Cartesian Product Research Articles

Variety of General Position Problems in Graphs

Let X be a vertex subset of a graph G. Then u,v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u, v\\in V(G)$$\\end{document} are X-positionable if V(P)∩X⊆{u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(P)\\cap X \\subseteq \\{u,v\\}$$\\end{document} holds for any shortest u, v-path P. If each two vertices from X are X-positionable, then X is a general position set. The general position number of G is the cardinality of a largest general position set of G and has been already well investigated. In this paper a variety of general position problems is introduced based on which natural pairs of vertices are required to be X-positionable. This yields the total (resp. dual, outer) general position number. It is proved that the total general position sets coincide with sets of simplicial vertices, and that the outer general position sets coincide with sets of mutually maximally distant vertices. It is shown that a general position set is a dual general position set if and only if its complement is convex. Several sufficient conditions are presented that guarantee that a given graph has no dual general position set. The total general position number, the outer general position number, and the dual general position number of arbitrary Cartesian products are determined.

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property}

A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian product. In this paper Siciak's result is generalized to the setting of Siciak-Zakharyuta functions with growth given by a compact convex set, along with discussing why this generalization does not work in the weighted setting.

A framework for generalizing toric inequalities for holographic entanglement entropy

We conjecture a multi-parameter generalization of the toric inequalities of [1]. We then extend their proof methods for the generalized toric inequalities in two ways. The first extension constructs the graph corresponding to the toric inequalities and the generalized toric conjectures by tiling the Euclidean space. An entanglement wedge nesting relation then determines the geometric structure of the tiles. In the second extension, we exploit the cyclic nature of the inequalities and conjectures to construct cycle graphs. Then, the graph can be obtained using graph Cartesian products of cycle graphs. In addition, we define a set of knots on the graph by following [1]. These graphs with knots then imply the validity of their associated inequality. We study the case where the graph can be decomposed into disjoint unions of torii. Under the specific case, we explore and prove the conjectures for some ranges of parameters. We also discuss ways to explore the conjectured inequalities whose corresponding geometries are d-dimensional torii (d > 2).

A Genetic Algorithm to Search the Space of Ehrhart h*-Vectors

We describe a genetic algorithm to find candidates for h * -vectors satisfying given properties in the space of integers vectors of finite length. We use an implementation of such algorithm to find a 52-dimensional lattice polytope having a non-unimodal h * -vector which is the Cartesian product of two lattice polytopes having unimodal h * -vectors. This counterexample answers negatively to a question by Ferroni and Higashitani.

Binary Operations on Directed Fuzzy Graphs

In this paper, the newly directed fuzzy graphs are generated using the operations such as union, join, composition and cartesian product of two directed fuzzy graphs, accompanied by their illustrative analysis. The degree of vertices in the newly generated directed fuzzy graphs is defined, and properties based on vertex degree are established. Additionally, the degrees of corresponding vertices in the composition and cartesian product of two directed fuzzy graphs are compared. The orientation of defined operations on fuzzy graphs is compared with the corresponding operations on directed fuzzy graphs and analyzed through illustrative examples.

Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all <i>n</i> ≥ 2 and the suborbits are length 1, (<i>n</i>-1), (<i>n</i>-1), (<i>n</i>-1), (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>3</sup>. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all <i>n</i> > 2 and suborbital graphs of the group action are undirected. It is shown that graphs <I>Γ<sub>2</sub></I> and <I>Γ</I><sub>3</sub> are regular of degree <i>n</i>-1, graphs <I>Γ</I><sub>4</sub>, <I>Γ</I><sub>5</sub> and <I>Γ</I><sub>6</sub> of degree (<i>n</i>-1)<sup>2</sup> and graph <I>Γ</I><sub>7</sub> is regular of degree (<i>n</i>-1)<sup>3</sup>. The suborbital graphs <i>Γ<sub>i</sub></i>(<i>i=</i>1, 2,…, 6) are disconnected, with the number of connected components equal to <i>n</i><sup>2</sup> while suborbital graph <I>Γ</I><sub>7</sub><i> </i>is connected for all <i>n</i> > 2.

The 2-rainbow domination number of Cartesian product of cycles

The 2-rainbow domination number of Cartesian product of cycles

Transitive path decompositions of Cartesian products of complete graphs

Transitive path decompositions of Cartesian products of complete graphs

The space of locally elliptic subgroups of a locally compact group

Let G be a locally compact group. We denote by Sub ( G ) the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper we establish that the subset Su b L E ( G ) of closed locally elliptic subgroups of G is Chabauty-closed if and only if the set com p n ( G ) consisting of the n-tuples of elements of G contained in a common compact subgroup, is closed in the Cartesian product G n , for every nonnegative integer n. Moreover, we prove that the subspace Su b L E ( G ) is Chabauty-closed in the case when the group G contains an open normal locally elliptic subgroup.

Hesitant Bipolar-Valued Intuitionistic Fuzzy Graphs for Identifying the Dominant Person in Social Media Groups

This work introduces the notion of a hesitant bipolar-valued intuitionistic fuzzy graph (HBVIFG), which reflects four different characterizations: membership with positive/negative aspects and non-membership with positive/negative aspects, incorporating multi-dimensional alternatives in all of its information. HBVIFG generalizes both HBVFG and BVHFG due to its diversified nature in observing four perspectives along with multiple attributes in a piece of information. Numerous studies, examples, and graphical representations emphasize the concept’s distinctiveness and importance. The following graph theory terms are defined: strong directed HBVIFG, full directed HBVIFG, directed spanning HBVIFSG, directed HBVIFSG, and partial directed hesitant bipolar-valued intuitionistic fuzzy subgraph (HBVIFSG). Examples of operations utilizing two HBVIFGs are Cartesian, direct, lexicographical, and strong products. A scenario is used to generate the mapping of relations, which includes homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism. We describe a directed HBVIFG application that employs an algorithm to determine the most dominant person and self-persistent person in a social system and a comparative study is also provided. The proposed method provides a more detailed framework for assessing the most dominant and self-persistent individual in a social network across multi-level attributes along with positive and negative side membership and non-membership grades in each element of a network.

Laplacian {−1,0,1}- and {−1,1}-diagonalizable graphs

A graph is called Laplacian integral if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets {−1,0,1} or {−1,1}. Such graphs include as special cases the recently-investigated families of Hadamard-diagonalizable and weakly Hadamard-diagonalizable graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call balanced, which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are {−1,0,1}-diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the {−1,0,1}- and {−1,1}-diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are {−1,0,1}- or {−1,1}-diagonalizable.

Neighbor Locating Coloring on Graphs: Three Products

Let G be a simple finite graph. A k -coloring of G is a partition π = { S 1 , ⋯ , S k } of V ( G ) so that each S i is an independent set and any vertex in S i takes color i . A k -coloring π = { S 1 , ⋯ , S k } of V ( G ) is a neighbor locating coloring if for any two vertices u , v ∈ S i , there is a color class S j for which one of them has a neighbor in S j and the other does not. The minimum k with this property is said to be the neighbor locating chromatic number of G , denoted by χ N L ( G ) . We initiate the study of the neighbor locating coloring of graphs resulting from three types of product of two graphs. We investigate the neighbor locating chromatic number of Cartesian, lexicographic, and corona products of two graphs. Finally, we untangle the neighbor locating chromatic number of any of the aforementioned three products of cycles, paths, and complete graphs.

Chromatic Coloring of Distance Graphs V

Let G = ( V , E ) be any graph. If there exists an injection f : V → Z , such that | f ( u ) – f ( v ) | is prime for every u v ∈ E , then we say G is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of n mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of n arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.

Non‐Hamiltonian Cartesian products of two even dicycles

AbstractIn this note it is proven that there exist infinitely many positive integers and such that the Cartesian product of a directed cycle of length and a directed cycle of length is non‐Hamiltonian. In particular, the Cartesian product of an 880‐dicycle and a 4368‐dicycle is non‐Hamiltonian. We also prove that there is no such graph on fewer than vertices, which is rather astonishing.

Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of <i>n</i> ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when <i>n</i> = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of <i>n</i> > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (<i>n</i> − 1).

Computing Zagreb Connection Indices for the Cartesian Product of Path and Cycle Graphs

Topological indices ( ) are a class of graph-based descriptors widely used in chemiformatics and Quantitative Structure-Activity Relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a function. Among the Zagreb connection indices have been extensively researched. This study delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of . These indices encompass the first , second , and third . Furthermore, comprehensive results of the modified first , modified second , and modified third are presented, along with the first multiplicative ZCI second multiplicative ZCI (SMZCI), and third multiplicative ZCI Moreover, modified first multiplicative ZCI ), modified second multiplicative ZCI and modified third multiplicative ZCI ( are also calculated. To provide precision, both the graphical and numerical analyses of the computed findings are aligned for the two Cartesian products. The foundation for mathematically modeling complex networks and chemical structures lies in graph theory. Topological indices ( ) are a class of graph-based descriptors widely used in chemiformatics and quantitative structure-activity relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a topological index function. Among these the Zagreb connection indices are extensively researched TIs. This article delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of . These indices encompass the first, Second and third. Furthermore, we present the comprehensive results of the modified , modified and modified , and also present the first multiplicative Zagreb connection index , and , along with their modified counterparts like modified first multiplicative Zagreb connection index , and . These analysis encompass two distinct types of graphs: cycle graphs and path graphs. To provide further precision, we align both the graphical and numerical analysis of our computed findings for these two Cartesian products.

Further results on enumeration of perfect matchings of Cartesian product graphs

Abstract Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics and chemistry. As the general problem is #P-complete, it is usually tackled by randomized heuristics and approximation schemes. Let G G and H H be two graphs. Denote by G × H G\times H the Cartesian product of graphs G G and H H . The number of perfect matchings of some types of Cartesian product G × H G\times H is investigated by the Pfaffian method, where G ≅ P 2 G\cong {P}_{2} , P 3 {P}_{3} , P 4 {P}_{4} or C 4 {C}_{4} , and H ≅ T H\cong T or H H is a non-bipartite graph with a unique cycle. In this article, we construct a Pfaffian orientation of the graph G × P 2 G\times {P}_{2} , where G G is a non-bipartite graph with k k odd cycles. We obtain the counting formula on the number of perfect matchings of G × P 2 G\times {P}_{2} .

Exploring Locality Sensitive Hashing as a Blocking Method for Large-Scale Administrative Datasets

ObjectivesLinking large-scale datasets is challenging due to the computational power required. This research explores using Locality-Sensitive-Hashing (LSH) as a novel blocking method to facilitate linking large administrative data. LSH uses an efficient technique of hashing data while preserving similarity, reducing the search space and processing power required to find links. ApproachSamples were taken from a gold-standard dataset and blocked using LSH. Various parameters were tested to establish LSH’s optimal performance at retrieving linked pairs and reducing the search space size. ResultsTesting the method on small datasets gave promising results, with the LSH method creating ~9,000 candidate pairs. This is a improvement over our traditional blocking method or cartesian product, which created ~70,000 and 23.4 million candidate pairs, respectively. We have therefore shown that LSH can significantly reduce the search-space size. A further advantage of the method is its capability in handling alternative variables such as those that may be present in longitudinal or composite data, without needing to manually anticipate the different combinations of variables which may occur. Variable formats to simulate agreement weighting were also considered, with encouraging results. Conclusion and ImplicationsOur research shows that LSH can be used to drastically reduce the search space when blocking for data linkage This suggests developing LSH as a blocking method could result in more effective blocking, quicker linkage, and no loss of quality compared to traditional methods. However, further research on increasing the method’s scale is necessary.

More on Whitney levels of some decomposable continua

In this paper, we show that there exists a non-D-continuum such that each positive Whitney level of the hyperspace of subcontinua of the continuum is both D⁎ and Wilder. We show that the property of being continuum-wise Wilder is not a Whitney property, while it is a Whitney reversible property. Furthermore, we introduce the new class of continua: closed set-wise Wilder continua. This class is larger than the class of continuum chainable continua and smaller than the class of continuum-wise Wilder continua. In addition to the above results, we show that the Cartesian product of two closed set-wise Wilder continua is close set-wise Wilder.

New constructions of MSRD codes

In this work, we provide four methods for constructing new maximum sum-rank distance (MSRD) codes. The first method, a variant of cartesian products, allows faster decoding than known MSRD codes of the same parameters. The other three methods allow us to extend or modify existing MSRD codes in order to obtain new explicit MSRD codes for sets of matrix sizes (numbers of rows and columns in different blocks) that were not attainable by previous constructions. In this way, we show that MSRD codes exist (by giving explicit constructions) for new ranges of parameters, in particular with different numbers of rows and columns at different positions.