Open Journal of Discrete Applied Mathematics (ODAM)

Graphs describe pairwise relations, while hypergraphs represent interactions involving more than two vertices. Super-HyperGraphs further permit vertices to be selected from iterated powersets, so that incidences can occur among nested objects such as teams, clusters, departments, portfolios, or control units. Many systems with such hierarchical organization are also time-dependent: their relations appear, disappear, or change activity over discrete or continuous time. Existing temporal graphs and temporal hypergraphs record temporal activation of edges or hyperedges, but they usually operate over a single-level vertex domain and therefore do not retain the identity of higher-level interacting objects. This paper develops the temporal \(n\)-Super-HyperGraph as a time-labeled higher-order structure for dynamic hierarchical connectivity. The first contribution is a precise definition based on a finite base set \(V_0\), an \(n\)-level supervertex family \(V\subseteq \mathcal{P}^n(V_0)\), a superedge family \(E\subseteq \mathcal{P}^{\ast}(V)\), a time domain \(T\), and an activity map \(\Lambda:E\to 2^T\). The second contribution is a hierarchy result showing that static \(n\)-Super-HyperGraphs, temporal hypergraphs, and temporal graphs are recovered by forgetting time or by imposing natural restrictions on \(n\) and edge cardinality. The third contribution is a collection of structural results proving that snapshots, time restrictions, temporal unions, temporal intersections, activity complements, and time shifts preserve the defining conditions of the model. The paper also presents a construction algorithm from static snapshots, proves its correctness, analyzes its complexity, and illustrates the interpretation of the model through project-management, logistics, and smart-building examples. These results give a rigorous mathematical basis for studying dynamic higher-order systems in which both temporal activation and hierarchical identity are essential.

Let \(G=(V(G),E(G))\) be a finite, simple, undirected graph. For a vertex \(v\in V(G)\), the closed neighborhood is denoted by \(N_G[v]\) and consists of \(v\) together with every vertex adjacent to \(v\). A dominator coloring of \(G\) is a proper vertex coloring in which every vertex dominates at least one color class; equivalently, for each \(v\in V(G)\) there exists a color class \(C\) such that \(C\subseteq N_G[v]\). The least number of colors required in such a coloring is the dominator chromatic number, denoted by \(\chi_d(G)\). This manuscript determines the dominator chromatic number for the modular products \(P_n\diamond P_m\) and \(C_n\diamond C_m\), where \(P_n\) is a path and \(C_n\) is a cycle. The results give closed expressions in terms of \(h=\min\{n,m\}\) and \(g=\max\{n,m\}\), including the exceptional small orders where the parity pattern of the product changes. The constructions identify the color classes that are forced by proper coloring and the additional singleton classes needed to satisfy the domination condition. Representative colorings of \(P_5\diamond P_5\) and \(C_4\diamond C_6\) illustrate how the decisive vertices in the second row control the transition from ordinary proper coloring to dominator coloring.

Closed forms are derived for nested finite sums of the form \[\sum_{a_{n-1}=c}^{a_n}\sum_{a_{n-2}=c}^{a_{n-1}}\cdots\sum_{a_0=c}^{a_1}x^{a_0},\] where \(a_n\) and \(c\) are integers and \(x\) is real or complex. This elementary identity is then used to evaluate multiple sums whose summands contain terms of the Horadam sequence \(\bigl(W_j(a,b;p,q)\bigr)\). The sequence is defined by \[W_0=a,\qquad W_1=b,\qquad W_j=pW_{j-1}-qW_{j-2}\quad(j\geq 2),\] where \(a,b,p,q\in\mathbb C\) with \(p\ne0\) and \(q\ne0\). The resulting identities include weighted sums involving Lucas sequences of the first and second kinds, Fibonacci and Lucas numbers, gibonacci sequences, and products of two and three shifted terms. The formulas show how the depth of summation is absorbed into binomial coefficients and shifted sequence indices, yielding compact expressions suitable for direct use in recurrence and summation problems.

The first degcity index \(\operatorname{DC}_{1}(G)\) of a connected graph \(G\) is the edge sum \[\operatorname{DC}_{1}(G)=\sum\limits_{uv\in E(G)}\bigl[e_G(u)+e_G(v)\bigr]\bigl[d_G(u)+d_G(v)\bigr],\] where \(d_G(u)\) and \(e_G(u)\) denote the degree and eccentricity of a vertex \(u\), respectively. The index combines local valency and global distance information in a single degree–eccentricity descriptor. This paper determines closed expressions for the first degcity index under six standard graph operations: disjoint union, join, Cartesian product, composition, symmetric difference and disjunction. The formulas separate the contributions of edges inherited from the factor graphs from the contributions created by the operation. The statements use the eccentricity behaviour in joins and the edge and degree relations in product-type operations, giving formulas that are consistent with the usual definitions of these graph operations.

