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On spectra and real energy of complex weighted digraphs. (English) Zbl 1456.05098

The authors studied spectral properties of complex weighted digraphs. Also, they showed that a complex weighted digraph \(D\) is balanced if and only if \(D\) and \(|D|\) have the same spectrum, where \(|D|\) is the absolute value weighted digraph of \(D\), that is, the digraph obtained by replacing the weight of each arc by its absolute value. They extended the concept of real energy to complex weighted digraphs and obtained extremal energy unicyclic complex weighted digraphs with cycle weight in the punctured disk \(\{z\in C : |z|\leq 1\}\{0\}\).
The authors considered a family of complex weighted digraphs \(D_{n,h}\), in which each digraph has order \(n\) and cycles of length \(h\geq 2\) only with constant complex weight \(c=a+ib\). For each \(D\in D_{n,h}\), the real energy of \(D\) is related to the real energy of the unweighted cycle of length \(h\) and in some special cases real energy can be compared using quasiorder relations on coefficients of the characteristic polynomial. Finally, they obtained upper bounds on the real energy which generalize those known for unweighted digraphs and signed digraphs.
The article has many avenues and interesting results. It is useful to researchers working on graphs and matrices.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C22 Signed and weighted graphs
05C76 Graph operations (line graphs, products, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
05C20 Directed graphs (digraphs), tournaments
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References:

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