Unicyclic signed graphs with minimal energy

Abstract

A connected signed graph with n vertices is said to be unicyclic if its number of edges is n. The energy of a signed graph S of order n with eigenvalues x1,x2,…,xn is defined as E(S)=∑j=1n|xj|. We obtain the integral representations for the energy of a signed graph. We show that even and odd coefficients of the characteristic polynomial of a unicyclic signed graph respectively alternate in sign. As an application of integral representation, we compute and compare the energy of unicyclic signed graphs. Finally, we characterize unicyclic signed graphs with minimal energy.