complete Bipartite Graphs

A signed graph Gσ is a pair (G,σ), where G is a graph, and σ : E(G) −→{−1,+1} is a function. Assume that m ≤ n are two positive integers. Let A =[ 0 B Bt 0 ] is the adjacency matrix of Kσm,n. In this talk we show that for every sign function σ, 2√mn ≤ E(Kσm,n) ≤ 2m√n, where E(Kσm,n) is the energy of Kσm,n. Also it is proved that the equality holds for the upper bound if there exists a Hadamard matrix of order n for which B is an m by n submatrix of H. Also if the equality holds, then every two distinct rows of B are orthogonal. We prove that for the lowerbound the equalityholds if andonly if Kσm,n isswitchingequivalentto Km,n.