Abstract: There are several graphs naturally associated with rings. The unitary Cayley graph of a ring R is the graph with vertex set R, where two elements x,y∈R are adjacent if and only if x−y is a unit of R. We show that the unitary Cayley graph CTn(F) of the ring Tn(F) of all upper-triangular matrices over a finite field F is isomorphic to a semistrong product of a complete graph and the antipodal graph of a Hamming graph. In particular, when |F|=2, the graph CTn(F) has a highly symmetric structure: it is the union of 2^{n−1} complete bipartite graphs. Moreover, we prove that the clique number and the chromatic number of C_Tn(F) are both equal to |F|, and we establish tight upper and lower bounds for the domination number of C_Tn(F).

Abstract: We model conscient entities as vertices of a Graph; and its edges, as the interaction between the them. We further introduce a two-layer multiplex network structure coupling the micro-level soul graph with a macro-level nation graph, enabling the study of how individual interactions aggregate to shape inter-nation relationships, and conversely, how geopolitical events influence individual states. The model includes concepts such as cultural entanglement, and virtue field restoration, providing a unified graph-theoretic treatment of both spiritual and geopolitical evolution. By combining deterministic evolution laws with graph Laplacian operators, the model captures the cyclical patterns of cooperation, fragmentation, and reunification across epochs. This work not only bridges ancient spiritual narratives with modern mathematical formalisms but also lays the foundation for quantitative simulations of the socio-political dynamics of humanity across Time.

Abstract: A Plithogenic Set extends the classical fuzzy, intuitionistic, and neutrosophic paradigms by assigning attribute-based membership and contradiction values to elements, and the same ideas naturally extend to graph-based structures such as the Plithogenic Graph. Neutrosophic sets, in turn, represent elements with independent degrees of truth, indeterminacy, and falsity on the unit interval, thus handling incomplete, inconsistent, and ambiguous information effectively. This paper investigates the Plithogenic Graph, the Plithogenic Vertex Graph, and the Plithogenic Edge Graph. In addition, it examines the Intuitionistic Fuzzy Vertex Graph and Edge Graph, the Neutrosophic Vertex HyperGraph and Neutrosophic Edge HyperGraph, and the Neutrosophic Vertex SuperHyperGraph and Neutrosophic Edge SuperHyperGraph.

Abstract: This paper develops a unified framework for uncertainty numbers by introducing ordered and ranking structures across six paradigms: fuzzy, neutrosophic, plithogenic, rough, granular, and functorial numbers. We define ordered variants via monotone boundary curves, chains of approximations, and diagrammatic (functorial) morphisms, and we prove that ordered neutrosophic and plithogenic numbers strictly generalize ordered fuzzy numbers. On the comparative side, we specify ranking functionals that are monotone with respect to natural dominance relations and show that ranking neutrosophic numbers recover classical rankings of fuzzy numbers, while ranking plithogenic numbers subsume both. We also formalize ordered and ranking versions of rough and granular numbers and establish retention of their native structures under projection. Finally, we introduce Ordered/Ranking Functorial Numbers, which organize all models as semiring-valued diagrams, yielding embedding and stability theorems and illustrative examples.

Abstract:

Interconnection networks, often modeled as graphs, are critical for high-performance computing systems due to their impact on performance metrics like latency and bandwidth. The dragonfly network, denoted as \( D(n,r) \), is a promising topology owing to its modularity, low diameter, and cost-effectiveness. Ensuring reliability and efficiency in these networks requires robust cycle embedding properties. The two-disjoint-cycle-cover pancyclicity ensures that the network can be partitioned into two vertex-disjoint cycles of any feasible length, which has practical implications for fault-tolerant routing and load balancing. Formally, a graph \( G \) is called two-disjoint-cycle-cover \( [a_1,a_2] \)-pancyclic if for any integer \( \ell \) satisfying \( a_1\leq \ell\leq a_2 \), there exist two vertex-disjoint cycles \( C_1 \) and \( C_2 \) in \( G \) such that \( |V(C_1)|=\ell and |V(C_2)|=|V(G)|-\ell \). While prior work has established Hamiltonicity and pancyclicity for \( D(n,r) \), the two-disjoint-cycle-cover problem remains unexplored. This paper fills this gap by proving that \( D(n,r) \) is two-disjoint-cycle-cover \( [3,\left\lfloor \frac{|V(D(n,r))|}{2} \right\rfloor] \)-pancyclic with \( n\geq 3 \) and \( r\geq 2 \), generalizing existing knowledge. Moreover, it can be obtained that \( D(n,r) \) is vertex-disjoint-cycle-cover. Our proof employs a constructive method with case analysis, ensuring the existence of such cycles.

