Inverses of non-bipartite unicyclic graphs with a unique perfect matching

Abstract

The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by $\mathcal{U}$, is considered in this article. This article describes the entries of the inverse of the adjacency matrix of a graph in $\mathcal{U}$. It is proved that the inverse graph of a graph in $\mathcal{U}$ is always non-bipartite. In this article, we characterize the graphs in $\mathcal{U}$ whose inverses are mixed graphs. Among all such graphs in $\mathcal{U}$ we identify those graphs whose inverses are quasi-bipartite. Furthermore, characterizations of unicyclic graphs in $\mathcal{U}$ possessing unicyclic and bicyclic inverses are also provided in this article.

Keywords:

• Adjacency matrix
• alternating path
• corona
• inverse graph
• mixed graph
• unique perfect matching

AMS CLASSIFICATIONS:

• 05C50
• 05C05
• 15A18
• 05C38
• 05C70
• 15A09

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 Science and Engineering Research Board (SERB), DST, Government of India, for providing financial support under the scheme MATRICS. This author acknowledges Council of Scientific and Industrial Research, India (CSIR), India, for providing financial assistance through JRF under grant number 09/796(0086)/2019-EMR-I.