Almost self-inverse graphs

ABSTRACT

Let G be a simple graph with adjacency matrix $A\left(G\right)$. The inverse of the graph G, when it exists, is the unique weighted graph whose adjacency matrix is similar to $A(G{)}^{-1}$ via a particular type of similarity. An invertible graph G is said to be an almost self-inverse graph if the inverse graph of G can be obtained from G just by joining two nonadjacent vertices. We present a characterization of almost self-inverse graphs of a specific class of graphs which contains the class of nonsingular trees.

KEYWORDS:

• Almost self-inverse graph
• self-inverse graph
• boxminus corona
• corona
• even type edge
• inverse graph
• minimal path
• odd type edge
• unique perfect matching

AMS CLASSIFICATIONS:

• 05C50
• 15A09

Acknowledgments

The authors sincerely thank the referees and the editors for their valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the first author was supported by the JC Bose Fellowship, Department of Science and Technology, Government of India and the second author research was supported by the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation [grant nos. 11561141001, 2219/15], the National Natural Science Foundation of China [grant no.11531001])