In this paper, the concepts of Pareto H -eigenvalue and Pareto Z -eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto H -eigenvalue (Pareto Z -eigenvalue). Furthermore, theminimumPareto H -eigenvalue (or Pareto Z -eigenvalue) of a symmetric ...

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n 2 ). In this paper we discuss the ...

Not long ago, Bagga, Beineke, and Varma [1] defined the super line multigraph of a simple graph Γ = (V,E) to be the graph Mr(Γ) whose vertex set is Pr(E), the class of r-element subsets of the edge set, and with an adjacency R ∼ R′ (where R,R′ ∈ Pr(E)) for every edge pair (e, f) with e ∈ R and f ∈ R′ such that e and f are adjacent in Γ. Thus, the number of edges joining R and R′ in Mr(Γ) is the...

In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least −1−τ , where τ is the golden ratio, can be described by a finite set of fat (−1 − τ)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least −1−τ is anH-line graph, where H is the set of isomorphism classes of maximal fat (−1−τ)-...

Let G be a connected graph on n vertices, and let D(G) be the distance matrix of G. Let ∂1(G) ≥ ∂2(G) ≥ · · · ≥ ∂n(G) denote the eigenvalues of D(G). In this paper, the connected graphs with ∂n−1(G) at least the smallest root of x3 − 3x2 − 11x− 6 = 0 are determined. Additionally, some non-isomorphic distance cospectral graphs are given.

Let G be a connected graph whose least eigenvalue λ(G) is minimal among the connected graphs of prescribed order and size. We show first that either G is complete or λ(G) is a simple eigenvalue. In the latter case, the sign pattern of a corresponding eigenvector determines a partition of the vertex set, and we study the structure of G in terms of this partition. We find that G is either biparti...

In a recent paper Sima, Van Huffel and Golub [Regularized total least squares based on quadratic eigenvalue problem solvers. BIT Numerical Mathematics 44, 793 812 (2004)] suggested a computational approach for solving regularized total least squares problems via a sequence of quadratic eigenvalue problems. Taking advantage of a variational characterization of real eigenvalues of nonlinear eigen...

Abstract. We consider semilinear eigenvalue problems for hemivariational inequalities at resonance. First we consider problems which are at resonance in a higher eigenvalue $\lambda_{k}$ (with $k\geq 1$ ) and prove two multiplicity theorems asserting the existence of at least $k$ pairs of nontrivial solutions. Then we consider problems which are resonant at the first eigenvalue $\lambda_{1}>0$ ...