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Graph Invertibility

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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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Authors and Affiliations

  1. Department of Mathematics, The University of Michigan-Flint, 303 E. Kearsley Street, Flint, MI, 48502, USA

    Cam McLeman

  2. Department of Mathematics, Willamette University, 900 State Street, Salem, OR, 97301, USA

    Erin McNicholas

Authors
  1. Cam McLeman
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  2. Erin McNicholas
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Correspondence to Cam McLeman.

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McLeman, C., McNicholas, E. Graph Invertibility. Graphs and Combinatorics 30, 977–1002 (2014). https://doi.org/10.1007/s00373-013-1319-7

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  • DOI: https://doi.org/10.1007/s00373-013-1319-7

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