Lexicographic Product Research Articles

Monophonic Eccentric Domination Numbers of Graphs

Let G be a (simple) undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a monophonic eccentric dominating set if every vertex in V (G) \ S has a monophonic eccentric vertex in S. The minimum size of a monophonic eccentric dominating set in G is called the monophonic eccentric domination number of G. It is shown that the absolute difference of the domination number and monophonic eccentric domination number of a graph can be made arbitrarily large. We characterize the monophonic eccentric dominating sets in graphs resulting from the join, corona, and lexicographic product of two graphs and determine bounds on their monophonic eccentric domination numbers.

Spectral determinations and eccentricity matrix of graphs

Let G be a connected graph on n vertices. For a vertex u∈G, the eccentricity of u is defined as ε(u)=max⁡{d(u,v)|v∈V(G)}, where d(u,v) denotes the distance between u and v. The eccentricity matrix E(G)=(ϵuv), whereϵuv:={d(u,v)if d(u,v)=min⁡{ε(u),ε(v)},0otherwise, has been firstly introduced in Chemical Graph Theory. In literature, it is also known as the DMAX-matrix. Graphs with the diameter equal to the radius are called self-centered graphs. Two non-isomorphic graphs are said to be M-cospectral with respect to a given matrix M if they have the same M-eigenvalues. In this paper, we show that, when n→∞, the fractions of non-isomorphic cospectral graphs with respect to the adjacency and the eccentricity matrix behave like those only concerning the self-centered graphs with diameter two. Secondly, we prove that a graph G has just two distinct E-eigenvalues if and only if G is an r-antipodal graph. Thirdly, we obtain many pairs of E-cospectral graphs by using strong and lexicographic products. Finally we formulate some problems waiting to be solved in order to build up a spectral theory based on the eccentricity matrix.

The H-join of arbitrary families of graphs - the universal adjacency spectrum

The H-join of a family of graphs G={G1,…,Gp}, also called generalized composition, H[G1,…,Gp], where all graphs are undirected, simple and finite, is the graph obtained by replacing each vertex i of H by Gi and adding to the edges of all graphs in G the edges of the join Gi∨Gj, for every edge ij of H. Some well known graph operations are particular cases of the H-join of a family of graphs G as it is the case of the lexicographic product (also called composition) of two graphs H and G, H[G]. During long time the known expressions for the determination of the entire spectrum of the H-join in terms of the spectra of its components (that is, graphs in G) and an associated matrix, related with the main eigenvalues of the components and the graph H, were limited to families G of regular graphs. In this work, with an approach based on the walk-matrix, we extend such a determination, as well as the determination of the characteristic polynomial, to the universal adjacency matrix of the H-join of families of arbitrary graphs. From the obtained results, the eigenvectors of the universal adjacency matrix of the H-join can also be determined in terms of the eigenvectors of the universal adjacency matrices of the components and an associated matrix.

Lindelöfness of lexicographic products of GO-spaces

In this paper, we consider when a lexicographic product of GO-spaces is Lindelöf. We mainly prove: (1) Let γ be any ordinal. Assume that for each α<γ, Xα is a Lindelöf GO-space, and for each α≥1, Xα has a minimum mα and maximum Mα. Then the lexicographic product ∏α<γXα of GO-spaces is also Lindelöf. (2) Let γ≤ω1. Assume that for each α<γ, Xα is a Lindelöf subspace of an ordinal. Then the lexicographic product ∏α<γXα of GO-spaces is also Lindelöf.

Geodesic transversal problem for join and lexicographic product of graphs

Geodesic transversal problem for join and lexicographic product of graphs

From Italian domination in lexicographic product graphs to w-domination in graphs

In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms of five different domination parameters of G . These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let N ( v ) denote the open neighbourhood of v ∈ V ( G ) , and let w = ( w 0 , w 1 , …, w l ) be a vector of nonnegative integers such that w 0 ≥ 1 . We say that a function f : V ( G ) → {0, 1, …, l } is a w -dominating function if f ( N ( v )) = ∑ u ∈ N ( v ) f ( u ) ≥ w i for every vertex v with f ( v ) = i . The weight of f is defined to be ω ( f ) = ∑ v ∈ V ( G ) f ( v ) . The w -domination number of G , denoted by γ w ( G ) , is the minimum weight among all w -dominating functions on G . Specifically, we show that γ I ( G ○ H ) = γ w ( G ) , where w ∈ {2} × {0, 1, 2} l and l ∈ {2, 3} . The decision on whether the equality holds for specific values of w 0 , …, w l will depend on the value of the domination number of H . This paper also provides preliminary results on γ w ( G ) and raises the challenge of conducting a detailed study of the topic.

Restrained Disjunctive Domination in Graphs under Some Binary Operations

A set S ⊆ V (G) is a disjunctive dominating set of a graph G if for every v ∈ V (G)\S, v is a neighbor of a vertex in S or S has at least two vertices each at distance 2 from v. We say that a disjunctive dominating set S of G is a restrained disjunctive dominating set if for each v ∈ V (G)\S there exists u ∈ V (G) \ S such that uv ∈ E(G) or there exist distinct vertices u, w ∈ V (G) \ S such that dG(u, v) = 2 = dG(w, v). The minimum cardinality γdr(G) of a restrained disjunctive dominating set of G is the restrained disjunctive domination number of G. In this paper, we characterize the restrained disjunctive dominating sets in some binary operations such as the join, corona and lexicographic product of graphs and, as a result, obtain the values of their corresponding restrained disjunctive domination numbers.

Some Properties of the Balanced Hypercube

As a variant of [Formula: see text], Huang and Wu in [IEEE Transactions on Computers 46 (1997) 484–490] introduced the balanced hypercube [Formula: see text] as an interconnection network topology for computing systems. In this paper, we show that [Formula: see text] is a lexicographic product of two graphs, and using this, we show that every minimum cyclic vertex-cut of [Formula: see text] isolates a 4-cycle, and prove that the edge neighbor connectivity of [Formula: see text] is [Formula: see text].

Vertex pancyclicity over lexicographic products

A lexicographic product of graphs G and H, denoted by is defined as a graph with the vertex set and an edge presents in the product whenever or (u 1 = u 2 and ). We investigate the sufficient conditions for vertex pancyclicity of lexicographic products of complete graphs Kn , paths Pn or cycles Cn with a general graph. We obtain that (i) if G 1 is a traceable graph of even order and G 2 is a graph with at least one edge, then is vertex pancyclic; (ii) if G 1 is hamiltonian and G 2 is a graph with at least one edge, then is vertex pancyclic.

Cubic Pythagorean Fuzzy Graphs

The purpose of the study is to explore graph theory based on cubic Pythagorean fuzzy sets. The concept of cubic Pythagorean fuzzy graphs (CuPFGs) is introduced in this research work. In addition, we define certain fundamental operations on CuPFGs including semistrong product, lexicographical product, and symmetric difference of two CuPFGs and demonstrate some of their key characteristics. Meanwhile, to investigate the preference of decision makers, cubic Pythagorean fuzzy preference relation is defined. Moreover, through a practical example, the applicability of our proposed work in multicriteria decision‐making is described, and to clarify the organization of the proposed method, a frame diagram is presented.

New Concepts on Quadripartitioned Bipolar Single Valued Neutrosophic Graph

The partition of indeterminacy function of the neutrosophic set into the contradiction part and the ignorance part represent the quadripartitioned single valued neutrosophic set. In this work, the new concept of quadripartitioned bipolar single valued neutrosophic graph is established, and the operations on it are studied. The Cartesian product, cross product, lexicographic product, strong product and composition of quadripartitioned bipolar single valued neutrosophic graph are investigated. The proposed concepts are illustrated with examples.

The Intellectualization of African Languages through Terminology and Lexicography: Methodological Reflections with Special Reference to Lexicographic Products of the University of KwaZulu-Natal

Terminology development and practical lexicography are crucial in language intel­lectualization. In South Africa, the Department of Sport, Arts and Culture, National Lexicography Units, universities, commercial publishers and other organizations have been developing terminol­ogy and publishing terminographical/lexicographical resources to facilitate the use of African languages alongside English and Afrikaans in prestigious domains. Theoretical literature in the field of lexicography (e.g., Bergenholtz and Nielsen (2006); Bergenholtz and Tarp (1995; 2010); Gouws 2020) has attempted to resolve traditional distinctions between lexicography and termi­nology while also addressing terminological imprecisions in the relevant scholarship. Taking the cue from such scholarship, this article reflects on the methodological approaches for developing lexicographical products for specific subject fields, i.e., resources that document and describe ter­minology from specialized academic and professional fields. Its focus is on the use of traditional methods vis-à-vis the application of electronic corpora and its technologies in the key practical tasks such as term extraction and lemmatization. The article notes that the limited availability of specialized texts in African languages hampers the development and deployment of advanced electronic corpora and its applications to improve the execution of terminological and lexico­graphical tasks, while also enhancing the quality of the products. The Illustrated Glossary of Southern African Architectural Terms (English–isiZulu), A Glossary of Law Terms (English–isiZulu) and the forth­coming isiZulu dictionary of linguistic terms are used for special reference. Keywords: intellectualization of African languages, lexicography, termi­nology, terminography, dictionary, subject field dictionaries, sub­ject field lexicography, glossary, electronic corpora

Differentiated Treatment of Cultural Items in Lexicographical Products: A Necessary Adaptation to the Digital Environment

The paper focuses on the lexicographical treatment of cultural objects. It argues that second-language learning requires second-culture learning and that digital technologies call for new solutions to both old and new challenges. As an example, it takes traditional Chinese musical instruments and starts with a critical analysis of their treatment in five Chinese–English dictionaries for both foreign learners and native speakers. It continues with some reflections on media convergence and its consequences for lexicography and reaches the conclusion that the one-size-fits-all dictionary must be replaced with a variety of lexicographical products on different platforms. Lexicographers' focus must therefore move from the dictionary to the database that supports these products. This leads to a discussion of equivalent and explanation types and the need to prepare four different database fields for equivalents and two for explanations. To exemplify this, the paper presents a lexicographical database with equivalents, explanations, and other types of culturally relevant items. It then uses a few examples to show how these lexicographical items stored in the database can be selectively employed on different platforms and adapted to specific user needs. The paper links directly to sound files and video clips with some of the discussed instruments. Finally, the paper provides some conclusions and perspectives for further improving the cultural dimension of learners' lexicography.

Comparative Study of Generalized Sum Graphs via Degree‐Based Topological Indices

In theoretical chemistry, topological indices (TIs) have important role to predict various physical and structural properties of the study under molecular graphs. Among all topological indices, Zagreb‐type indices have been used more effectively in the chemical literature. In this paper, we have computed first Zagreb, second Zagreb, forgotten, and hyper Zagreb indices of the generalized Q‐sum graph in the form of different TIs of its basic graphs, where α ≥ 1 is a positive integer. This family of graphs is obtained by the lexicographic product of the graph Qα(H1) and H2, where Qα(H1) is constructed with the help of the generalized line superposition operation Qα on H1. As a conclusion, we also checked the correlation between predefined graph H1[H2] under the operation of lexicographic product H1 and H2 with newly defined generalized Q‐sum graphs using linear regression models of various degree‐based TIs.

Connected domination game: Predomination, Staller-start game, and lexicographic products

The connected domination game was introduced in 2019 by Borowiecki, Fiedorowicz and Sidorowicz as another variation of the domination game. We answer a problem from their paper regarding the relation between the number of moves in a game where Dominator/Staller starts the game. Additionally, we study the relation to the diameter and present graphs with small game connected domination number. We also determine the value on the lexicographic products, and consider the effect of predomination of a vertex.

Induced-Paired Dominating Sets in Lexicographic Product of Graphs

Induced-Paired Dominating Sets in Lexicographic Product of Graphs

Weakly connected 2-domination in the lexicographic product of graphs

Weakly connected 2-domination in the lexicographic product of graphs

Aspekte van vernuwing in die Afrikaanse leksikografie: Standaard- en pedagogiese woordeboeke as barometer

Innovative Aspects of Afrikaans Lexicography: Standard and Pedagogical Dictionaries as Barometer. The aim of this contribution is to offer an overview of the areas of most significant innovation in the Afrikaans lexicography over the past two decades, with specific reference to standard and pedagogical dictionaries, since they represent the typological categories in which most development is visible. However, reference will also be made to other dictionary types and contexts where relevant. Firstly, the current state of affairs with regard to the Afrikaans lexicography will be outlined in terms of existing lexicographic products. Secondly, some comments will be offered on the professionalisation of Afrikaans lexicography — a topic which has not really been dealt with in metalexicographical research. The roles of the Bureau of the Woordeboek van die Afrikaanse Taal and commercial dictionary publishers will also be highlighted. Thereafter, secondary and specific lexicographic processes will be discussed, with reference to corpus lexicography and lemma selection, macrostructural innovation, and microstructural innovation. The discussion of the latter will include integrated microstructures in bilingual dictionaries, lexicographic definitions in monolingual dictionaries, cotext entries, and aspects of grammatical data. The state of affairs and future of the Afrikaans e-lexicography will also be addressed. Finally, some comments on special dictionaries will be offered. Keywords: commercial lexicography, corpus lexicography, e-lexicography, explanatory dictionary, lexicographic definition, macrostructure, microstructure, pedagogical lexicography, school dictionary, translation dictionary

On star and acyclic coloring of generalized lexicographic product of graphs

&lt;abstract&gt;&lt;p&gt;A $ star \; coloring $ of a graph $ G $ is a proper vertex coloring of $ G $ such that any path of length 3 in $ G $ is not bicolored. The $ star \; chromatic \; number $ $ \chi_s(G) $ of $ G $ is the smallest integer $ k $ for which $ G $ admits a star coloring with $ k $ colors. A $ acyclic \; coloring $ of $ G $ is a proper coloring of $ G $ such that any cycle in $ G $ is not bicolored. The $ acyclic \; chromatic \; number $ of $ G $, denoted by $ a(G) $, is the minimum number of colors needed to acyclically color $ G $. In this paper, we present upper bound for the star and acyclic chromatic numbers of the generalized lexicographic product $ G[h_n] $ of graph $ G $ and disjoint graph sequence $ h_n $, where $ G $ exists a $ k- $colorful neighbor star coloring or $ k- $colorful neighbor acyclic coloring. In addition, the upper bounds are tight.&lt;/p&gt;&lt;/abstract&gt;

Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

A topological index (TI) is a numerical descriptor of a molecule structure or graph that predicts its different physical, biological, and chemical properties in a theoretical way avoiding the difficult and costly procedures of chemical labs. In this paper, for two connected (molecular) graphs G 1 and G 2 , we define the generalized total-sum graph consisting of various (molecular) polygonal chains by the lexicographic product of the graphs T k G 1 and G 2 , where T k G 1 is obtained by applying the generalized total operation T k on G 1 with k ≥ 1 as some integral value. Moreover, we compute the different degree-based TIs such as first Zagreb, second Zagreb, forgotten Zagreb, and hyper-Zagreb. In the end, a comparison among all the aforesaid TIs is also conducted with the help of certain statistical tools for some particular families of generalized total-sum graphs under lexicographic product.