Abstract: For every prime p and every integer a, the backward finite difference δp(a) := a^p − (a − 1)^p equals the cyclotomic binary form Φp(a, a − 1), where Φp(X, Y) is the homogenisation of the p-th cyclotomic polynomial, and hence equals the norm NQ(ζp)/Q(a − ζp(a − 1)). For p = 3 this specialises to the identity δ3(a) = NZ[ω](a − ω(a − 1)), where ω = e^(2πi/3), connecting the individual cubic finite difference obtained by differencing the classical sum formula of Nicomachus of Gerasa (~100 CE) with the Eisenstein norm that appears in Euler's factorisation of a^3 + b^3. We develop this identity in three directions: (a) General cyclotomic framework. For each prime p, every prime divisor q of δp(a) satisfies q ≡ 1 (mod p), imposing an arithmetic sieve whose density ~1/(p−1) grows increasingly severe with p. (b) Arithmetic density. The values {δ3(a)}a≥1 form a thin subfamily of the Löschian numbers (norms in Z[ω]), with counting function ~√(N/3) versus the Landau-Ramanujan asymptotic CN/√log N for all Löschian numbers up to N. (c) Three-language equivalence. For the cubic case we prove a precise equivalence among: (i) divisibility of δ3(a), (ii) multiplicative order modulo q, and (iii) splitting of q in Z[ω]. We also give an elementary proof of the base case 1 + b^3 = c^3 (no positive-integer solutions), and derive 3-adic constraints on any hypothetical solution to a^3 + b^3 = c^3 via the Lifting-the-Exponent Lemma, without invoking unique factorisation in Z[ω].

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 so- lution be computed fast enough? we formulate three problems in graph imbedding to show P ≠ NP.

Abstract: The construction of partial minimum spanning trees being NP-hard, several heuristic algorithms have already been formulated. Many of these heuristics (such as Kruskal's) use shortest paths to connect the components of the tree. In this work, we present an approximate construction algorithm for the minimum Steiner tree (the optimal tree for diffusion multicast). This construction is based on graph-related structures more advantageous than shortest paths. The algorithm uses connections like simple Steiner trees if necessary. These simple trees can be represented by hyperedges. A hyper metric closure can also be used.

Abstract: We show that in a vector space over Z₃, the union of any four linear bases is an additive basis, thus proving the Additive Basis Conjecture for p=3 and providing an alternative proof of the weak 3-flow conjecture.

Abstract: Assembly Theory, as developed by Cronin and co-workers, assigns to an object an assembly index: the minimal number of binary join operations required to build at least one copy of the object from a specified set of basic building blocks, allowing reuse of intermediate components. For strings over a finite alphabet, the canonical assembly index can be defined in the free semigroup (Σ+, ·) with universal binary concatenation and a “no-trash” condition, and its exact computation has been shown to be NP-complete. In this paper we propose an extension of the canonical, string-based formulation which augments pure concatenation with templated assembly steps. Intermediate objects may contain a distinguished wildcard symbol ∗ that represents a compressible block. Templates are restricted to block-compressed substrings of the target string and can be instantiated by inserting previously assembled motifs into one or many wildcard positions, possibly in parallel. This yields a new complexity measure, the templated assembly index, which strictly generalises the canonical index while preserving its operational character. We formalise the model, clarify its relation to the canonical assembly index and to classical problems such as the smallest grammar problem, and discuss the computational complexity of determining the templated assembly index. Finally, we sketch potential applications in sequence analysis, modularity detection, and biosignature design.

Abstract: This study examines the δ-complements of graphs—a specific type of graph complement whose adjacency depends on the adjacency of the vertices with identical degrees in the original graph. More specifically, we study this type of complement regarding the domination number. We provide sharp Nordhaus-Gaddum-type bounds on the domination number of a graph and its δ-complement. We also provide sharp bounds on the domination numbers of the δ-complements of joined graphs and Cartesian product graphs.

Abstract: The goal of this paper is to combine algebraic quantum field theory with methods from modal logic and the causal set program. We introduce a factorially damped past-influence operator on causal sets, formulate locally covariant nets over Alexandrov intervals, and give discrete analogues of the Haag–Kastler axioms. We close by sketching a route toward dynamics via sequential growth and incidence-algebra localization.

Abstract: A hypergraph generalizes an ordinary graph by allowing an edge to connect any nonempty subset of the vertex set. By iterating the powerset operation one step further, one obtains nested (higherorder) vertex objects and, consequently, a finite SuperHyperGraph whose vertices and edges may themselves be set-valued at multiple levels. Thus, many hierarchical graph structures exist in the literature. Moreover, not only in graph theory but also in broader fields—such as through concepts like Decision Trees and Tree Soft Sets—it is well known that tree structures are effective tools for representing hierarchical concepts. In this paper, we define a new class of graphs called Tree-Vertex Graphs. In this framework, a tree structure is imposed on the vertex set, and the edge set is defined in a manner consistent with the tree-structured vertex set. The tree structure therefore serves as a key concept for representing hierarchical graphs.

Abstract: A MetaGraph is a higher-level graph whose vertices are themselves graphs, with edges representing specified relations among these graphs. An Iterated MetaGraph extends this notion recursively: its vertices are MetaGraphs, thereby forming a hierarchical structure of graphs-of-graphs across multiple levels. In this paper, we introduce and study specific classes of MetaGraphs, including Graph-in-Tree and Cycle-in-Cycle structures. In addition, we define a new class of graphs, called Spiral Graphs, and investigate their fundamental properties.

Abstract: In this paper, we investigate the lexicographic and colexicographic orderings of m-ary vectors of length n, as well as the mirror (left-recursive) reflected Gray code, complementing the classical m-ary reflected Gray code. We present efficient algorithms for generating vectors in each of these orders, each achieving constant amortized time per vector. Additionally, we propose algorithms implementing the four fundamental functions in generating combinatorial objects—successor, predecessor, rank, and unrank—each with time complexity Θ(n). The properties and the relationships between these orderings and the set of integers {0,1,…,mn−1} are examined in detail. We define explicit transformations between the different orders and illustrate them as a digraph very close to the complete symmetric digraph. In this way, we provide a unified framework for understanding ranking, unranking, and order conversion. Our approach, based on emulating the execution of nested loops, proves to be powerful and flexible, leading to elegant and efficient algorithms that can be extended to other combinatorial generation problems. The mirror m-ary Gray code introduced here has potential applications in coding theory and related areas. By providing an alternative perspective on m-ary Gray codes, we aim to inspire further research and applications in combinatorial generation and coding.

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.