The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by U, is considered in this article. This article describes the entries of the inverse of the adjacency matrix of a graph in U. It is proved that the inverse graph of a graph in U is always non-bipartite. In this article, we characterize the graphs in U whose inverses are mixed graphs. Among all such graphs in U we identify those graphs whose inverses are quasi-bipartite. Furthermore, characterizations of unicyclic graphs in U possessing unicyclic and bicyclic inverses are also provided in this article.
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