Graph Invertibility
Abstract
Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs. We examine the question of whether there exists a "litmus subgraph" whose bipartiteness determines invertibility. As an application of our invertibility criteria, we quickly describe all invertible unicyclic graphs. Finally, we describe a general combinatorial procedure for iteratively constructing invertible graphs, giving rise to large new families of such graphs.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2011
- DOI:
- 10.48550/arXiv.1108.3588
- arXiv:
- arXiv:1108.3588
- Bibcode:
- 2011arXiv1108.3588M
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 18 pages