Geodesic
Unicyclic graphs with bicyclic inverses
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 1133-1143
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A graph is nonsingular if its adjacency matrix $A(G)$ is nonsingular. The inverse of a nonsingular graph $G$ is a graph whose adjacency matrix is similar to $A(G)^{-1}$ via a particular type of similarity. Let $\mathcal {H}$ denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in $\mathcal {H}$ which possess unicyclic inverses. We present a characterization of unicyclic graphs in $\mathcal {H}$ which possess bicyclic inverses.
A graph is nonsingular if its adjacency matrix $A(G)$ is nonsingular. The inverse of a nonsingular graph $G$ is a graph whose adjacency matrix is similar to $A(G)^{-1}$ via a particular type of similarity. Let $\mathcal {H}$ denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in $\mathcal {H}$ which possess unicyclic inverses. We present a characterization of unicyclic graphs in $\mathcal {H}$ which possess bicyclic inverses.
DOI : 10.21136/CMJ.2017.0429-16
Classification : 05C50, 15A09
Keywords: adjacency matrix; unicyclic graph; bicyclic graph; inverse graph; perfect matching 
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Panda, Swarup Kumar. Unicyclic graphs with bicyclic inverses. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 1133-1143. doi: 10.21136/CMJ.2017.0429-16

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