New formulas for the n moment μn(a, b, c, d; q) of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured. Résumé. Nous présentons de nouvelles formules pour les n-moments μn(a, b, c, d...

Despite numerous efforts extolling the virtues of formal modeling (Epstein 2008; Schank 2001; Smith & Conrey 2007; Marewksi & Olsson 2009; Farrell & Lewandowsky 2010; Weinhardt & Vancouver 2012; Smaldino et al. 2015), there remains widespread resistance among social and behavioral scientists to adopt formal modeling in their general research approach. In addition to the technical challenge pose...

Motivated by Gauss’s first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a...

We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as (3, 3)hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite (3, 3)-hypergraphs. The proof tech...

The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let Gn,m be the set of connected graphs of order n and with m edges. In this note we determined the extremal graphs from Gn,m with n ≤ m ≤ 2n−4 minimizing the matching energy. Also we determined the minimal matching energy of graphs from Gn,m where m = n−1+t and 1 ≤ t ≤ β−1 and w...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average...

Let $S(G)$ be the Seidel matrix of a graph $G$ of order $n$ and let $D_S(G)=diag(n-1-2d_1, n-1-2d_2,ldots, n-1-2d_n)$ be the diagonal matrix with $d_i$ denoting the degree of a vertex $v_i$ in $G$. The Seidel Laplacian matrix of $G$ is defined as $SL(G)=D_S(G)-S(G)$ and the Seidel signless Laplacian matrix as $SL^+(G)=D_S(G)+S(G)$. The Seidel signless Laplacian energy $E_{SL^+...

Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this article, we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove in this framework that every recursive definition of a graph po...

The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph, which is proposed first by Gutman and Wagner [The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177–2187]. And they gave some properties and asymptotic results of the matching energy. In this paper, we characterize the trees with n vertices whose complements have t...