Inverse of the distance matrix of a block graph

Abstract

A connected graph G, whose 2-connected blocks are all cliques (of possibly varying sizes) is called a block graph. Let D be its distance matrix. By a theorem of Graham, Hoffman and Hosoya, we have det(D) ≠ 0. We give a formula for both the determinant and the inverse, D ⁻¹ of D.

Keywords:

• distance matrix
• determinant
• Laplacian matrix

AMS Subject Classifications::

• 15A09
• 15A15
• 15A24

Acknowledgements

Some results in this work were in their conjecture form, tested using the computer package ‘Sage’. We thank the authors for generously releasing their software as an open-source package. We sincerely thank the referees for bringing several inaccuracies to our notice, resulting in a considerable improvement in the presentation. R.B. Bapat gratefully acknowledges the support of the JC Bose Fellowship, Department of Science and Technology, Government of India. S. Sivasubramanian thanks Professor Murali K. Srinivasan for his support in making a trip to ISI Delhi possible and the Stat-Math unit of ISI Delhi for their hospitality.