Distance matrix of a multi-block graph: determinant and inverse

Abstract

A connected graph is called a multi-block graph if each of its blocks is a complete multipartite graph. We consider the distance matrix of multi-block graphs with blocks whose distance matrices have nonzero cofactor. In this case, if the distance matrix of a multi-block graph is invertible, we find the inverse as a rank one perturbation of a multiple of a Laplacian-like matrix. We also provide the inverse of the distance matrix for a class of multi-block graphs in which the distance matrix of one of its block has zero cofactor.

Keywords:

• m-partite graphs
• Laplacian-like matrix
• distance matrix
• determinant
• cofactor

MSC:

• 05C12
• 05C50

Acknowledgements

We take this opportunity to thank the anonymous reviewers for their critical reading of the manuscript and suggestions which have immensely helped us in getting the article to its present form. The authors would like to thank one of the anonymous reviewers for the constructive suggestions and inputs for the proof of Propositions 4.1 and 4.3. We also thank A.K. Lal for his helpful suggestions and comments. Some of the computations were verified using the computer package ‘Sage’. We thank the authors of ‘Sage’ for generously releasing their software as an open-source package.

Disclosure statement

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

Additional information

Funding

Sumit Mohanty would like to thank the Department of Science and Technology, Science and Engineering Research Board, India, for financial support through the projects MATRICS [MTR/2017/000458].