On nonsingular trees and a reciprocal eigenvalue property

Authors: Barik, S.¹; Neumann, M.²; Pati, S.¹

Source: Linear and Multilinear Algebra, Volume 54, Number 6, December 2006 , pp. 453-465(13)

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Abstract:

Let G be a simple connected graph. Let A ( G ) be the adjacency matrix of G . We give a combinatorial description of A ( G ) −1 of a bipartite graph G with a unique perfect matching. As a corollary, we obtain the combinatorial description of the inverse of a nonsingular tree. A graph is said to have property (R) if is an eigenvalue of G wheneverλis an eigenvalue of G . Further, ifλand have the same multiplicity, for each eigenvalueλthen it is said to have the property (SR). We characterize all trees with property (R) and show that it is exactly the class of all trees with property (SR). The class of trees with property (R) is also identical to the class of corona trees, namely, the trees which can be obtained from other trees by adding pendants to all vertices. Other equivalent conditions for a tree to have property (R) are also given.

Keywords: Tree; Adjacency matrix; Property (R); Property (SR); Inverse graph; Alternating path

Document Type: Research article

DOI: http://dx.doi.org/10.1080/03081080600792897

Affiliations: 1: Department of Mathematics, IIT Guwahati, PIN-781039, Guwahati, India 2: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009, USA

Publication date: 2006-12-01