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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]

Let G be a graph, and let $ A(G) $ A (G) be the adjacency matrix of G. The computation of permanent of $ A(G) $ A (G) is #p-complete. Computing permanent of $ A(G) $ A (G) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph $ K_{p, p} $ K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from $ K_{p, p} $ K p , p with six edges deleted are determined by their permanental polynomials. [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]

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by Gn,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured that among all graphs in Gn,m, n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star Sn with m−n + 1 additional edges all connected to the same vertices for m ≤ n + ⌊ (n − 7) / 2 ⌋ , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy. [ABSTRACT FROM AUTHOR]

In [A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra. 2010;58(3-4):387–390], Butler constructed a family of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. In this article, we extend this construction for generating larger classes of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. Also, we provide a couple of constructions of non-bipartite graphs, which are cospectral for the adjacency matrices but not necessarily for the normalized Laplacian matrices. [ABSTRACT FROM AUTHOR]

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph G is a cograph if and only if no induced subgraph of G has an eigenvalue in the interval. It is also shown that the multiplicity of any eigenvalue of a cograph G does not exceed the sum of multiplicities of 0 and as eigenvalues of G. We introduce a partial order on the vertex set of graphs G in terms of inclusions among the open and closed neighbourhoods of vertices, and conjecture that the multiplicity of any eigenvalue of a cograph G except for does not exceed the maximum size of an antichain with respect to that partial order. In two extreme cases (in particular for threshold graphs), the conjecture is shown to be true. Finally, we prove that bipartite -free graphs have no eigenvalue in the intervals and (0, 1 / 2). [ABSTRACT FROM AUTHOR]

A graphis nonsingular if its adjacency matrixis nonsingular. A nonsingular graphis said to have an inverseifis signature similar to a nonnegative matrix. An-graphis a bipartite graph with a unique perfect matching. We introduce the ‘even’ness of a nonmatching edge and show that under certain conditions a connected-graph has an inverse if and only ifis bipartite, whereis the set of all even nonmatching edges. This extends some known results, providing us with a larger class of graphs possessing inverses. [ABSTRACT FROM PUBLISHER]

In this note we show how to construct two distinct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices by 'unfolding' a base bipartite graph in two different ways. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph. Let A(G) be the adjacency matrix of G . We give a combinatorial description of A ( G )-1 of a bipartite graph G with a unique perfect matching. As a corollary, we obtain the combinatorial description of the inverse of a nonsingular tree. A graph is said to have property (R) if 1/λis an eigenvalue of G whenever  λ  is an eigenvalue of G . Further, if  λ  and have the same multiplicity, for each eigenvalue  λ  then it is said to have the property (SR). We characterize all trees with property (R) and show that it is exactly the class of all trees with property (SR). The class of trees with property (R) is also identical to the class of corona trees, namely, the trees which can be obtained from other trees by adding pendants to all vertices. Other equivalent conditions for a tree to have property (R) are also given. [ABSTRACT FROM AUTHOR]