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. In this article we show P does not equal NP using power sets.

Abstract: This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.

Abstract: Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely Frisch's identity and Klamkin's identity, appear as immediate consequences of these polynomial identities. We subsequently establish several combinatorial identities, including a generalization of each of Frisch's identity and Klamkin's identity. Finally, we develop a scheme for deriving combinatorial identities associated with polynomial identities of a certain type.

Abstract: We introduce a new axiom, the Chebiam Continuum Axiom (CCA), which provides a novel perspective on the Continuum Hypothesis (CH). By integrating computational methods with classical set theory, we develop forcing techniques that construct models where the CCA holds and demonstrate its consistency with ZFC. Our approach leverages large cardinal properties to establish a hierarchical structure on the power set of א0, revealing an intricate stratification between א0 and 2א0. This stratification suggests that the classical formulation of CH as a binary question may be inadequate. We prove that the CCA is independent of ZFC but compatible with large cardinal axioms, offering a new framework that reconciles seemingly contradictory intuitions about the continuum. Our computational simulations provide empirical support for the theoretical results, suggesting that CCA captures essential properties of the continuum that extend beyond the traditional scope of CH.

Abstract: In this paper, we initiate the study of total triple Roman domination, in which we aim to ensure that each vertex of the graph is protected by at least three units, either located on itself or its neighbors, while guaranteeing that none of its neighbors remain unprotected. Formally, a total triple Roman dominating function is a labeling f of the vertices of the graph with labels {0,1,…,4} such that f(N[v])≥|AN(v)|+3, where AN(v) denotes the set of active neighbors of vertex v, i.e., those assigned a positive label. We investigate the algorithmic complexity of the associated decision problem, establish sharp bounds regarding graph structural parameters, and obtain the exact values for several graph families.

Abstract: In this paper, we focus on the internal structural characteristics of permutation tableaux and their correspondence with linked partitions. We begin by introducing new statistics or permutation tableaux, designed to thoroughly describe various positional relationships among the topmost 1’s and the rightmost restricted 0’s. Subsequently, we develop two marking algorithms for permutation tableaux, each from the perspective of columns and rows. Additionally, we introduce tugging and rebound transformations, which elucidate the generative relationship from original partitions to linked partitions. As a result,we demonstrate that the construction of these two marking algorithms in permutation tableaux provides a straightforward method for enumerating the crossing number and nesting number of the corresponding linked partitions.

Abstract: Generalized closeness is a recently defined centrality measure. Decay stability, based on generalized closeness, was studied in [Coroničova Hurajova et al. Math. Scand., 2018, 123: 39-50] and several conjectures on decay stability of certain graph classes were put forward. Here we disprove two conjectures by showing that the Cartesian and strong products of paths are not decay stable.

Abstract:

We introduce the notion of p-adic spherical codes (in particular, p-adic 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, obtained by Pfender, extends to p-adic Hilbert spaces.

Abstract: An \textit{independent set} in a graph comprises vertices that are not adjacent to one another, whereas a \textit{clique} consists of vertices where all pairs are adjacent. For a given graph $G$, let the following notations be defined: the number of vertices in $G$ is $n(G)=n$, the cardinality of a maximum independent set in $G$ is $\alpha(G)-\alpha$, the size of the largest clique in $G$ is $\omega(G)=\omega$, the cardinality of the intersection of all maximum independent sets in $G$ is $\xi(G)=\xi$. As the main finding of the article, we present an upper bound on the number of maximum independent sets as follows: \[ s_{\alpha} \leq \sum_{k=0}^{\alpha- \xi} \binom{n - \alpha - \omega + 1}{k} + (\omega - 1) \cdot \sum_{k=0}^{\alpha- \xi - 1} \binom{n - \alpha - \omega + 1}{k} \] \[ \leq \omega \cdot 2^{\min(\alpha - \xi, n - \alpha - \omega - \xi + 1)}. \] As an application of our findings, we explore a series of inequalities that connect the number of longest increasing subsequences with the number of longest decreasing subsequences in a given sequence of integers.

Abstract:

Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index, Mo(G), is defined as: Mo(G) = Σuv∈E(G) nu(e|G) − nv(e|G), where nu(e|G) and nv(e|G) represent the number of vertices closer to vertex u than v, and closer to v than u, respectively, for an edge e = uv. The inverse Mostar index problem has gained significant attention recently. In their work, Alizadeh et al. [Solving the Mostar index inverse problem, J. Math. Chem. 62 (5) (2024) 1079–1093], proposed the following open problem: “Which nonnegative integers can be realized as Mostar indices of c-cyclic graphs, for a given positive integer c?" Subsequently, one of the present authors [On the inverse Mostar index problem for molecular graphs, Trans. Comb. 14 (1) (2024) 65–77] conjectured that, except for finitely many positive integers, all other positive integers can be realized as the Mostar index of a c-cyclic graph, where c ≥ 3. In this paper, we fully resolve this open problem by proving the aforementioned conjecture.

