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We study the Laplacian energy of threshold graphs, inspired by the recent results of Vinagre, Del-Vecchio, Justo and Trevisan [22] . In particular, we compute the degree sequences of threshold graphs that maximize (or minimize) the Laplacian energy for a fixed number of vertices and edges. The analysis involves combinatorial methods using Ferrers diagrams and ideas from majorization theory. Some new inequalities for threshold degree sequences are obtained in this process. In the review process a referee pointed out that, recently and independently, Helmberg and Trevisan [14] obtained very similar results, and we discuss this connection. [ABSTRACT FROM AUTHOR]

Abstract: We present a family of polynomials with nonnegative coefficients that are symmetric and have only real nonpositive zeros. Therefore, they are also log-concave, unimodal and have nonnegative γ-vectors. This family of polynomials arises by applying the operator to the rational function . Two distinguished examples are the classical Eulerian polynomials and the Narayana polynomials. We use two special linear bases, besides the standard basis, to provide various characterizations for these polynomials and their coefficients, in terms of generalized Stirling numbers of the second kind. The type of generalized Stirling numbers we deal with, emerge from the normal ordering of . We generalize several formulas known for Eulerian and Narayana polynomials. Likewise, we provide an explicit formula to compute the so-called γ-vector of any symmetric polynomial of the type consider in this paper. Further related polynomials are obtained by applying the operator to a general Möbius transformation. Along the way, associated families of novel nonnegative integer sequences are put in evidence as a byproduct of our polynomial characterizations. We provide explicit formulas for some of these integral sequences. [Copyright &y& Elsevier]

Abstract: Let be the eigenvalues of a graph G. For any , the k-th spectral moment of G is defined by . We use the fact that is also the number of closed walks of length k in G to show that among trees T whose degree sequence is D or majorized by D, is maximized by the greedy tree with degree sequence D (constructed by assigning the highest degree in D to the root, the second-, third-, …highest degrees to the neighbors of the root, and so on) for any . Several corollaries follow, in particular a conjecture of Ilić and Stevanović on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Marković and Glišić on the Estrada index of such trees, which is defined as . [Copyright &y& Elsevier]

Abstract: A configuration of the lit-only -game on a finite graph is an assignment of one of two states, on or off, to all vertices of . Given a configuration, a move of the lit-only -game on allows the player to choose an on vertex s of and change the states of all neighbors of s. Given any integer k, we say that is k-lit if, for any configuration, the number of on vertices can be reduced to at most k by a finite sequence of moves. Assume that is a tree with a perfect matching. We show that is 1-lit and any tree obtained from by adding a new vertex on an edge of is 2-lit. [Copyright &y& Elsevier]

Abstract: We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the commutative case. We show that the creation operators on the full Fock space and the co-ordinate multipliers on the Drury–Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility. [Copyright &y& Elsevier]

Abstract: In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directly connected with the theory of quasi-definite discrete Sobolev orthogonal polynomials. It opens a lot of procedural dilemmas which we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new. [Copyright &y& Elsevier]

Abstract: We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link between the applications. [Copyright &y& Elsevier]

Abstract: We consider an infinite lower triangular matrix and a sequence called the (a,b)-sequence such that every element except lying in column 0 can be expressed aswhere a and b are integers with and . This concept generalizes the A-sequence of a Riordan matrix. As a result, we explore several structural properties of Riordan matrices by means of (a,b)-sequences. In particular, if b <0 then this leads to an extended Riordan matrix which is a bilaterally infinite matrix. [Copyright &y& Elsevier]

Abstract: Truncations of completely alternating sequences are entirely characterized. The completely hyperexpansive completion problem is solved for finite sequences of (positive) numbers in terms of positivity of attached matrices. Solutions to the problem are written explicitly for sequences of two, three, four, five and six numbers. As an application, an explicit solution of the subnormal completion problem for five numbers is given. [Copyright &y& Elsevier]