Employing \(q\)-commutive structures, we develop binomial analysis and combinatorial applications induced by an important operator in
analogue Fourier analysis associated with well-known \(q\)-series of L.J. Rogers.

In [H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance
pooling designs, Discrete Math. \(243 (2002) 167-170\)], by using subspaces
in a vector space Ngo and Du constructed a family of well-known pooling
designs. In this paper, we construct a family of pooling designs by using
bilinear forms on subspaces in a vector space, and show that our design and
Ngo-Du’s design have the same error-tolerance capability but our design is
more economical than Ngo-Du’s design under some conditions.

A transverse Steiner quadruple system \((TSQS)\) is a triple \((X, \mathcal{H}, \mathcal{B})\) where \(X\) is a \(v\)-element set of points, \(\mathcal{H} = \{H_1, H_2, \ldots, H_r\}\) is a partition of \(X\) into holes, and \(\mathcal{B}\) is a collection of transverse \(4\)-element subsets with respect to \(\mathcal{H}\), called blocks, such that every transverse \(3\)-element subset is in exactly one block. In this article, we study transverse Steiner quadruple systems with \(r\) holes of size \(g\) and \(1\) hole of size \(u\). Constructions based on the use of \(s\)-fans are given, including a construction for quadrupling the number of holes of size \(g\). New results on systems with \(6\) and \(11\) holes are obtained, and constructions for \(\text{TSQS}(x^n(2n)^1)\) and \(\text{TSQS}(4^n2^1)\) are provided.

Let \(G\) be a subgraph of the complete graph \(K_{r+1}\) on \(r+1\) vertices, and let \(K_{r+1} – E(G)\) be the graph obtained from \(K_{r+1}\) by deleting all edges of \(G\). A non-increasing sequence \(\pi = (d_1, d_2, \ldots, d_n)\) of nonnegative integers is said to be potentially \(K_{r+1} – E(G)\)-graphic if it is realizable by a graph on \(n\) vertices containing \(K_{r+1} – E(G)\) as a subgraph. In this paper, we give characterizations for \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{r+1} – E(G)\)-graphic for \(G = 3K_2, K_3, P_3, K_{1,3}\), and \(K_2 \cup P_2\), which are analogous to Erdős-Gallai’s characterization using a system of inequalities. These characterizations partially answer one problem due to Lai and Hu [10].

An unoriented flow in a graph is an assignment of real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. This is equivalent to a flow in a bidirected graph where all edges are extraverted. A nowhere-zero unoriented \(k\)-flow is an unoriented flow with values from the set \(\{\pm 1, \ldots, \pm( k-1)\}\). It has been conjectured that if a graph admits a nowhere-zero unoriented flow, then it also admits a nowhere-zero unoriented \(6\)-flow. We prove that this conjecture holds true for Hamiltonian graphs, with \(6\) replaced by \(12\).

Let \(G\) be a graph with vertex set \(V(G)\), \(d_G(u,v)\) and \(\delta_G(v)\) denoteas the topological distance between vertices \(u\) and \(v\) in \(G\), and \(d_G(v)\) as the degree of vertex \(v\) in \(G\),respectively. The Schultz polynomial of \(G\) is defined as \(H^+(G) = \sum\limits_{u,v \subseteq V(G)} (\delta _G(u)+\delta _G(v))x^{d_G(u,v)}\), and the modified Schultz polynomial of \(G\) is defined as \(H^*(G) = \sum\limits_{u,v \subseteq V(G)}(\delta _G(u)+\delta _G(v)) x^{d_G(u,v)}\). In this paper, we obtain explicit analytical expressions for the expected values of the Schultz polynomial and modified Schultz polynomial of a random benzenoid chain with $n$ hexagons. Furthermore, we derive expected values of some related topological indices.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\). The orientation number of \(G\), denoted by \(\vec{d}(G)\), is defined as \(\min\{d(D) \mid D \in \mathcal{D}(G)\}\), where \(d(D)\) denotes the diameter of the digraph \(D\). In this paper, we prove that \(\vec{d}(P_3 \times K_5) = 4\) and \(\vec{d}(C_8 \times K_3) = 6\), where \(\times\) is the tensor product of graphs.

In this paper, we consider the domination number, the total domination number, the restrained domination number, the total restrained domination number and the strongly connected domination number of lexicographic product digraphs.

Radio labeling is a variation of Hale’s channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph \(G\) subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph \(G\) is a function \(c: V(G) \to \mathbb{Z}_+\) such that \[d(u, v) + |c(u) – c(v)| \geq 1 + \text{diam}(G)\] for every two distinct vertices \(u\) and \(v\) of \(G\), where \(d(u, v)\) is the distance between \(u\) and \(v\). The \emph{span} of a radio labeling is the maximum integer assigned to a vertex. The \emph{radio number} of a graph \(G\) is the minimum span, taken over all radio labelings of \(G\). This paper establishes the radio number of the Cartesian product of a cycle graph with itself,( i.e., of \(C_n \Box C_n\)).

In this note we present an application of \(q\)-Lucas theorem, from which the \(q\)-binomial rational root theorem obtained by K. R. Slavin can be deduced as a special case.