Total graph of a signed graph

TL;DR

This paper extends the concept of total graphs to signed graphs, analyzing their spectral properties, stability under switching, and key measures like balance and frustration, with explicit spectrum calculations in regular cases.

Contribution

It introduces a new construction for total graphs of signed graphs and studies their spectral and structural properties, including stability and eigenvalues.

Findings

Total signed graphs are stable under switching.

Spectra of adjacency matrices are computed for regular cases.

Some total graphs have exactly two main eigenvalues.

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 definitions of the line signed graph, we define 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.