Spectra of signed graphs and related oriented graphs (Q6650515)

An oriented or a signed graph is connected, regular, or bipartite if the same holds for its underlying graph. Similarly, a matching (or a perfect matching) refers to the matching in the underlying graph.\N\NThe theory of spectra of oriented graphs is strongly connected to the theory of spectra of signed graphs, and many problems concerning spectral parameters of oriented graphs can be transferred to the framework of signed graphs, where the entire theory of real symmetric matrices can be employed.\N\NFor every oriented graph \(G^\prime\), there exists a bipartite signed graph \(H^\prime\) such that the spectrum of \(H^\prime\) contains the full information about the spectrum of the skew adjacency matrix of \(G^\prime\). This approach allows the author to transfer certain problems related to the skew eigenvalues of oriented graphs into the framework of signed graphs. In this framework, the theory of real symmetric matrices can be utilized.\N\NThe main result is a crucial relation between the spectrum of an oriented graph and the spectrum of a related signed graph.\N\NIn this work, the author continues his previous research by relating the characteristic polynomials, eigenspaces, and energy of \(G^\prime\) to those of \(H^\prime\). At the same time, he addresses several open problems concerning the skew eigenvalues of oriented graphs.\N\NThe author addresses the problems [\textit{C. Adiga} et al., Linear Algebra Appl. 432, No. 7, 1825--1835 (2010; Zbl 1217.05131)] related to the existence of skew-symmetric conference matrices that do not give the maximum energy of the corresponding oriented graph.\N\NIt is an interesting paper that is useful to researchers working in this area.