On the inverse of unicyclic 3-coloured digraphs

Unicyclic 3-colored digraphs with bicyclic inverses

“…They characterized non-bipartite, unicyclic graphs U with unique perfect matching for which U + is unicyclic and bicyclic. The notion of inverse of a 3-colored digraph was first studied in [7]. The authors adapted the following definition of inverse in [7].…”

“…The notion of inverse of a 3-colored digraph was first studied in [7]. The authors adapted the following definition of inverse in [7].…”

“…The weighted directed graph denoted by G + is the graph on the same vertex set as that of G constructed as follows: two distinct vertices i and j are adjacent in G + if and only if α ij ̸ = 0. The graph G + is known as the inverse graph of G, see [7]. One can view A(G) −1 as the adjacency matrix A(G + ) of G + .…”

“…Furthermore, the study of the inverses of the class of unicyclic 3-colored digraphs with the cycle weight −1 is quite similar to the unweighted undirected case. Thus in [7], the authors studied the inverses of unicyclic 3-colored digraphs with the cycle weight ±i. They considered the class of such unicyclic 3-colored digraphs having a unique perfect matching, denoted by U g .…”

“…They considered the class of such unicyclic 3-colored digraphs having a unique perfect matching, denoted by U g . In [7], authors characterized the 3-colored digraphs in U g whose inverses are again 3-colored digraphs. They supplied a complete characterization of unicyclic 3-colored digraphs in U g possessing unicyclic inverses.…”