Revista de la
Unión Matemática Argentina
Spectra of subdivisions of signed graphs, signed $R$-graphs, and related products
Mir Riyaz ul Rashid, Shariefuddin Pirzada, Tahir Shamsher, and Zoran Stanić

Volume 69, no. 2 (2026), pp. 499–526    

Published online (final version): September 30, 2026

https://doi.org/10.33044/revuma.4483

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Abstract

The subdivision of an ordinary graph is obtained by inserting a vertex into every edge, while its $R$-graph is obtained by adding a new vertex to every edge and joining it to both ends. We study the corresponding constructions for signed graphs, prove that both are stable under switching, resolve their balance, and determine the spectrum (of the adjacency matrix) of the signed $R$-graph in the regular case. We also introduce two corona-like products based on signed subdivisions and four based on signed $R$-graphs. We determine their characteristic polynomials and spectra, together with their Laplacian characteristic polynomials and spectra, either in general or under appropriate regularity assumptions. Finally, for the generalized subdivision introduced in [Ars Math. Contemp. 23 (2023), 3–9], we express its spectrum in terms of the Laplacian spectrum of the original signed graph, thereby answering a problem posed there. These results provide constructions of cospectral and Laplacian cospectral signed graphs.

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