2011/08/17 by Cam McLeman, Erin McNicholas, McLeman, Cam +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1108.3588
openalex publication_date 2011/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.