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We study the eigenvalues of the unique connected anti-regular graph A n . Using Chebyshev polynomials of the second kind, we obtain a trigonometric equation whose roots are the eigenvalues and perform elementary analysis to obtain an almost complete characterization of the eigenvalues. In particular, we show that the interval Ω = [ − 1 − 2 2 , − 1 + 2 2 ] contains only the trivial eigenvalues λ = − 1 or λ = 0 , and any closed interval strictly larger than Ω will contain eigenvalues of A n for all n sufficiently large. We also obtain bounds for the maximum and minimum eigenvalues, and for all other eigenvalues we obtain interval bounds that improve as n increases. Moreover, our approach reveals a more complete picture of the bipartite character of the eigenvalues of A n , namely, as n increases the eigenvalues are (approximately) symmetric about the number − 1 2 . We also obtain an asymptotic distribution of the eigenvalues as n → ∞ . Finally, the relationship between the eigenvalues of A n and the eigenvalues of a general threshold graph is discussed. [ABSTRACT FROM AUTHOR]

We consider only simple graphs. A graph G is said to be nonsingular if its adjacency matrix A ( G ) is nonsingular. The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G ) via a diagonal matrix of ±1s. An invertible graph G is said to be a self-inverse graph if G is isomorphic to its inverse. Let H be the class of connected bipartite graphs with unique perfect matchings. We consider the problem of characterizing the graphs in H which are self-inverse graphs. In this article, we answer this problem for the class of unicyclic graphs. The study of the strong reciprocal eigenvalue property is closely related to the study of graphs which are isomorphic to their inverse graphs. A nonsingular graph G is said to satisfy the property (SR) if the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) and they both have the same multiplicities. In this article, we show that if a unicyclic graph G in H has property (SR), then G is invertible and the inverse graph of G is unicyclic. As an application, we show that a noncorona unicyclic graph in H with property (SR) can have one of the five specified structures. Finally, our discussions lead to the following problem. Does there exist a unicyclic graph G ∈ H which has property (SR) but G is not self-inverse? [ABSTRACT FROM AUTHOR]

We consider only simple graphs. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. On the class H g of such graphs G the equivalence of the following statements are known. i) The reciprocal of the spectral radius of the adjacency matrix A ( G ) is the least positive eigenvalue of the adjacency matrix. ii) The graph is isomorphic to its inverse, where the inverse of a graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G ) via a diagonal matrix of ±1s. iii) The graph has the reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) . iv) The graph has the strong reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) and they both have the same multiplicities. v) The graph is a corona graph, that is, it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex. It is known that the statements iii) and iv) are not equivalent, if we allow nonbipartite graphs. However, the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings, where we relax the condition of bipartiteness of the contracted graph. In this article, we establish the equivalence of the first four statements for a class larger than H g . We then supply a constructive structural description of those graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let α 0 ( G ) be the smallest α for which A α ( G ) is positive semidefinite. It is known that α 0 ( G ) ≤ 1 / 2 . The main results of this paper are: (1) if G is d -regular then α 0 = − λ min ( A ( G ) ) d − λ min ( A ( G ) ) , where λ min ( A ( G ) ) is the smallest eigenvalue of A ( G ) ; (2) G contains a bipartite component if and only if α 0 ( G ) = 1 / 2 ; (3) if G is r -colorable, then α 0 ( G ) ≥ 1 / r . [ABSTRACT FROM AUTHOR]

In this paper the unique graph with maximal index (i.e. the largest eigenvalue of the adjacency matrix) is identified among all connected bipartite graphs of order n and size n + k , under the assumption that k ≥ 0 and n ≥ k + 5 . [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetković and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ , we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ . We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ = 0 , ± 1 . [ABSTRACT FROM AUTHOR]

For a fixed real number r we give several necessary and/or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix, or signless Laplacian matrix, less then or equal to r . [ABSTRACT FROM AUTHOR]

Assuming a uniform random model of selecting creation sequences, we show that almost every connected threshold graph has more negative than positive eigenvalues. We show that no threshold graphs have eigenvalues in ( − 1 , 0 ) . Sequences of equienergetic graphs are given including the striking result that for all n ≥ 3 , there exist n − 1 threshold graphs of order n 2 , pairwise noncospectral, each equienergetic to K n 2 . For all n > 8 , we find an n -vertex hyperenergetic threshold graph. [ABSTRACT FROM AUTHOR]

Abstract: Consider a graph on vertices with adjacency matrix and degree sequence . A universal adjacency matrix of is any matrix in Span with a nonzero coefficient for , where and and are the identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur. [Copyright &y& Elsevier]