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 \(n \to \infty\).

In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, D. S. Kim and T. Kim have studied some identities of Frobenius-Euler polynomials arising from umbral calculus \((see[6])\).

Let \(H\) be a subgraph of \(G\). An \(H\)-design \((V, \mathcal{C})\) of order \(v\) and index \(\lambda\) is embedded into a \(G\)-design \((X, \mathcal{B})\) of order \(v+w\), \(w \geq 0\), and index \(\lambda\), if \(\mu \leq \lambda\), \(V \subseteq X\) and there is an injective mapping \(f: \mathcal{C} \rightarrow \mathcal{B}\) such that \(B\) is a subgraph of \(f(B)\) for every \(B \in \mathcal{C}\).
For every pair of positive integers \(v\) and \(\lambda\), we determine the minimum value of \(w\) such that there exists a balanced incomplete block design of order \(v+w\), index \(\lambda \geq 2\) and block-size \(4\) which embeds a \(K_3\)-design of order \(v\) and index \(\mu = 1\).

Let \(S\) be a finite, nonempty set of nonzero integers which contains no squares. We obtain conditions both necessary and sufficient for \(S\) to have the following property: for infinitely many primes \(p\), \(S\) is a set of quadratic nonresidues of \(p\). The conditions are expressed solely in terms of purely external (respectively, internal) combinatorial properties of the set II of all prime factors of odd multiplicity of the elements of \(S\). We also calculate by means of certain purely combinatorial parameters associated with \(\prod\) the density of the set of all primes \(p\) such that \(S\) is a set of quadratic residues of \(p\) and the density of the set of all primes \(p\) such that \(S\) is a set of quadratic nonresidues of \(p\).