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Based on the Stirling triangle of the second kind, the Whitney triangle of the second kind and one triangle of Riordan, we study a Stirling–Whitney–Riordan triangle $$[T_{n,k}]_{n,k}$$ satisfying the recurrence relation: $$\begin{aligned} T_{n,k}= & {} (b_1k+b_2)T_{n-1,k-1}+[(2\lambda b_1+a_1)k+a_2+\lambda ( b_1+b_2)] T_{n-1,k}+\\&\lambda (a_1+\lambda b_1)(k+1)T_{n-1,k+1}, \end{aligned}$$ where initial conditions $$T_{n,k}=0$$ unless $$0\le k\le n$$ and $$T_{0,0}=1$$ . We prove that the Stirling–Whitney–Riordan triangle $$[T_{n,k}]_{n,k}$$ is $$\mathbf{x} $$ -totally positive with $$\mathbf{x} =(a_1,a_2,b_1,b_2,\lambda )$$ . We show that the row-generating function $$T_n(q)$$ has only real zeros and the Turan-type polynomial $$T_{n+1}(q)T_{n-1}(q)-T^2_n(q)$$ is stable. We also present explicit formulae for $$T_{n,k}$$ and the exponential generating function of $$T_n(q)$$ and give a Jacobi continued fraction expansion for the ordinary generating function of $$T_n(q)$$ . Furthermore, we get the $$\mathbf{x} $$ -Stieltjes moment property and 3- $$\mathbf{x} $$ -log-convexity of $$T_n(q)$$ and show that the triangular convolution $$z_n=\sum _{i=0}^nT_{n,i}x_iy_{n-i}$$ preserves Stieltjes moment property of sequences. Finally, for the first column $$(T_{n,0})_{n\ge 0}$$ , we derive some properties similar to those of $$(T_n(q))_{n\ge 0}.$$

In this paper, we consider a generalized ordered Bell polynomial P n ( q ) defined by the following exponential generating function ∑ n ≥ 0 P n ( q ) n ! t n = e γ t ( β β + β ′ q − β ′ q e t β ) 1 + γ ′ β ′ . Using the method of exponential Riordan arrays and orthogonal polynomials, we give the Jacobi continued fraction expansion of ∑ n ≥ 0 P n ( q ) t n . Then we get the q-Stieltjes moment property, strong q-log-convexity, 3-q-log-convexity and Hankel determinants of { P n ( q ) } n ≥ 0 .

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

The aim of this paper is to develop analytic techniques to deal with Hankel-total positivity of sequences. We show two nonlinear operators preserving Stieltjes moment property of sequences. They actually both extend a result of Wang and Zhu that if ( a n ) n ≥ 0 (a_n)_{n\geq 0} is a Stieltjes moment sequence, then so is ( a n + 2 a n − a n + 1 2 ) n ≥ 0 (a_{n+2}a_{n}-a^2_{n+1})_{n\geq 0} . Using complete monotonicity of functions, we also prove Stieltjes moment properties of the sequences ( Γ ( n 0 + a i + 1 ) Γ ( k 0 + b i + 1 ) Γ ( ( n 0 − k 0 ) + ( a − b ) i + 1 ) ∏ j = 0 m 1 d j i + e j ) i ≥ 0 \left ( \frac {\Gamma (n_{0}+ai+1)}{{\Gamma (k_{0}+bi+1)} {\Gamma ((n_0-k_0)+(a-b)i+1)}}\prod _{j=0}^m\frac {1}{d_ji+e_j}\right )_{i\geq 0} and ( ∑ k ≥ 0 α k λ k n ) n ≥ 0 \left (\sum _{k\ge 0}\frac {\alpha _k}{\lambda _{k}^{n}}\right )_{n\geq 0} . Particularly in a new unified manner our results imply the Stieltjes moment properties of binomial coefficients ( p n + r − 1 n ) \binom {pn+r-1}{n} and Fuss-Catalan numbers r p n + r ( p n + r n ) \frac {r}{pn+r}\binom {pn+r}{n} proved by Mlotkowski, Penson, and Zyczkowski, and Liu and Pego, respectively, and also extend some results for log-convexity of sequences proved by Chen-Guo-Wang, Su-Wang, Yu, and Wang-Zhu, respectively.

Given a sequence of polynomials$\{x_k(q)\}_{k \ges 0}$, define the transformation$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$for$n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function ofyn(q) and that ofxn(q). We also prove that the transformation preservesq-TPr+1(q-TP) property of the Hankel matrix$[x_{i+j}(q)]_{i,j \ges 0}$, in particular forr= 2,3, implying ther-q-log-convexity of the sequence$\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of typesAandB, derangement polynomials typesAandB, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strongq-log-convexity of derangement polynomials typeB, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strongq-log-convexity.

