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.
