Computer Science and Mathematics

Abstract: Voronov, Neopryatnaya and Dergachev constructed 5-chromatic unit-distance graphs on the sphere \( S^2(r_1), r_1=\cos\frac{\pi}{10} \), with \( 372 \) vertices, and on \( S^2(r_2) \), \( r_2=\cos\frac{3\pi}{10} \), with \( 972 \) vertices, and noted that the first graph stays 5-chromatic after deleting any single vertex. By SAT-based vertex minimization we find a 5-chromatic subgraph of the first graph with \( 231 \) vertices and \( 938 \) edges, and one of the second with \( 961 \) vertices and \( 4028 \) edges. Both are vertex-critical and contain no Moser spindle; the smaller one is the best found by our search, not a proved minimum. All coordinates are given exactly in nested radicals and verified in exact arithmetic; non-4-colourability is certified by DRAT proofs checked with DRAT-TRIM.

Abstract: We introduce a unified framework for computing nested sums involving products of q-binomial coefficients and q-integers, and organize their evaluation into explicit calculation formulas. The core of the framework is that every such sum admits three equivalent expansions in q-binomial coefficients, linked by explicit transformations. These transformations yield inversion formulas, orthogonality relations, and closed-form evaluations for weighted infinite series. The framework recovers many classical q-binomial theorems as special cases and systematically generates new summation formulas through parameter specialization. In the classical limit q → 1, the machinery reduces to ordinary binomial sums. It gives explicit formulas for the Stirling numbers of both kinds (with parameters N and N − M) and for the Eulerian numbers. In the modular setting, the same formalism provides a unified treatment of a broad family of congruences for nested sums modulo primes.

Abstract: Bradač, Gishboliner and Sudakov asked whether \(R(H, K_{n,n}) = O(n^2)\) for every \(2\)-degenerate graph \(H\). The same bound is predicted by the question of Erdős, Faudree, Rousseau and Schelp on Ramsey size-linear graphs (Erdős Problem \#566) for graphs in which every \(j\)-vertex subgraph spans at most \(2j-3\) edges. We prove \(R(\widetilde{W}_k, K_{n,n}) \le (8k+16)n^2\) for the wheel \(W_k\) minus a spoke, for every \(k \ge 4\), and \(R(J, K_{n,n}) \le 20n^2\) for a graph \(J\) on eight vertices. For every bipartite graph \(F\) we prove \(R(\widetilde{W}_5, F) = O(e(F)\sqrt{\log e(F)})\), and we also obtain \(R(\widetilde{W}_5, K_n) = O(n^{5/2}/\log^{3/2} n)\). If \(\widetilde{W}_5\) is Ramsey size-linear, then \(W_5\) is an explicit minimally non-Ramsey size-linear graph. The proofs combine minimum-degree cores of graphs whose complement contains no \(K_{n,n}\), double counting of pairs with few common neighbours, and the fact that in a \(\widetilde{W}_5\)-free graph the edges lying in at least four triangles form a \(C_4\)-free graph inside every neighbourhood. We also introduce construction sequences, which give \(R(G, K_{n,n}) = O(n^2)\) for every graph admitting one. A computer search shows that \(38\) of the \(49\) minimal open cases of Problem \#566 on at most eight vertices admit such a sequence.

Abstract: The concept of tangent spaces that is utilized in physics to analyze geometric structure and motion in a smooth manifold are applied to finite simple connected graphs. This note develops the incidence structure of a graph that represents the tangent structure of a graph, develops vertex incidence spaces that are vector spaces at each vertex consisting of vectors tangent to that vertex, and reveals the connection operator of a graph. From the graph tangent structure perspective, an example of communication and information flow in a collaboration network that utilizes parallel transport is discussed along with a discussion of additional graph dimensions. Several concepts are left to this note's readers for further development such as possible tangent structure polynomials based on vertex degree and a possible application to knot theory. As with any new perspective, the full realm of applications is in the future.

Abstract: Enterprise knowledge is commonly fragmented across operational systems, process models, and documents, limiting its use for traceable automated reasoning. This paper introduces the Business Graph, a formal model that represents an organization as a typed, temporal, and evidence-aware graph. Its meta-ontology and relation algebra establish consistent semantics for querying and inference, while temporal, epistemic, and provenance information makes results traceable to their sources and derivations. The proposed architecture separates language understanding and presentation from symbolic execution through explicit contracts, BGQL, and lineage-preserving reasoning. A Python reference implementation validates graph and contract semantics, executes queries over in-memory and persistent snapshots, and produces evidence-bound results with immutable execution identity. The concept and its implementation are examined through synthetic delayed-order and payment-service scenarios that exercise explanation, dependency traversal, missing-knowledge handling, and grounded result presentation. Automated tests verify the implemented semantics, while provider-neutral rendering and retrieval protocols are exercised with interchangeable external language and retrieval models rather than tied to a particular model family. The experiments show that contract validation prevents unsupported references and that adding local graph structure improves retrieval coverage, while also revealing that grounded generation may remain incomplete and that relevant retrieval does not by itself establish answer sufficiency. These results support the Business Graph as a computable foundation for auditable AI-assisted business reasoning, while larger independent evaluations remain future work.