Let R be a ring with identity. The nil-clean graph of R is a graph, denoted by G_NC(R), whose vertex-set is the set R, and where two distinct vertices x and y are adjacent if and only if x + y is a nil-clean element of R. An element r ∈ R is called a nil-clean element if it can be decomposed a sum of an idempotent and a nilpotent element of R. Let G be a finite undirected graph. An automorphism φ of G is a permutation on the vertex-set V(G) such that the graph preserves adjacency, that is, φ(v₁) is adjacent to φ(v₂) if and only if v₁ is adjacent to v₂. The set of all automorphisms of G together with the composition operation of permutations forms the automorphism group of G. In this paper, we firstly compute the order of the automorphism groups of nil-clean graphs for the ring ℤ_n. And then we determine the structure of the automorphism groups of G_NC(ℤ_n) for n = p^k, pq, 2^kp^l, where p, q are distinct primes and k, l are positive integers.

A subfamily of Dyck words called tight Dyck words is seen to correspond, via a “castling” procedure, to the vertex set of an ordered tree T. From T, a “blowing” operation recreates the whole family ol Dyck words. The vertices of T can be elementarily updated all along T. This simplifies an edge-supplementary arc-factorization view of Hamilton cycles of odd and middle-levels graphs found by T. Mütze et al. This takes into account that the Dyck words represent: (a) the cyclic and dihedral vertex classes of odd and middle-levels graphs, respectively, and (b) the cycles of their 2-factors, as found by T. Mütze et al.

For any real number α, the general energy of a graph is defined as the sum of the α-th powers of the nonzero singular values of its adjacency matrix. This definition unifies several classical spectral invariants, such as the graph energy and spectral moments. In this paper, we establish bounds on the general energy of graphs. These bounds, in turn, yield new estimates for the ordinary energy and spectral moments, and lead to a more general relationship between these quantities.

This paper introduces the concept of edge sum labeling in hypergraphs, where the edges of a hypergraph \(\mathcal{H}\) are assigned distinct positive integers such that the sum of the labels of all edges incident to any vertex is itself an edge label of \(\mathcal{H}\). Moreover, if the sum of the labels of any collection of edges equals the label of another edge in \(\mathcal{H}\), those edges must be incident to at least one common vertex. Additionally, we define and investigate zero edge sum hypergraphs, exploring their unique properties and presenting various results related to this new class of hypergraphs.

The basic building block of a modern block cipher is the substitution box (S-box), which provides the nonlinearity that is required when fending off advanced cryptanalytic methods. The novel approach introduced in the present work is computationally efficient, but it is still a robust algorithm for generating an 8 8 S-box through an operation of a specifically defined bijective mapping over the Galois Field \(GF(2^{8})\). The proposed S-box was strictly tested on a broad range of standard security criteria to prove its cryptographic integrity. A good performance has been identified in the analysis with a nonlinearity of 112 and linear approximation probability (LAP) of 0.0625, and the outstanding element of differential approximation probability (DAP) of 0.0156. Using a strong cryptographic construction, the new AES structure implements an S-box whose avalanche-like properties are best shown by a low value of the strict avalanche criterion (SAC) 0.4995 and strong bit independence (BIC) scores. The experimental results have supported the hypothesis that the proposed S-box has a much greater resistance to both the differential and the linear attacks compared with the state-of-the-art algebraic, heuristic, and chaos-based designs. In order to show how applicable the S-box can be in practical terms, a framework of image encryption incorporates the use of the S-box in it, whereby it operates as the basis element of a block cipher. The resulting cipher image achieved an entropy of 7.9978, which demonstrates a very high degree of randomness and strong resistance to statistical attacks. The feature article introduces a significant step towards systematizing cryptographic design through the introduction of a sound and carefully defined framework for the construction of high-security S-boxes.

Girth-regular graphs with equal girth, regular degree and chromatic index are studied for the determination of 1-factorizations with each 1-factor intersecting every girth cycle. Applications to hamiltonian decomposability and to 3-dimensional geometry are given.Applications are suggested for priority assignment and optimization problems.