Computer Science and Mathematics

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.

Abstract: In a finite graph, an independent set is a group of vertices that are not connected by edges. The independence, or independent number, tells how big the largest collection of vertices is where no two nodes are next to each other. When optimizing network architecture, error correcting codes, and resource allocation, the independence number a crucial metric in graph theory plays a significant role. In this article we investigate independence within structured graph families, specifically Toeplitz graphs and circulant graphs. For these graphs, solutions are computationally tractable because of periodic structure which makes them indispensable for combinatorial optimization, modeling cyclic communication protocols, and parallel grid based systems. For k = 2,3,4,5, this problem is solved for C_n < 1,k >, in 2008 (ISITAE) [33], but it remains open for bigger families of circulant graphs that are subject to NP-Hard and for Toeplitz graphs. Moreover, this research solve an open problem for Toeplitz and generalized circulant graphs that highlight an in depth study of the independence numbers that fluctuate across a broad spectrum of generating values and their precise bounds.

Abstract: This paper introduces a new novel graph parameter, called the \( \textit{De-Jaenisch number} \) of a graph. The parameter is inspired by the classical Queens problem of Carl Ferdinand de Jaenisch which in itself is inherently linked to domination theory in graphs. The De-Jaenisch number quantifies the stability of minimal dominating sets under vertex insertions along edges. We formally define the lower and upper De-Jaenisch numbers, \( \psi^-(G) \) and \( \psi^+(G) \) and establish foundational properties of these parameters. Exact values are determined for families of graphs. The results presented here lay the groundwork for further theoretical developments and applications of this parameter in domination theory and related areas.

Abstract: The Traveling Salesman Problem (TSP) is a representative NP-hard combinatorial optimization problem whose search space grows rapidly with the number of cities. Classical ant colony optimization (ACO) is attractive because of its distributed construction process and positive-feedback search, but its performance deteriorates on large instances owing to probability dilution, pheromone concentration, search stagnation, and the cost of optimizing a complete tour with a single colony. This paper proposes Collaborative Attention Ant Colony Optimization (CAACO), which integrates sparse coordinate-based graph attention, random-offset collaborative subpath optimization, a memetic MAX–MIN Ant System, and adaptive double-bridge stagnation escape. The method is evaluated against FACO, BCACO, and CCACO on nine symmetric Euclidean TSP instances, with every algorithm executed independently 30 times per instance. CAACO obtains the lowest mean GAP on all nine instances, with an average instance-level mean GAP of 3.193%, compared with 12.587%, 22.177%, and 23.826% for the three baselines. Per-instance two-sided Mann–Whitney U tests with Holm correction show significant differences in all 27 comparisons (pHolm < 10⁻¹⁰), and the Vargha–Delaney effect size is A₁₂=1.000 throughout. These results demonstrate a consistently favorable solution-quality distribution for the proposed framework, while the runtime analysis shows a quality–efficiency trade-off relative to the efficiency-oriented baseline. A 30-run ablation study further shows that removing attention, dynamic slicing, and double-bridge escape increases the average mean GAP from 3.199% to 4.525%, 5.863%, and 3.813%, respectively; all 27 module-level comparisons remain significant after Holm correction.

Abstract: In this paper, we introduce and study the positive-influence Roman dominating function (PIRDF), a novel graph-theoretic concept that unifies Roman domination with positive-influence domination. A labeling f : V(G) → {0,1,2} is a PIRDF if it satisfies the Roman domination condition and the set of positively labeled vertices forms a positive-influence dominating set, meaning every unlabeled vertex has at least half of its neighbors positively labeled. The positive-influence Roman domination number, \( \gamma_R^{PI}(G) \), is defined as the minimum weight of a PIRDF of G. We establish several tight bounds. We prove that the associated decision problem is NP-complete for general graphs, and show that it is solvable in linear time for graph classes of bounded clique-width via a LinEMSOL₁ formulation. Exact values of \( \gamma_R^{PI}(G) \) are computed for several standard graph families, including paths, cycles, complete graphs, complete bipartite graphs, star graphs, and friendship graphs. In addition, degree-based bounds are derived, and graphs attaining the lower bound \( \gamma_R^{PI}(G) = 1 + \lceil \delta/2 \rceil \) are fully characterized.