Abstract: Fuzzy sets, introduced by Zadeh, and their extensions—including intuitionistic fuzzy sets and neutro-sophic sets—provide flexible mathematical frameworks for representing graded, hesitant, incomplete, and even mutually inconsistent information. These uncertainty models have been extended to network structures through fuzzy graphs, intuitionistic fuzzy graphs, and neutrosophic graphs. A common feature of such models is that uncertainty is typically assigned to vertices and/or edges, whereas the orientation of an edge is usually treated as fixed. This raises a natural question: can uncertainty be associated not only with vertices or edges, but also with the orientation itself ? More specifically, can one represent an interaction whose direction is not completely determined—for example, an edge that may be oriented from u to v to one degree, from v to u to another degree, or may instead be regarded as undirected or bidirected under uncertainty? To address this type of orientation un-certainty, the concepts of the Fuzzy-Oriented Mixed Graph and the Neutrosophic-Oriented Mixed Graph have recently been introduced. In this paper, we extend these orientation-uncertainty frameworks from ordinary graphs to higher-order network structures. Specifically, we introduce the FuzzDi-rected HyperGraph, the Single-Valued NeutroDirected HyperGraph, the FuzzDirected n-SuperHyperGraph, and the Single-Valued NeutroDirected n-SuperHyperGraph. These constructions provide mathematical frameworks for representing fuzzy and neutrosophic uncertainty in the orientations of hyperedges and superhyperedges, thereby extending orientation-uncertain graph modeling to hypergraph and SuperHyperGraph settings.

Abstract: A graph \(G\) is Ramsey size-linear if \(R(G,H)=O(e(H))\) for every graph \(H\) without isolated vertices. Erdős, Faudree, Rousseau and Schelp asked whether every graph in which each \(k\)-vertex subgraph spans at most \(2k-3\) edges is Ramsey size-linear (Erdős Problem #566). We prove that every graph with no \(K_4\)-minor is Ramsey size-linear, with \(R(G,H)\leq 624\,v(G)\,e(H)\). This covers all \(2\)-trees, which attain the density \(2k-3\), and extends results of Erdős et al. for graphs \(K_1+T\) with \(T\) a tree and for connected graphs with at most \(v(G)+1\) edges. The proof combines the triangle-elimination process and an averaging argument of Bradač, Gishboliner and Sudakov. As consequences, a connected graph with at most \(v(G)+2\) edges is Ramsey size-linear unless it contains \(K_4\) or \(K_4^*\), and each of the pairs \(\{K_4^*,W_4\}\) and \(\{W_5^-,W_5\}\) contains a minimally non-Ramsey size-linear graph. We also make explicit further structures produced by the proof of Bradač, Gishboliner and Sudakov and, by a computer search over graphs on at most eight vertices, reduce the open cases of the question to \(49\) minimal graphs governed by eleven cores.

Abstract: Let \( \mathcal{D}_n^{(r)} \)be the set of \( r \)-fans of Dyck paths of semilength \( n \). We prove that promotion on this set exhibits cyclic sieving with the \( q \)-multi-Catalan polynomial \( g_{n,r}(q)=\prod_{1\le i\le j\le n-1}\frac{[i+j+2r]_q}{[i+j]_q} \). This proves the conjecture of Pappe, Pfannerer, Schilling and Simone, equivalently the staircase case of Hopkins' rowmotion conjecture, for every size and height. The proof uses the promotion–rotation theorem for spin invariants and a classical orthogonal-group integral. A principal-specialisation calculation evaluates every rotation trace and resolves the signs at roots of unity. The same character calculation gives the energy-polynomial congruence, explicit fixed-point products, and a cyclic-module comparison with a symplectic representation. We determine the full dihedral character of promotion and evacuation and factor both reflection counts. Combining the staircase theorem with known rowmotion embeddings and transpose symmetries proves all-height cyclic sieving for shifted staircases and determines the fixed points of all rowmotion–transpose subgroups on square plane partitions. A gap model treats propellers and dihedral root posets. For chains of \( V \)'s, we prove polynomiality, all even-order identities, and several infinite families of odd-order identities. Exact polyhedral and Ehrhart computations prove the full piecewise-linear \( E_6 \) minuscule CSP, the order-six and order-nine identities for \( E_7 \), and all root identities for \( V(4) \) and \( V(8) \). The remaining odd-order identities and uniform coefficient positivity for general \( V(n) \) are stated separately.

