Spectra of subdivisions of signed graphs, signed $R$-graphs and related products

Abstract

The subdivision is a bipartite graph built from an ordinary graph by inserting a vertex into every edge, and an R-graph is obtained by adding a new vertex to every edge and joining it to the ends of the corresponding edge. In this paper we deal with similar constructions for signed graphs. Both are stable under switching, and the question on balance is completely resolved. In the regular case, the spectrum of the adjacency matrix of signed R-graph is computed. We also introduce two corona-like products based on the subdivision of a signed graph and four similar products based on the signed R-graph operation. For each of them we compute the characteristic polynomial along with the spectrum of the adjacency matrix and the spectrum of the Laplacian matrix either in general case or in case when one constituent is just regular or simultaneously regular and net-regular. In addition, we consider an other operation, called the generalized subdivision, introduced in [Ars Math. Contemp. 23 (2023), 3–9] and compute the spectrum of its adjacency matrix in terms of the Laplacian spectrum of the corresponding signed graph. In this way, we positively address a problem posed in the same reference. Our results can be interesting in the context of signed graphs sharing the same spectrum, since they provide constructions of such signed graphs in case of the ordinary spectrum as well as in case of the Laplacian spectrum.