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A graphis nonsingular if its adjacency matrixis nonsingular. A nonsingular graphis said to have an inverseifis signature similar to a nonnegative matrix. An-graphis a bipartite graph with a unique perfect matching. We introduce the ‘even’ness of a nonmatching edge and show that under certain conditions a connected-graph has an inverse if and only ifis bipartite, whereis the set of all even nonmatching edges. This extends some known results, providing us with a larger class of graphs possessing inverses. [ABSTRACT FROM PUBLISHER]

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 1/λis an eigenvalue of G whenever  λ  is an eigenvalue of G . Further, if  λ  and have the same multiplicity, for each eigenvalue  λ  then 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. [ABSTRACT FROM AUTHOR]