Abstract: We study guarded recursive operators on spaces of formal languages equipped with weighted first-disagreement ultrametrics generated by arbitrary decreasing depth profiles. We show that all admissible profiles preserve the same compact complete finite-depth topology while producing substantially different quantitative contraction behavior. A uniform decay profile is introduced to measure the relative decrease of metric weights under guarded disagreement shifts. We determine the exact Lipschitz modulus of canonical guarded contexts and prove that the same decay quantity is the optimal universal Lipschitz coefficient for finite guarded wrapping operators with a prescribed minimal guard length. This leads to an exact universal Banach contractivity criterion and to a sharp guard-length threshold separating uniformly contractive and nonuniform regimes. Independently of Banach contractivity, positive guardedness is shown to guarantee a unique fixed language, global Picard convergence, and explicit finite-depth stabilization for every admissible weight profile. Representative geometric, stretched-exponential, polynomial, and block-structured profiles illustrate all possible finite and infinite decay-threshold regimes.

Abstract: We derive explicit formulas that lift vertex and edge orbits from the twin quotient of a finite simple undirected graph to the graph itself. The edge formula is the new contribution. It is driven by quotient edge orbits rather than by the vertex orbits of their endpoints, and it carries a separate family of internal clique-edge orbits. The underlying group decomposition is known for connected graphs, and we give a short proof that needs no connectedness assumption. Colouring each quotient vertex by class size and internal adjacency type gives a split surjection onto the coloured quotient group, with kernel the product of the symmetric groups on the twin classes. Every formula applies to each subgroup of that group, so partial symmetry information yields lower bounds on orbit sizes and upper bounds on orbit counts.

Abstract: Let \(G = (V, E)\) be a graph. An independent dominating set in \(G\) is a subset of \(V\) such that it is both independent and dominating.The independent domination number of \(G\) is the minimum size of an independent dominating set in \(G\).In this note, we present an upper bound for the independent domination number of a connected graph.

Abstract: Let \(G = (V, E)\) be a graph. A connected dominating set of \(G\) is a subset of \(V\) that is a dominating set and induces a connected subgraph of \(G\).The connected domination number of \(G\) is the minimum size of all connected dominating sets of \(G\).In this note, we present sufficient conditions based on the connected domination number for some Hamiltonian properties of graphs.

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? In this article we present the idea of facial digits of a positive integer and formulate a problem in recasting positive integers in terms of their facial digits to show P \( \neq \) NP.

Abstract: Prior work established Chollet’s conjecture for arbitrary complex Hermitian positive semidefinite matrices only for orders q ≤ 4. We prove Chollet’s permanent conjecture through order six: for 1 ≤ q ≤ 6 and complex Hermitian positive semidefinite q × q matrices A, B, per(A ◦ B) ≤ per(A) per(B). After reducing to the self-conjugate formulation, we encode a normalized border by two permanent polynomials whose relevant coefficients are represented by nonnegative sums of squared norms over orthogonal symmetric-tensor sectors. Evaluating the doubled polynomial along (a, b) = (t, t²) at t = √2 closes the induction, and a permutation-monomial Cauchy–Schwarz inequality transfers the result to arbitrary pairs.

Abstract: We determine the small Ramsey degree of the ordered edge in the class of finite linearly ordered split graphs with induced order-preserving embeddings: it is exactly three. For the upper bound, a chosen clique–independent-set partition turns a split graph into an ordered free-amalgamation structure. The three expansions of an ordered edge, of types QQ,QI, IQ, are homogenized successively and the partition is then forgotten. For sharpness, an ordered four-vertex path has a unique split partition and forces all three types in every host. We also prove that the corresponding degree in the full class of ordered chordal graphs is at least two, and exhibit exact failures of free amalgamation and of amalgamation after naming a perfect elimination ordering. These diagnostics explain why the same argument does not settle the chordal problem, which remains open.

Abstract: We prove that the optimal sphere packing density in \( \mathbb{R}^7 \) is \( \pi^3/105 \), achieved uniquely by the scaled root lattice \( \sqrt{2}\,E_7 \). The argument is local: it suffices to show that every Voronoi cell of any unit-ball packing in \( \mathbb{R}^7 \) has volume at least~\( 16 \), with equality characterised by the \( E_7 \) kissing configuration. The Weyl group \( W(E_7) \) of order~\( 2{,}903{,}040 \) partitions the configuration space into \( 115 \) chamber types; for each, positivity of the free-volume surplus \( F_\Omega(\theta) \)on \( (0,\pi/2) \) is certified by an exact rational \( 7\times7 \) \( LDL^\top \) decomposition (minimum pivot \( 823/3360 \)) together with a complete polynomial positivity argument covering the full angular range. The large-angle range follows from a geometric analysis of the freed-root body associated to each cap-cutting direction. The proof is unconditional and self-contained, carried out in exact rational arithmetic throughout.

