Search

Let (G, w) be a weighted graph with the adjacency matrix A. The group inverse of (G, w), denoted by (G # , w #) is the weighted graph with the weight w # (v i v j) of an edge v i v j in G # is defined as the ijth entry of A # , the group inverse of A. We study the group inverse of singular weighted trees. It is shown that if (T, w) is a singular weighted tree, then (T # , w #) is again a weighted tree if and only if (T, w) is a star tree, which in turn holds if and only if (T # , w #) is graph isomorphic to (T, w). A new class T w of weighted trees is introduced and studied here. It is shown that the group inverse of the adjacency matrix of a positively weighted tree in T w is signature similar to a non-negative matrix. [ABSTRACT FROM AUTHOR]

The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by U , is considered in this article. This article describes the entries of the inverse of the adjacency matrix of a graph in U . It is proved that the inverse graph of a graph in U is always non-bipartite. In this article, we characterize the graphs in U whose inverses are mixed graphs. Among all such graphs in U we identify those graphs whose inverses are quasi-bipartite. Furthermore, characterizations of unicyclic graphs in U possessing unicyclic and bicyclic inverses are also provided in this article. [ABSTRACT FROM AUTHOR]

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]

Abstract: An undirected unweighted graph is called nonsingular if its adjacency matrix is nonsingular. A graph is said to have property (R) if with every eigenvalue of , is also an eigenvalue. If, further, the multiplicity of and as eigenvalues of are the same, then is said to have property (SR). In a previous paper the classes of unweighted trees with property (R) and (SR) where shown to be the same and that this common class coincides with the class of corona trees, that is, trees which have been obtained from smaller trees by adding a pendent edge to each vertex. In this paper we continue the study of nonsingular trees in two ways. First we characterize the set of all graphs whose inverses are nonsingular trees. Second, we investigate the extent of property (R) for weighted trees and characterize the property for all trees with 8 vertices or more, under some conditions on the weights. [Copyright &y& Elsevier]

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]