On the complement of a signed graph

Abstract

Given a signed graph $(\Gamma ,\sigma )$ and a spanning tree of Γ, we define pseudo-potentials on Γ, which coincide with usual potential functions in the balanced case. Using a pseudo-potential, we are able to define a signature on the complement ${\Gamma }^c$ of Γ, in such a way that the signed complete graph obtained by taking the union of Γ and ${\Gamma }^c$ is stable under switching equivalence, and providing a solution to an open problem in the literature of signed graphs. Then, we introduce three new notions of signed regularity, we characterize them in terms of the adjacency matrix of $(\Gamma ,\sigma )$, and we show that under such regularity hypotheses the spectrum of the signed complete graph can be described in terms of the spectra of $(\Gamma ,\sigma )$ and of its signed complement. As an application of our machinery, we define a signed version of a generalization of the classical NEPS of graphs, whose signature is stable under switching equivalence. In particular, this construction allows to give a switching stable definition of the lexicographic product of signed graphs, for which the spectrum is explicitly determined in the regular case.