Abstract: This article offers formulas for computing various q-binomial nested sums, give three forms of results, reveals the three forms of q-binomial and their interrelationships. It is a powerful tool for q-analysis, which can prove and generalize many classic conclusions in a simple way. This article also utilized it to obtain a large number of new results, including formulas for q-Eulerian numbers and polynomials. By taking the limit of q to 1, it can calculate general nested sums and analyze binomial coefficients.

Abstract: We introduce Combinatorial-Topological Entropy (CTE), a structural measure quantifying the intrinsic complexity of combinatorial topologies, including simplicial complexes and hypergraphs. Unlike classical entropy, CTE does not depend on probability distributions but instead uses simplex dimensions, adjacency hierarchies, and connectivity patterns. We formalize a CTE incorporating parameters α and β to weight simplex size and adjacency influence. Using illustrative examples, including tetrahedra, hypergraphs, and higher-dimensional simplicial complexes, we demonstrate the measure’s sensitivity to structural features. Our results show CTE distinguishes between different combinatorial configurations, supporting its role as a structural invariant. Heatmaps visualize trends across α and β, demonstrating adjacency and size effects.

Abstract: The P versus NP problem, a conjecture formulated by Stephen Cook in 1971, is one of the deepest and most challenging problems in contemporary mathematics and theoretical computer science. A concise mathematical formulation of the problem reads: is P = NP?. In longer phrasing, this asks: given a problem instance, if some additional data can be recognized fast enough as logically implying the existence of a solution (to the instance), then can a solution be computed fast enough without the aid of any such additional data? In this article explain why P does not equal NP using a problem centered on bifurcating power set

Abstract: A finite hypergraph generalizes the classical graph model by allowing hyperedges that can connect any nonempty subset of vertices. Building on this foundation, a finite SuperHyperGraph is obtained through iterative application of the powerset construction, thereby creating nested families of vertex and edge sets that capture multi-layered relationships. Graph labeling assigns numbers or symbols to vertices and/or edges of a graph under rules, modeling constraints, optimization, or communication. In this paper, we define and study the mathematical properties of Graph Labeling, HyperGraph Labeling, SuperHyperGraph Labeling, Graph MultiLabeling, HyperGraph MultiLabeling, and SuperHyperGraph MultiLabeling.

Abstract: Utilizing several methods, this note shows that, in any collaboration network analysis, paper exclusion not only creates a loss of information, but can lead to incorrect interpretation of network structure because interpretation of vertex degree in the authors only graph is not well defined. Because the bipartite authors with papers graph is the actual social network, metric dimension is used to show that the relative distance structure of the bipartite graph is often defined by the structure of the papers, not that of the authors. Due to the NP-hard nature of metric dimension, methods that increase computational efficiency for the bipartite authors and papers graph are explored. With a departmental collaboration focus, public data for 245 professors from mathematics, physics and biology departments of three U.S. public universities is analyzed with network structure compared using metric dimension. By discipline, an average graph is defined with average graphs constructed from the collected data for the authors only structure, for the bipartite authors with papers structure and for papers only. Social analysis of the collected data shows that a 27\% change in the total number of hubs, along with identifying different professors as hubs, when the authors only graphs are compared to bipartite graphs reiterating the need for paper inclusion in any collaboration study.

Abstract: Graph theory provides a rigorous foundation for representing relationships and connectivity through vertices and edges. Hypergraphs extend this framework by introducing hyperedges that connect more than two vertices. Superhypergraphs further enhance the model via iterated powerset constructions, capturing hierarchical and self-referential structures among hyperedges. A spanning tree is a connected, acyclic subgraph that covers all vertices of a graph with exactly |V| − 1 edges. A spanning hypertree is a connected, Berge-acyclic subhypergraph of a uniform hypergraph that spans all vertices with hypertree structure. In this paper, we study the notion of a spanning superhypertree as the natural spanning tree concept within superhypergraphs. We also discuss several concrete realworld examples of spanning superhypertrees and analyze their structural properties.

Abstract: A Hyperstructure is built on the concept of the powerset, offering a framework to model interactions among elements of a set. Extending this idea, a Superhyperstructure utilizes the n-th powerset to represent hierarchical systems with multiple layers, enabling richer abstractions and more complex relationships. In this paper, we investigate Multiary Hyperstructures and Multiary Superhyperstructures, which generalize these constructions. A multiary hyperstructure extends algebraic systems, allowing operations with multiple inputs producing set-valued outputs, thereby modeling uncertainty and multi-participant interactions. A multiary superhyperstructure further lifts multiary hyperstructures onto higher powerset levels, encoding hierarchical, layered relationships with generalized superhyper-operations across domains.

