A \(T\)-shape tree \(T(l_1, l_2, l_3)\) is obtained from three paths \(P_{l_1+1}\), \(P_{l_2+1}\), and \(P_{l_3+1}\) by identifying one of their pendent vertices. A generalized \(T\)-shape tree \(T_s(l_1, l_2, l_3)\) is obtained from \(T(l_1, l_2, l_3)\) by appending two pendent vertices to exactly \(s\) pendent vertices of \(T(l_1, l_2, l_3)\), where \(1 \leq s \leq 3\) is a positive integer. In this paper, we firstly show that the generalized \(T\)-shape tree \(T_2(l_1, l_2, l_3)\) is determined by its Laplacian spectrum. Applying similar arguments for the trees \(T_1(2l_1, l_2, l_3)\) and \(T_3(l_1, 2l_2, l_3)\), one can obtain that any generalized \(T\)-shape tree on \(n\) vertices is determined by its Laplacian spectrum.

In this paper, we use the \(q\)-difference operator and the Andrews-Askey integral to give a transformation for the Al-Salam-Carlitz polynomials. As applications, we obtain an expansion of the Carlitz identity and some other identities for Al-Salam-Carlitz
polynomials .

In this paper we define new generalizations of the Lucas numbers,which also generalize the Perrin numbers. This generalization is based on the concept of \(k\)-distance Fibonacci numbers. We give in-terpretations of these numbers with respect to special decompositions and coverings, also in graphs. Moreover, we show some identities for these numbers, which often generalize known classical relations for the Lucas numbers and the Perrin numbers. We give an application of the distance Fibonacci numbers for building the Pascal’s triangle.

This paper introduces the new notions of \(\delta-\alpha-\)open sets and the \(\delta-\alpha-\)continuous functions in the topological spaces and investigates some of their properties.

Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (n_1g) \cdots (n_lg)\), where \(g \in G\) and \(n_1, \ldots, n_l \in [1, \text{ord}(g)]\), and the \({index}\) \(\text{ind}(S)\) of \(S\) is defined to be the minimum of \((n_1 + \cdots + n_l)/\text{ord}(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) when \(G = \langle g \rangle\) is a cyclic group of a prime order and \(S\) has the form \(S = g^2{(n_2g)}(n_3g){(n_4)}\). It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime order \(p \geq 31\), then every minimal zero-sum sequence \(S\) of the above-mentioned form has index \(1\), except in the case that \(S = g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)\).

The paper presents two sharp upper bounds for the largest Laplacian eigenvalue of mixed graphs in terms of the degrees and the average \(2\)-degrees, which improve and generalize the main results of Zhang and Li [Linear Algebra Appl.\(353(2002)11-20]\),Pan (Linear Algebra Appl.\(355(2002)287-295]\),respectively. Moreover, we also characterize some extreme graphs which attain these upper bounds. In last, some examples show that our bounds are improvement on some known bounds in some cases.

Cagman \(et\; al\). introduced the concept of a fuzzy parameterized fuzzy soft set(briefly, \(FPFS)\) which is an extension of a fuzzy set and a soft set. In this paper, we introduce the concepts of \(FPFS\) filters and \(FPFS\) implicative filters of lattice implication algebras and obtain some related results. Finally, we define the concept of \(FPFS\)-aggregation operator of lattice implication algebras.

We propose a practical linear time algorithm for the LONGEST PATH problem on \(2\)-trees.

By means of a \(q\)-binomial identity, we give two generalizations of Prodinger’s formula, which is equivalent to the famous Dilcher’s formula.

In this paper, we consider a random mapping \(\hat{T}_{n,\theta}\) of the finite set \(\{1,2,\ldots,n\}\) into itself, for which the digraph representation \(\hat{G}_{n,\theta}\) is constructed by: (1) selecting a random number \(\hat{L}_n\) of cyclic vertices, (2) constructing a uniform random forest of size \(n\) with the selected cyclic vertices as roots, and (3) forming `cycles’ of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter \(\theta\). We investigate \(\hat{k}_{n,\theta}\), the size of a `typical’ component of \(\hat{G}_{n,\theta}\), and we obtain the asymptotic distribution of \(\hat{k}_{n,\theta}\) conditioned on \(\hat{L}_n = m(n)\). As an application of our results, we show in Section 3 that provided \(\hat{L}_n\) is of order much larger than \(\sqrt{n}\), then the joint distribution of the normalized order statistics of the component sizes of \(G_{n,\theta}\) converges to the Poisson-Dirichlet \((\theta)\) distribution as \(n \to \infty\).