γ-Inverse graph of some mixed graphs

Abstract

Let GG be a graph. Then, the inverse graph G−1{G}^{-1} of GG is defined to be a graph that has adjacency matrix similar to the inverse of the adjacency matrix of GG, where the similarity matrix is ±1\pm 1 diagonal matrix. In this article, we introduced a generalization of this definition that serves the mixed graphs where the definition applied for the α\alpha -Hermitian adjacency matrices of mixed graphs. Furthermore, for a class of unicyclic graphs, we were able to find an inverse mixed graph for a graph GG, where it was proven that G−1{G}^{-1} does not exist.

Keywords

• mixed graphs
• α-hermitian adjacency matrix
• inverse matrix
• mixed graphs
• bipartite mixed graphs
• unicyclic bipartite mixed graphs
• perfect matching
• 05c20
• 05c50
• 15a30