Abstract: In 1970, Dana Scott proposed his highly influential ''mathematical theory of computation'' to define the relationship between the text of a program and what the program computes (or denotes) --- the ''semantics'' of the program. Scott used a complete lattice based on the ''information ordering'', with the bottom element representing undefined --- a program failing to terminate normally, thus producing no information. The top element, however, was unused. Hence most subsequent applications of denotational semantics have used mathematical structures that avoid top elements. We suggest that the information ordering is relevant not only to semanticists, but also to working programmers, as a basis for determining if a program component or a computation is correct according to their intentions. We also suggest that a return to the use of complete lattices is called for if we wish to broaden formal semantics to allow it to encompass programmer intentions. That is because often those intentions permit more than one runtime behaviour for a given input. For example, several declarative debugging tools allow a programmer to declare that a runtime call is ''inadmissible'', loosely meaning the called program component should never be used with such input. If the input is garbage, the programmer does not care what garbage is output --- all results are acceptable, which can be modeled by the top element of a complete lattice. In this paper we explore the connections between the information ordering, correctness of computations and programs, and debugging. We present a general theory and describe several instances where the intention for what our logic/functional code computes plus what it actually computes can be described by elements in a complete lattice. This extends both the theoretical basis and practical flexibility of declarative debugging and reasoning about partial correctness and gives an attractive mathematical framework that encompasses our intentions, our programs and what they compute.

Abstract: We give a certified finite verification of the literal vertex formulation currently displayed as Erd\H{o}s Problem \#580. Namely, for every $1\le n\le 19$, an \(n\)-vertex graph having at least \(\lceil n/2\rceil\) vertices of degree at least \(\lceil n/2\rceil\) contains every tree on at most \(\lfloor n/2\rfloor\) vertices. The only order requiring new computer-assisted analysis is \(n=18\). An edge-minimal counterexample is reduced to a host partition \(V(G)=L\sqcup S\) with $|L|=|S|=9$, degree exactly $9$ on \(L\), and \(S\) independent. Exact embedding theorems cover $42$ of the $47$ non-isomorphic trees on nine vertices. The five remaining trees reduce, by deleting their leaves, to four rooted cores. For each core we construct a Boolean formula whose models are precisely the reduced hosts avoiding that rooted core. All four formulas are unsatisfiable. The deposited data include complete CNF instances and DRUP refutations. The archived traces were validated by reverse unit propagation, all four formulas are independently solved as unsatisfiable by a second SAT solver, a standalone semantic audit reconstructs the complete formulas---including all $3735$ base clauses and every avoidance clause---without importing the production generator, the complete $47$-tree witness table is generated from the classifier, and regeneration from the published encoder reproduces all four CNFs byte for byte. This is a finite partial result concerning trees on at most \(n/2\) vertices; it does not settle the stronger classical formulation asking for trees with at most \(n/2\) edges.

Abstract: Gallai’s theorem states that whenever the Euclidean plane is colored with finitely many colors, every finite configuration of points admits a monochromatic homothetic copy. This paper presents a detailed exposition of a proof originally published by Ernst Witt in 1952 and subsequently expanded by Alexander Soifer. Additional intermediate steps are supplied throughout, yielding a self-contained and mechanically verifiable proof. The argument is formulated recursively through finite configurations associated with a double induction and thereby makes explicit the finite structures underlying the theorem.

Abstract: The recently introduced Hyperbolic Sombor index HSO(G) is defined as HSO(G)=∑_uv∈E(G)√d_u²+d_v²/min{d_u,d_v}. In this paper, we investigate the mathematical properties of this novel vertex-degree-based topological index for general graphs. We first establish tight upper and lower bounds on HSO(G) in terms of the variable Euler–Sombor index EU(λ,G) for all λ ≥ −2. Furthermore, we derive new bounds connecting HSO(G) with three other well-known degree-based invariants: the second Zagreb index M₂(G), the Elliptic Sombor index ESO(G), and the Forgotten Sombor index FSO(G).

Abstract: Quantum resource estimates for the elliptic-curve discrete logarithm problem (ECDLP) now shape cryptographic migration planning, blockchain security analysis, and fault-tolerant architecture design. Recent work has moved in two complementary directions: Babbush et al. give improved secp256k1 resource estimates supported by zero-knowledge attestation while withholding sensitive circuit details, whereas Luo et al. publish an explicit reversible modular-inversion construction based on the extended Euclidean algorithm, reducing the logical-qubit footprint of prime-field ECDLP and identifying gate count, depth, and architecture-aware implementation as natural optimization targets. This note proposes a third disclosure model: verifiable resource certificates for public reversible arithmetic blocks. A certificate records a circuit commitment, gate basis, resource counts, input-output specification, deterministic test generation, correctness transcript, and optional proof artifact. We specialize the framework to modular inversion blocks |x⟩|0⟩ → |x⟩|x⁻¹ mod p⟩, for \(x\in\mathbb F_p^\times\), which are central to affine-coordinate quantum ECDLP implementations. We prove a basic soundness bound for hash-derived randomized testing and outline a prototype verifier. The goal is not a new quantum attack, but reproducible, comparable, and independently auditable quantum-ECDLP arithmetic claims.