Abstract: We investigate the modular total H-irregularity strength, a modular variant of the total H-irregularity strength of graphs. We establish a sufficient condition under which these two graph invariants coincide. As applications of this result, we determine the exact values of the modular total H-irregularity strength for several graph families. In each case, the obtained value attains the corresponding lower bound, proving that the bound is sharp.

Abstract: We prove that the optimal sphere packing density in \(\mathbb{R}^4\) is \(\Delta_4=\pi^2/16\), achieved uniquely by the \(D_4\) root lattice. The proof is local: every Voronoi cell of a unit-ball packing satisfies \(\mathrm{vol}(V_c)\ge 8\), with equality precisely when the active neighbours form the \(D_4\) kissing configuration at distance~$2$. The Voronoi domain decomposes into $176$ combinatorial chamber types under the Weyl group \(W(D_4)\) of order~$192$. For each type the free-volume surplus \(F_\Omega(\theta)\ge 0\) is established by two complementary certificates: \emph{Level~A}: an exact rational \(LDL^\top\) decomposition of a $6\times 6$ Hessian, with minimum pivot \(\tfrac{160}{13}\) at chamber C004, certifying the sum-of-squares identity \(F_\Omega^{(2)}(\tau)=\tau^\top H_\Omega\tau\ge 0\); and \emph{Level~B}: an explicit polynomial certificate combining a sum-of-squares identity \(H(c)=18(c-\tfrac{11}{18})^2+\tfrac{23}{18}\), a discriminant argument, and Bernstein expansions with positive rational coefficients (minimum \(\tfrac{1088}{625}\) on Piece~I; minimum \(\approx 0.765\) on Piece~II). All claims are verified independently by two computer algebra systems using exact rational arithmetic throughout. This resolves the four-dimensional sphere packing problem.

Abstract: Let \(G\) be an \(r\)-regular graph of order n, r ≥ 1, and form \(G^{(\mathbf{m})}\) by attaching \(m_v\) pendants to each core vertex \(v\). For a symmetric degree-based weight \(\varphi\), define \[ I_\varphi(H) = \sum_{uv \in E(H)} \varphi\bigl(d_H(u), d_H(v)\bigr). \] Fix the thorn multiset \(\{m_v : v \in V(G)\}\), but allow its values to be reassigned among core vertices. A multilevel cut decomposition isolates placement dependence. The cut between the core classes of degrees \(a\) and \(b\) has coefficient \[ -\tfrac12 \, \Delta\varphi(a,b), \quad \text{where } \Delta\varphi(a,b) = \varphi(a,a) - 2\varphi(a,b) + \varphi(b,b). \] For a fixed core and a finite nonempty set \(A\) of attainable degrees, every thorn multiset whose core-vertex degrees lie in \(A\) is reassignment-invariant if and only if \(G\) is complete or the restricted weight matrix is a symmetric sum matrix. If invariance is required for every connected \(r\)-regular core, where r ≥ 2, only the sum-matrix case remains. With two thorn levels, \(I_\varphi\) is affine in the induced-edge count \(e_G(S)\) of the \(k\) vertices receiving the larger number. This relation determines the value set and exact range and gives a criterion for fixed-cardinality invariance. When the coefficient is nonzero, the extremal-placement problem is NP-hard on cubic cores. For fixed \(k\), the mean and variance of the induced-edge count depend on the core only through its order and regular degree. For \(3 \le k \le n-3\), the third factorial moment determines the triangle count, and the index distribution does so as well whenever the affine coefficient is nonzero. For cycles and equal-part complete multipartite cores, the ranges have closed forms. The ordinary, reduced, and Euler Sombor indices are maximized when the heavy-vertex set induces as few core edges as possible, whereas the elliptic Sombor index is maximized when it induces as many as possible. The forgotten index is placement-invariant.

Abstract: The classical theory of q-binomial coefficients is rich with celebrated identities, yet a unified framework for evaluating nested sums involving products of such coefficients has remained elusive. This paper provides such a framework. Our main result establishes that every nested sum of this type admits three equivalent q-binomial expansions, linked by explicit transformations that reveal inversion formulas, orthogonality relations, and other deep structural connections. The framework does much more than compute: it unifies the proofs of many classical q-binomial theorems—Rothe, MacMahon, Gauss, Jacobi, Cauchy—with remarkable brevity, revealing them as special cases of a single mechanism, while simultaneously serving as a systematic identity generator through arbitrary parameter specializations that yield vast families of new summation formulas. The method is algorithmic, widely applicable, and transforms q-analysis into a field where new results can be produced systematically rather than discovered case by case.