On the inverse of unicyclic 3-coloured digraphs

Abstract

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight ±i and with a unique perfect matching, denoted by ${\mathcal{U}}_g$, is considered in this article. We characterize the 3-coloured digraphs in ${\mathcal{U}}_g$ whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in ${\mathcal{U}}_g$ whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in ${\mathcal{U}}_g$ are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in ${\mathcal{U}}_g$ possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight ±i satisfying the strong reciprocal eigenvalue property.

Keywords:

• Adjacency matrix
• corona
• inverse graph
• 3-coloured digraph
• weighted directed graph

AMS classifications:

• 05C50
• 15A18
• 15A09
• 05C20
• 05C22

Acknowledgements

The authors sincerely thank the referees and editors for carefully reading the manuscript and their suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This author acknowledges SERB, DST, Government of India, for providing financial support under the scheme MATRICS with grant number [MTR/2018/000352]. This author acknowledges CSIR, India, for providing financial assistance through JRF under grant number [09/796(0086)/2019-EMR-I].