𝕮-inverse of graphs and mixed graphs

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Abstract This article introduces a generalization of the concept of inverse graphs applicable to both graphs and mixed graphs. Given a graph G G with adjacency matrix A ( G ) A\left(G) , the inverse graph G − 1 {G}^{-1} is defined such that its adjacency matrix is similar to the inverse of A ( G ) A\left(G) through a diagonal matrix with entries of ± 1 \pm 1 . While this diagonal matrix may or may not exist for graphs with nonsingular adjacency matrices, our study extends the concept to include mixed graphs as well. It has been proven that for certain unicyclic graphs, such a diagonal matrix does not exist. Motivated by this, we generalized the definition of inverse graphs to include mixed graphs, allowing us to find inverse mixed graphs for a class previously shown to lack one.