On weight-symmetric 3-coloured digraphs

Abstract

We consider particular weighted directed graphs with edges having colour red, blue or green such that each red edge has weight 1, each blue edge has weight $-1$ and each green edge has weight $\mathrm{i}$ (the imaginary unit). Such a directed graph is called a 3-coloured digraph (for short, a 3-CD). Every mixed graph and every signed graph can be interpreted as a 3-CD. We first study some structural properties of 3-CDs by means of the eigenvalues of the adjacency matrix. In particular, we give spectral criteria for singularity of such digraphs. Second, we consider weight-symmetric 3-CDs, i.e. those 3-CDs that are switching isomorphic to their negation. It follows that the class of weight-symmetric 3-CDs is included in the class of 3-CDs whose spectrum is symmetric (with respect to the origin). We give some basic properties and several constructions of weight-symmetric 3-CDs and establish constructions of 3-CDs which have symmetric spectrum but are not weight-symmetric.

Keywords:

• Weighted directed graph
• 3-coloured digraph
• adjacency matrix
• weight-symmetry
• self-invertibility
• graph products

MSC:

• 05C50
• 15B57
• 05C22

Acknowledgments

We are very grateful to the anonymous referees for the evaluation of our paper and for the constructive critics.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the first author was supported by Iranian National Science Foundation (INSF) [contract no. 98021291]. The research of the second author was supported by the Serbian Ministry of Education, Science and Technological Development via the University of Belgrade, Faculty of Mathematics.