Open Distance-Pattern Uniform Graphs

All graphs considered in this paper are finite, simple, undirected and connected. For graph theoretic terminology we refer to Harary [6]. The concept of open distance-pattern and open distance-pattern uniform graphs weresuggested by B.D. Acharya

A Characterization of Maximal Outerplanar-Open Distance Pattern Uniform Graphs

ليكن A ⊆ V(H)  لأي رسم بياني H ، كل عقدة w من H يتم تصنيفها باستخدام مجموعة من الأرقام ;  ، حيث تشير d (w، v) إلى المسافة بين العقدة w والعقدة v في H ، والمعروفة باسم نمط المسافة A المفتوح . يُعرف الرسم البياني H بأنه الرسم البياني لنمط المسافة المفتوحة (odpu) - الرسم البياني ، إذا كانت هناك مجموعة فرعية غير فارغة A ⊆ V(H)  مع  هي نفسها لجميع . هنا تُعرف  باسم النمط الموحد لنمط المسافة المفتوحة (odpu-) للرسم البياني H و A يُعرف بمجموعة- odpu من H. سوف يعرف الحد الادنى من رؤوس الكاردينال لاي مجموعة- odpu من H  ان وجد كعدد – odpu للرسم  البياني H . في هذه المقالة ، نعطي توصيفًا للرسوم البيانية الخارجية القصوى للمخطط -odpu . كما وجدنا أيضًا أن الرقم الفردي المحتمل للرسم البياني اما يكون اثنان او خمسة فقط.Let A ⊆ V(H) of any graph H, every node w of H be labeled using a set of numbers; , where d(w,v) denotes the distance between node w and the node v in H, known as its open A-distance pattern. A graph H is known as the open distance-pattern uniform (odpu)-graph, if there is a nonempty subset A ⊆V(H) together with  is the same for all . Here  is known as the open distance pattern uniform (odpu-) labeling of the graph H and A is known as an odpu-set of H. The minimum cardinality of vertices in any odpu-set of H, if it exists, will be known as the odpu-number of the graph H. This article gives a characterization of maximal outerplanar-odpu graphs. Also, it establishes that the possible odpu-number of an odpu-maximal outerplanar graph is either two or five only

Analysis of Social Network Data Mining for Security Intelligence Privacy Machine Learning

The Modern communication on the Internet platform is most responsive through social media. Social media has changed and is still reshaping how we share our thoughts and emotions in communication. It has introduced a constant real-time communication pattern that was before unheard of. Young and old, organizations, governmental agencies, professional associations, etc., all have social media accounts that they use exclusively for communication with other users. Social media also acts as a powerful network engine that connects users regardless of where they are in the world. The development of global communication will greatly benefit from the availability of this new communication platform in the future. Consequently, there is a pressing need to research usage trends. Therefore, it is vital to investigate social media platform usage trends in order to develop automated systems that intelligence services can use to help avert national security incidents. Through the use of social media data mining, this research study suggests an automated machine learning model that can improve speedy response to crises involving national and International security

Equality of domination and transversal numbers in hypergraphs

A subset <i>S</i> of the vertex set of a hypergraph ℋ is called a dominating set of ℋ if for every vertex <i>v</i> not in <i>S</i> there exists <i>u ∈ S</i> such that <i>u</i> and <i>v</i> are contained in an edge in ℋ. The minimum cardinality of a dominating set in ℋ is called the domination number of ℋ and is denoted by γ(ℋ). A transversal of a hypergraph ℋ is defined to be a subset <i>T</i> of the vertex set such that <i>T ⋂ E ≠ Ø</i> for every edge <i>E</i> of ℋ. The transversal number of ℋ, denoted by <i>t</i>.(ℋ), is the minimum number of vertices in a transversal. A hypergraph is of rank <i>k</i> if each of its edges contains at most <i>k</i> vertices. The inequality <i>t</i>(ℋ) = γ(ℋ) is valid for every hypergraph ℋ without isolated vertices. In this paper, we investigate the hypergraphs satisfying <i>t</i>(ℋ) = γ(ℋ), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class ϴ ^p₂. Structurally we focus our attention on hypergraphs in which each subhypergraph ℋ¹ without isolated vertices fulfills the equality <i>t</i>(ℋ¹) = (ℋ¹). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer <i>k</i>, there are only a finite number of forbidden subhypergraphs of rank <i>k</i>, and each of them has domination number at most <i>k</i>

Computational Analysis of Some More Rectangular Tessellations of Kekulenes and Their Molecular Characterizations

Cycloarene molecules are benzene-ring-based polycyclic aromatic hydrocarbons that have been fused in a circular manner and are surrounded by carbon–hydrogen bonds that point inward. Due to their magnetic, geometric, and electronic characteristics and superaromaticity, these polycyclic aromatics have received attention in a number of studies. The kekulene molecule is a cyclically organized benzene ring in the shape of a doughnut and is the very first example of such a conjugated macrocyclic compound. Due to its structural characteristics and molecular characterizations, it serves as a great model for theoretical research involving the investigation of π electron conjugation circuits. Therefore, in order to unravel their novel electrical and molecular characteristics and foresee potential applications, the characterization of such components is crucial. In our current research, we describe two unique series of enormous polycyclic molecules made from the extensively studied base kekulene molecule, utilizing the essential graph-theoretical tools to identify their structural characterization via topological quantities. Rectangular kekulene Type-I and rectangular kekulene Type-II structures were obtained from base kekulene molecules arranged in a rectangular fashion. We also employ two subcases for each Type and, for all of these, we derived ten topological indices. We can investigate the physiochemical characteristics of rectangular kekulenes using these topological indices