A new series of four-associate class partially balanced incomplete block designs in two replications has been proposed. The blocks of these designs are of two different sizes. The blocks can be divided into two groups such that every treatment appears in each group exactly once, and any two blocks belonging to two different groups have a constant number of treatments in common, i.e., these designs are affine resolvable.

Let \( 0<k\in\mathbb{Z} \). Let the star 2-set transposition graph \( ST^2_k \) be the \( (2k-1) \)-regular graph whose vertices are the \( 2k \)-strings on \( k \) symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such \( 2k \)-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph \( PC^2_k \) has the same vertex set of \( ST^2_k \) and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of \( ST^2_k \). For \( 1<k\in\mathbb{Z} \), we show that \( ST^2_k \) and \( PC^2_k \), among other intermediate transposition graphs, have total colorings via \( 2k-1 \) colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of \( ST^2_k \) and \( PC^2_k \), and partitions into \( 2k-1 \) such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.

This paper investigates the Turan-like problem for \(\mathcal{K}^-_{r + 1}\)-free \((r \geq 2)\) unbalanced signed graphs, where \(\mathcal{K}^-_{r + 1}\) is the set of unbalanced signed complete graphs with \(r+1\) vertices. The maximum number of edges and the maximum index for \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs are given. Moreover, the extremal \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs with the maximum index are characterized.

In this paper, we give a classification of all Mengerian \(4\)-uniform hypergraphs derived from graphs.

The \( n \)-dimensional Möbius cube \( MQ_n \) is an important variant of the hypercube \( Q_n \), which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of \( MQ_n \), and shows that if \( MQ_n \) (\( n \geq 5 \)) contains at most \( n-2 \) faulty vertices and/or edges then, for any fault-free edge \( uv \) in \( MQ_n^i (i=0,1) \) and any integer \( \ell \) with \( 7-i \leqslant \ell \leqslant 2^n – f_v \), there is a fault-free cycle of length \( \ell \) containing the edge \( uv \), where \( f_v \) is the number of faulty vertices. The result is optimal in some senses.

In a recent paper Cameron, Lakshmanan and Ajith [6] began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable hypergraphs encoding the generating properties of a finite group. In particular, answering a question asked in their paper, we classified the finite solvable groups whose generating hypergraph is the basis hypergraph of a matroid.

Let \( G \) be a graph, the zero forcing number \( Z(G) \) is the minimum of \( |Z| \) over all zero forcing sets \( Z \subseteq V(G) \). In this paper, we are interested in studying the zero forcing number of quartic circulant graphs \( C_{p}\left(s,t\right) \), where \( p \) is an odd prime. Based on the fact that \( C_{p}\left(s,t\right) \cong C_{p}\left(1,q\right) \), we give the exact values of the zero forcing number of some specific quartic circulant graphs.

Behera and Panda defined a balancing number as a number b for which the sum of the numbers from \(1\) to \(b – 1\) is equal to the sum of the numbers from \(b + 1\) to \(b + r\) for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.

The secure edge dominating set of a graph \( G \) is an edge dominating set \( F \) with the property that for each edge \( e \in E-F \), there exists \( f \in F \) adjacent to \( e \) such that \( (F-\{f\})\cup \{e\} \) is an edge dominating set. In this paper, we obtained upper bounds for edge domination and secure edge domination number for Mycielski of a tree.

In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected (\(HC\)) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.