Total graph of a signed graph

Abstract

The total graph is built by joining the graph to its line graph by means of the incidences. We introduce a similar construction for signed graphs. Under two similar defnitions of the line signed graph, we defne the corresponding total signed graph and we show that it is stable under switching. We consider balance, the frustration index and frustration number, and the largest
eigenvalue. In the regular case we compute the spectrum of the adjacency matrix of the total graph and the spectra of certain compositions, and we determine some with exactly two main eigenvalues.