Abstract:

We introduce the notion of noncommutative spherical codes (in particular, noncommutative kissing number problem). We show that the one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes, derived by Pfender extends to Hilbert C*-modules.

Abstract: Hyperstructures and their hierarchical extensions—SuperHyperStructures—offer a flexible framework for representing multi-layered and intricate systems [1,2]. This paper explores several extended variants of the classical SuperHyperStructure, including Rough, Soft, Fuzzy, and Functorial SuperHyper-Structures. Some of these investigations revisit earlier concepts, while others broaden the theoretical scope. The overall aim is not only to provide new insights but also to promote the study and dissemination of SuperHyperStructures and their related families within the broader mathematical community.

Abstract: In this work, a Structure is interpreted broadly as a mathematical system that may originate in Set Theory, Logic, Social Science, Business Management, Probability and Statistics, Algebra, Geometry, and related areas. A MetaStructure is conceived as a higher-level framework in which collections of mathematical structures are treated as single objects governed by uniform meta-operations. An Iterated MetaStructure is obtained by repeatedly applying the MetaStructure construction, thereby generating successive layers in which “structures of structures” form a hierarchical tower. This paper investigates whether concepts such as Ontology, Computing, Puzzle, Logic, Ethics, Data, and Governance can be systematically extended within the framework of MetaStructures and Iterated MetaStructures.

Abstract: This paper introduces ARCI′ (Adjusted Rural Connectivity Index), a novel graph-theoretic metric designed to evaluate and optimize sparse networks by balancing construction cost and population coverage. Traditional minimum spanning tree (MST) approaches focus solely on minimizing cost, often neglecting social factors such as population distribution. ARCI′ incorporates a penalty factor to weigh network cost against population served, allowing flexible prioritization. We analyze ARCI′ mathematically, study its sensitivity to the penalty parameter, and demonstrate its utility through computational experiments on synthetic network data. Our results highlight ARCI′ as a promising metric for multi-objective network design in rural infrastructure, sensor placement, and related fields.

Abstract: We propose a dynamic extension of the Petford-Welsh coloring algorithm that estimates the chromatic number of a graph without requiring k as an input. The basic algorithm is based on the model that is closely related to the Boltzmann machines that minimize the Ising model Hamiltonian. The method begins with a minimal coloring and adaptively adjusts the number of colors based on solution quality. We evaluate our approach on a variety of graphs from the DIMACS benchmark suite using different initialization strategies. The results show that the algorithm designed is not only capable of providing near optimal solutions but also is very robust. We demonstrate that our approach can be surprisingly effective on real-world instances, although more adaptive or problem-specific strategies may be needed for harder cases. The main advantage of the proposed randomized algorithm is its inherent parallelism that may be explored in future studies.

Abstract: The P versus NP problem, a conjecture formulated by Stephen Cook in 1971, is one of the most challenging problems in contemporary mathematics and theoretical computer science. A concise mathematical formulation of the problem reads: is P = NP? In longer phrasing, this asks: given a problem instance, if some additional data can be recognized fast enough as logically implying the existence of a solution (to the instance), then can a solution be computed fast enough without the aid of any such additional data? In this article we present the idea of graph coteries that are sets whose members are sets with speci ed graph-theoretic properties. Then we formulate three problems in graph coteries to show P does not equal NP.

Abstract: We investigate binary sequences generated by non-Markovian rules with memory length $\mu$, similar to those adopted in Elementary Cellular Automata. This generation procedure is equivalente to a shift register and certain rules produce sequences with maximal periods, known as de Bruijn sequences. We introduce a novel methodology for generating de Bruijn sequences that combines: (i) a set of derived properties that significantly reduce the space of feasible generating rules, and (ii) a neural network-based classifier that identifies which rules produce de Bruijn sequences. Experiments for large values of $\mu$ demonstrate the approach’s effectiveness and computational efficiency.

Abstract: This study investigates the properties and applications of Eulerian and Hamiltonian graphs within complex networks, with a particular focus on their roles in transportation, communication, and biological systems. The primary objective is to develop a deeper mathematical understanding of these graph structures and their implications for optimizing network performance. Key methodologies employed include the implementation and analysis of Hierholzer’s and Fleury’s algorithms for detecting Eulerian circuits, as well as dynamic programming, backtracking, and approximation algorithms for Hamiltonian cycles. Our results reveal significant enhancements in network optimization, showcasing improvements in traversal strategies and connectivity while effectively minimizing operational costs. Furthermore, the study identifies computational challenges associated with large-scale networks and proposes heuristic methods to address these issues. The findings provide valuable insights and recommendations for designing more efficient and resilient network infrastructures, demonstrating the practical applicability of graph theory in solving real-world problems in various domains.