An injective coloring of a graph \(G\) is an assignment of colors to the vertices of \(G\) so that any two vertices with a common neighbor receive distinct colors. A graph \(G\) is said to be injectively \(k\)-choosable if any list \(L(v)\) of size at least \(k\) for every vertex \(v\) allows an injective coloring \(\phi(v)\) such that \(\phi(v) \in L(v)\) for every \(v \in V(G)\). The least \(k\) for which \(G\) is injectively \(k\)-choosable is the injective choosability number of \(G\), denoted by \(\chi_i^l(G)\). In this paper, we obtain new sufficient conditions to ensure \(\chi_i^l(G) \leq \Delta(G) + 1\). We prove that if \(mad(G) \leq \frac{12k}{4k+3}\), then \(\chi_i^l(G) = \Delta(G) + 1\) where \(k = \Delta(G)\) and \(k \geq 4\). Typically, proofs using the discharging technique are different depending on maximum average degree \(mad(G)\) or maximum degree \(\Delta(G)\). The main objective of this paper is finding a function \(f(\Delta(G))\) such that \(\chi_i^l(G) \leq \Delta(G) + 1\) if \(mad(G) < f(\Delta(G))\), which can be applied to every \(\Delta(G)\).

The traditional parameter used as a measure of vulnerability of a network modeled by a graph with perfect nodes and edges that may fail is edge connectivity \(\lambda\). For the complete bipartite graph \(K_{p,q}\), where \(1 \leq p \leq q\), \(\lambda(K_{p,q}) = p\). In this case, failure of the network means that the surviving subgraph becomes disconnected upon the failure of individual edges. If, instead, failure of the network is defined to mean that the surviving subgraph has no component of order greater than or equal to some preassigned number \(k\), then the associated vulnerability parameter, the component order edge connectivity \(\lambda_c^{(k)}\), is the minimum number of edges required to fail so that the surviving subgraph is in a failure state. We determine the value of \(\lambda_c^{(k)}(K_{p,q})\) for arbitrary \(1 \leq p \leq q\) and \(4 \leq k \leq p+q\). As it happens, the situation is relatively simple when \(p\) is small and more involved when \(p\) is large.

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.