Abstract: The \( 3 \times 3 \times 3 \) cube is the subject of numerous works and is examined several times as a group. Not only that, the God's Number (the maximum required number of rotations of the optimal solution path) is the subject of many works and has often been calculated and improved by new algorithms . The calculation of the God's Number of the \( 3 \times 3 \times 3 \) cube is much more demanding and therefore also more interesting than that of the \( 2 \times 2 \times 2 \) cube. The motivation of this work is to apply the knowledge of the \( 3 \times 3 \times 3 \) cube in \( 2 \times 2 \times 2 \) cube. In contrast to the \( 3 \times 3 \times 3 \) cube, the \( 2 \times 2 \times 2 \) cube has fewer pieces and no edge cubies - this makes the smaller cube less complex. However, the \( 2 \times 2 \times 2 \) cube can be rotated due to the lack of center cubies, which means that the top side can be changed at will. In this work, the knowledge of the group of the \( 3 \times 3 \times 3 \) cube is adapted and thus the \( 2 \times 2 \times 2 \) cube is represented using group theory to gain knowledge about the cube which includes calculating the number of possible cube configurations and creating a concept for finding the optimal solution path.

Abstract:

The generalized matrix of a graph G is defined as M(G) = A(G) − tD(G) (t ∈ R, A(G) and D(G) respectively denote the adjacency matrix and the degree matrix of G), and the generalized characteristic polynomial of G is merely the characteristic polynomial of M(G). Let K_m,n be the complete bipartite graph. Then the K_m,n-complement of a subgraph G in K_m,n is defined as the graph obtained by removing all edges of an isomorphic copy of G from K_m,n. In this paper, by using a determinant expansion on the sum of two matrices (one of which is a diagonal matrix), a general method for computing the generalized characteristic polynomial of the K_m,n-complement of a bipartite subgraph G was provided. Furthermore, when G is a graph with rank no more than 4, the explicit formula for the generalized characteristic polynomial of the K_m,n-complements of G is given.

Abstract: We classify in an explicit way those line spreads of projective 5-space over a field that have the property that the given spread induces a spread in the 3-space generated by any pair of spread lines. We determine their fix groups and conclude that there exist such spreads with trivial fix group. Also, we characterise regular line spreads among all line spreads of projective 3-space by their projectivity group, and also by a weakening of the regularity condition.

Abstract: Ramsey theory is applied to the analysis of operators acting on the functions belonging to the L^2 Hilbert space. The operators form the vertices of the bi-colored graph. If the operators commute, they are connected by a red link; if the operators do not commute they are connected with a green link. Thus, the complete, bi-colored graph emerges and the Ramsey theory becomes applicable. If the graph contains six vertices/operators, at least one monochromatic triangle will necessarily appear in the graph. Thus, the triad of operators forming the read triangle possesses the common set of eigenfunctions. The extension of introduced approach to infinite sets of operators is addressed. Applications of the introduced approach to problems of classical and quantum mechanics are suggested.

Abstract: We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and combinatorial identities are derived.

Abstract: Formal Calculation provides formulas for computing nested sums, offers results for all three forms, and explores the connections between them. It is a powerful tool for studying various numbers in a unified manner, making special numbers of many kinds appear ordinary. Additionally, this paper generalizes Wilson's theorem, Wolstenholme's theorem, and the Eulerian polynomial, demonstrating that they are merely special cases.

Abstract: In this paper we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35,59,95 and 97 points and as a consequence we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35,59,95 and 97 points provided that a conjecture included in the literature is true. As another consequence, we also improve the lower bound on the maximum number of halving pseudolines for sets in the plane with 35 points. These examples and the recursive bounds for the maximum number of halving lines for sets with an odd number of points achieved, give a new insight in the study of the rectilinear crossing number problem, one of the most challenging tasks in Discrete Geometry. With respect to this problem, it is conjectured that, for all n multiple of 3, there are 3-symmetric sets of n points for which the rectilinear crossing number is attained.

Abstract: We show a simple bijection $P$ between permutations $S_n$ of length $n$ and underdiagonal paths of size $n$, the last being lattice paths made of up $U=(1,1)$, down $D=(1,-1)$, west $W=(-1,1)$ steps, running from $(0,0)$ to $(2n,0)$, and such that: (1) the path is weakly bounded by the lines $y=0$ and $x=y$; (2) a $D$ (resp. $W$) step cannot be followed by a $W$ (resp. $D$) step. The aim of this paper is to study and enumerate families of underdiagonal paths which are defined by restricting the bijection $P$ to subclasses of $S_n$ avoiding some vincular patterns. For a given pattern $\tau$, let $S(\tau)$ be the family of permutations avoiding $\tau$, and $P(\tau)$ the family of underdiagonal paths corresponding to permutations in $S(\tau)$, precisely $P(\tau)=\{ P(\pi): \pi \in S(\tau) \}.$ We will consider patterns $\tau$ of length $3$ and $4$, and, when it is possible, we will provide a characterization of the underdiagonal paths of $P(\tau)$ in terms of geometrical constrains, or equivalently, the avoidance of some factors. Finally, we will provide a recursive growth of these families by means of generating trees and then their enumerative sequence.

Abstract: Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein.

Abstract:

This paper presents four novel schemes for secure data encryption utilizing mathematical transformations. The first scheme employs matrix manipulations and partitioning of numerical values derived from plain text characters, constructing a cipher through matrix additions. The second scheme introduces division operations and additional transformation layers for enhanced data obscurity. The third scheme incorporates tridiagonal matrices, creating a structured encryption process. The fourth scheme refines this approach with quotient-remainder operations and symmetric keys to construct a tridiagonal matrix cipher. Detailed algorithms for encryption and decryption are provided, with examples illustrating practical applications. This work contributes to mathematical cryptography by offering enhanced security methods for digital communication.