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arXiv · 1108.3588

Graph Invertibility

Cam McLeman·Erin McNicholas
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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.

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Cam McLeman, Erin McNicholas. 2011-08-17. Graph Invertibility. https://arxiv.org/abs/1108.3588

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