Abstract: We give a source-scoped proof of Conjecture 19 from Written on the Wall II, as stated in the cited WOWII source materials, for finite nontrivial connected simple graphs. For such a graph G, let b(G) denote the maximum order of an induced bipartite subgraph, let ecc(v) denote the eccentricity of a vertex v, and let lambda(G) denote the largest independence number of an open neighborhood. We show that b(G) is at least the floor of the sum of lambda(G) and the average vertex eccentricity. The strict-average case uses the WOWII 13 diameter-local independence bound, for which we include both a paper proof and a formally checked Lean proof. The new mathematical ingredient for WOWII 19 is the equality-case witness, which constructs an induced bipartite subgraph from a diametral geodesic and a maximum independent neighborhood. A supplementary Lean artifact verifies both the WOWII 19 statement and the WOWII 13 bound used in the strict branch. We make no global literature-priority assertion beyond resolving the conjecture in the cited WOWII presentation.

Abstract: A continuous graph, or metric graph, is obtained from a finite graph by replacing each edge by a unit interval and using the induced shortest-path metric. Recent work of Grigoriev and Faulkner formulated continuous analogues of several classical graph-optimization problems, including independent set, vertex cover, chromatic number, and treewidth. In this paper we solve exactly the distance-two independent-set problem for continuous complete multipartite graphs. Let \[ G=K_{n_1,\ldots,n_k},\qquad N=\sum_{i=1}^k n_i,\qquad M=\max_{1\le i\le k} n_i, \] where $k\ge 2$ and \(n_i\ge 1\). We prove that \[ \alpha_1(\Gamma(G))=\max\left\{M,\floor{N/2}\right\}, \] where \(\alpha_1\) is the maximum cardinality of a set of points in \(\Gamma(G)\) whose pairwise metric distances are at least two. The formula shows that the optimum is governed by two support mechanisms: selecting vertices from a largest part, or using interior points supported on the edges of a maximum matching; choosing the midpoints gives a canonical optimal realization of the matching construction. We also give an exact constructive algorithm, EMCIA, for computing the value and producing a labelled optimal configuration. The value is computed in $O(k)$ time; a labelled optimal configuration is produced in $O(N\log k)$ time using a priority queue, or in $O(N)$ time after bucket sorting when the part sizes are bounded. Reproducible numerical illustrations show the transition between the largest-part and matching-supported regimes and include a quantitative balance-entropy diagnostic for the input part-size distribution.

Abstract: This paper introduces a ladder-based traversal framework for representing repeated symbolic structures within symbolic and numerical sequences. Repeated structures are interpreted as ordered traversal paths between hierarchical rung states, for example 1→11→111, permitting repetition to be analysed as directed traversal behaviour rather than solely as static substring occurrence. The framework defines traversal states using traversal direction, rung progression between adjacent repetition orders, traversal-gap structure, and climb behaviour. Valid traversal is restricted to transitions between neighbouring rung states only. The model additionally permits chained traversal paths spanning multiple repetition orders, overlapping embeddings in which symbolic occurrences participate in multiple traversals, and recursive overlap between rung states. Unlike conventional repetition frameworks that primarily characterise repeated substrings through periodicity or maximal interval structure, the proposed framework provides a ladder-based and geometric interpretation of symbolic repetition in which repeated structures may be analysed as interconnected traversal systems rather than purely linear strings. Potential applications include symbolic sequence fingerprinting, traversal-density analysis, overlap statistics, traversal entropy measures, and Monte-Carlo comparison of traversal behaviour within random and non-random symbolic systems.

Abstract: A graph G=(V,E) together with a positive real-valued weight-function w:V→R+ or w:E→R+ is called a weighted graph and is denoted by (G;w). In this paper, we introduce the concepts of connectivity, edge-connectivity, and restricted edge-connectivity for a weighted graph (G;w), and we prove general bounds analogous to those in the unweighted case. Furthermore, we study the connectivity and edge-connectivity of the line graph and the P2-path graph of a weighted graph, establishing upper and lower bounds for each of these parameters.