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

Inverses of non-bipartite unicyclic graphs with a unique perfect matching