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.

Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move on \(G\) consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of \(G\), denoted \(f(G)\), is the smallest integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex via pebbling moves. In this paper, we calculate the \(t\)-pebbling number of the graph \(D_{n,C_{2m}}\). Furthermore, we verify the \(q\)-\(t\)-pebbling number to demonstrate that \(D_{n,C_{2m}}\) possesses the \(2t\)-pebbling property.

Most. of pooling designs are always constructed by the “containment matrix”. But we are interested in considering non-containment
relationship. In [J. Guo, K. Wang, Pooling designs with surprisingly high degree of error correction in a finite vector space, Discrete Appl Math], Guo and Wang gave a construction by the use of non-containment relationship. In this paper, we generalize Guo-Wang’s designs and obtain a new family of pooling designs. Our designs and Guo-Wang’s designs have the same numbers of items and pools,but the error-tolerance property of our designs is better than that of Guo-Wang’s designs.

A \(k\)-edge labeling of a graph \(G\) is a function \(f: E(G) \to \{0, \ldots, k-1\}\). Such a labeling induces a labeling on the vertex set \(V(G)\) by defining \(f(v) := \sum f(e) \pmod{k}\), where the summation is taken over all edges \(e\) incident on \(v\). For an edge labeling \(f\), let \(v_f(i)\) (resp., \(e_f(i)\)) denote the number of vertices (resp., edges) receiving the label \(i\). A graph \(G\) is said to be \(E_k\)-cordial if there exists a \(k\)-edge labeling \(f\) of \(G\)such that \(|v_f(i) – v_f(j)| \leq 1\) and \(|e_f(i) – e_f(j)| \leq 1\) for all \(0 \leq i, j \leq k-1\). A wheel \(W_n\) is the join of the cycle \(C_n\) on \(n\) vertices and \(K_1\). A Helm \(H_n\) is obtained by attaching a pendent edge to each vertex of the cycle of the wheel \(W_n\). We prove that (i) Helms, (ii) one-point unions of helms, and (iii) path unions of helms are \(E_3\)-cordial.

In this paper, we prove that the graphs \(P_n\) (\(n \geq 3\)), \(C_n\) (\(n \geq 3\), \(n \not\equiv 4 \pmod{8}\)), and \(K_n\) (\(n \geq 3\)) are \(E_4\)-cordial graphs. Additionally, we show that every graph of \(\geq 3\) is a subgraph of an \(E_4\)-cordial graph.