1. Inverses of weighted graphs.
- Author
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Panda, S.K. and Pati, S.
- Subjects
- *
WEIGHTED graphs , *INVERSE relationships (Mathematics) , *BIPARTITE graphs , *MATRICES (Mathematics) , *GEOMETRIC vertices - Abstract
We consider only connected bipartite graphs G with a unique perfect matching M . Let G w be the weighted graph obtained from G by giving weights to its edges using the positive weight function w : E ( G ) → ( 0 , ∞ ) such that w ( e ) = 1 for each e ∈ M . An unweighted graph G may be viewed as a weighted graph with the weight function w ≡ 1 . A weighted graph G w is nonsingular if its adjacency matrix A ( G w ) is nonsingular. The inverse of a nonsingular weighted graph G w is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G w ) via a diagonal matrix of ±1s. By G / M we denote the graph obtained from G by contracting each edge of M to a single vertex. It is known that if G / M is bipartite, then G w is invertible for each weight function w. In this article, we consider the converse of this result and show that if G w is invertible for each w, then the graph G / M is bipartite. We also supply exact conditions under which G w is invertible for one w will force that the graph G / M is bipartite. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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