In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\'ery numbers. For any nonnegative integer $n$, define \begin{gather*}f_n:=\sum_{k=0}^n\binom nk^3,\ D_n:=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}, b_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k,\ A_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2. \end{gather*} For $n=0,1,2,\ldots$, we show that $6^{-n}|f_{i+j}|_{0\leq i,j\leq n}$ and $12^{-n}|D_{i+j}|_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\leq i,j\leq n}$ and $24^{-n}|A_{i+j}|_{0\leq i,j\leq n}$ are always integers., Comment: 13 pages, final published version

Let G be a simple graph with n vertices and let C ( G ; x ) = ∑ k = 0 n ( − 1 ) n − k c ( G , k ) x k denote the Laplacian characteristic polynomial of G . Then if the size | E ( G ) | is large compared to the maximum degree Δ ( G ) , Laplacian coefficients c ( G , k ) are approximately normally distributed (by central and local limit theorems). We show that Laplacian coefficients of the paths, the cycles, the stars, the wheels and regular graphs of degree d are approximately normally distributed respectively. We also point out that Laplacian coefficients of the complete graphs and the complete bipartite graphs are approximately Poisson distributed respectively.

Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrencefor n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not only preserves the strong q-log-convexity property, but also preserves the strong q-log-concavity property. Finally, we demonstrate that the strong q-log-convexity property is preserved by the Stirling transformation and Whitney transformation of the second kind, which extends some known results for the strong q-log-convexity property.

Many combinatorial numbers can be placed in the following generalized triangular array $[T_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} T_{n,k}=\lambda(a_0n+a_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1}+\frac{d(da_1-b_1)}{\lambda}(n-k+1)T_{n-1,k-2} \end{equation*} with $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$ for suitable $a_0,a_1,a_2,b_0,b_1,b_2,d$ and $\lambda$. For $n\geq0$, denote by $T_n(q)$ the generating function of the $n$-th row. In this paper, we develop various criteria for $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ based on the Jacobi continued fraction expression of $\sum_{n\geq0}T_n(q)t^n$, where $\textbf{x}$ is a set of indeterminates consisting of $q$ and those parameters occurring in the recurrence relation. With the help of a criterion of Wang and Zhu [Adv. in Appl. Math. (2016)], we show that the corresponding linear transformation of $T_{n,k}$ preserves Stieltjes moment properties of sequences. Finally, we present some related examples including factorial numbers, Whitney numbers, Stirling permutations, minimax trees and peak statistics., Comment: Advances in Applied Mathematics, 130 (2021) 102232, 33pp

Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle and some triangles related to binomial coefficients, rook polynomials and Laguerre polynomials. We not only get total positivity of these lower-triangles, and $q$-Stieltjes moment properties and $3$-$q$-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve Stieltjes moment property. In particular, we solve a conjecture of Sokal on $q$-Stieltjes moment property of rook polynomials., Comment: It will appear in SIAM J. Discrete Math

The aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

In this paper, we study various unimodality properties of generalized Stirling numbers introduced by Hsu and Shiue. We obtain the log-convexity of generalized Bell numbers, the reality of zeros and the q -log-convexity of generalized Bell polynomials. We also show that certain linear transformations associated with generalized Stirling numbers preserve the log-convexity.

We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches., Comment: revised Theorem 2.1 (iii), 10 pages

Let { T n , k } n , k ≥ 0 \{T_{n,k}\}_{n,k\ge 0} be an array of nonnegative numbers satisfying the recurrence relation T n , k = ( a 1 k 2 + a 2 k + a 3 ) T n − 1 , k + ( b 1 k 2 + b 2 k + b 3 ) T n − 1 , k − 1 \begin{equation*} T_{n,k}=(a_1k^2+a_2k+a_3)T_{n-1,k}+(b_1k^2+b_2k+b_3)T_{n-1,k-1} \end{equation*} with T n , k = 0 T_{n,k}=0 unless 0 ≤ k ≤ n 0\le k\le n . We obtain some results for the total positivity of the matrix ( T n , k ) n , k ≥ 0 \left (T_{n,k}\right )_{n,k\ge 0} , Pólya frequency properties of the row and column generating functions, and q q -log-convexity of the row generating functions. This allows a unified treatment of the properties above for some triangular arrays of the second kind, including the Stirling triangle, Jacobi-Stirling triangle, Legendre-Stirling triangle, and central factorial numbers triangle.

In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph Gn,k, where Gn,k is obtained from the complete graph Kn-k by attaching paths of almost equal lengths to all vertices of Kn-k. We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see [A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233–240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius.

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials., Comment: 14 pages

This note investigates the least eigenvalues of connected graphs with n vertices and maximum degree �, and characterizes the unique graph whose least eigenvalue achieves the minimum among all the connected graphs with n vertices and maximum vertex degree � > n .

In this paper, we characterize the unique graph whose least eigenvalue achieves the minimum among all graphs with n vertices and domination number γ . Thus we can obtain a lower bound on the least eigenvalue of a graph in terms of the domination number.

In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices.