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Let G be a simple graph with the adjacency matrix $ A(G) $ A (G). Let $ \tau (G) $ τ (G) denote the smallest positive eigenvalue of $ A(G) $ A (G). In 1990, Pavlíková and Kr $ \breve{c} $ c ˘ -Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let $ \mathscr {U} $ U denote the class of all connected bipartite unicyclic graphs with a unique perfect matching, and for each $ m\geq ~3 $ m ≥ 3 , let $ \mathscr {U}_n $ U n be the subclass of $ \mathscr {U} $ U with graphs on n = 2m vertices. We first obtain the classes of unicyclic graphs U in $ \mathscr {U} $ U such that $ \tau (U)\leq \sqrt {2}-1 $ τ (U) ≤ 2 − 1. We then find the unique graph $ U_o^n $ U o n (resp. $ U_e^n $ U e n ) having the maximum τ value among all graphs in $ \mathscr {U}_n $ U n when m is odd (resp. when m is even). Finally, we prove that $ U_o^6 $ U o 6 (the graph obtained from a cycle of order 4, by adding two pendants to two adjacent vertices) is the graph with maximum τ value among all graphs in $ \mathscr {U} $ U . As a consequence, we obtain a sharp upper bound for $ \tau (U) $ τ (U) when $ U\in \mathscr {U} $ U ∈ U . [ABSTRACT FROM AUTHOR]

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]

In this article, we undertake the problem of finding the first four trees on a fixed number of vertices with the maximum smallest positive eigenvalue. Let T n , d denote the class of trees on n vertices with diameter d. First, we obtain the bounds on the smallest positive eigenvalue of trees in T n , d for d = 2 , 3 , 4 and then upper bounds on the smallest positive eigenvalue of trees are obtained in general class of all trees on n vertices. Finally, the first four trees on n vertices with the maximum, second maximum, third maximum and fourth maximum smallest positive eigenvalue are characterized. [ABSTRACT FROM AUTHOR]

It was shown in 2006 that among the nonsingular trees T (whose adjacency matrix A(T) is nonsingular), the corona trees (trees that are obtained by taking any tree T and then adding a new pendant vertex at each vertex of T) are the only ones which satisfy the reciprocal eigenvalue property (λ is an eigenvalue of A(T) if and only if 1 λ is an eigenvalue of A(T), where their multiplicities are allowed to be different). A general question remained open. Can there be a tree which has at least one zero eigenvalue and whose nonzero eigenvalues satisfy the reciprocal eigenvalue property? In this note, we show that there are no such trees with at least two vertices. The proof is a beautiful application of the product of graphs. [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]