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From equation (3), it follows that Graph HT ht Corollary 2. Then for all i Graph HT ht I , we have i Graph HT ht . Then i Graph HT ht Proof. [Extracted from the article]

The class of unicyclic non-bipartite graphs with unique perfect matching, denoted by U , is considered in this article. This article provides a complete characterization of unicyclic graphs in U which possess tricyclic inverses. [ABSTRACT FROM AUTHOR]

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight ±i and with a unique perfect matching, denoted by U g , is considered in this article. We characterize the 3-coloured digraphs in U g whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in U g whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in U g are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in U g possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight ±i satisfying the strong reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article describes the structure of the graph minimizing the algebraic connectivity among all connected graphs made with some given blocks with fixed number of pendant blocks, the blocks that have exactly one point of articulation. As an application, we conclude that over all graphs made with given blocks, the algebraic connectivity is minimum for a graph whose block structure is a path. [ABSTRACT FROM AUTHOR]