The Signed Domination Number of Cartesian Products of Directed Cycles

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function is called a signed dominating function (SDF) iffor each vertex v. The weight w(f ) of f is defined by. The signed domination number of a digraph D is gs(D) = min{w(f ) : f is an SDF of D}. Let Cmn denotes the Cartesian product of directed cycles of length m and n. In this paper, we determine the exact value of signed domination number of some classes of Cartesian product of directed cycles. In particular, we prove that: (a) gs(C3n) = n if n 0(mod 3), otherwise gs(C3n) = n + 2. (b) gs(C4n) = 2n. (c) gs(C5n) = 2n if n 0(mod 10), gs(C5n) = 2n + 1 if n 3, 5, 7(mod 10), gs(C5n) = 2n + 2 if n 2, 4, 6, 8(mod 10), gs(C5n) = 2n + 3 if n 1,9(mod 10). (d) gs(C6n) = 2n if n 0(mod 3), otherwise gs(C6n) = 2n + 4. (e) gs(C7n) = 3n

Irreversible k-threshold conversion number of some graphs

Purpose – This paper aims to study Irreversible conversion processes, which examine the spread of a one way change of state (from state 0 to state 1) through a specified society (the spread of disease through populations, the spread of opinion through social networks, etc.) where the conversion rule is determined at the beginning of the study. These processes can be modeled into graph theoretical models where the vertex set V(G) represents the set of individuals on which the conversion is spreading. Design/methodology/approach – The irreversible k-threshold conversion process on a graph G=(V,E) is an iterative process which starts by choosing a set S_0?V, and for each step t (t = 1, 2,…,), S_t is obtained from S_(t−1) by adjoining all vertices that have at least k neighbors in S_(t−1). S_0 is called the seed set of the k-threshold conversion process and is called an irreversible k-threshold conversion set (IkCS) of G if S_t = V(G) for some t = 0. The minimum cardinality of all the IkCSs of G is referred to as the irreversible k-threshold conversion number of G and is denoted by C_k (G). Findings – In this paper the authors determine C_k (G) for generalized Jahangir graph J_(s,m) for 1 < k = m and s, m are arbitraries. The authors also determine C_k (G) for strong grids P_2? P_n when k = 4, 5. Finally, the authors determine C_2 (G) for P_n? P_n when n is arbitrary. Originality/value – This work is 100% original and has important use in real life problems like Anti-Bioterrorism

Eternal Domination of Generalized Petersen Graph

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m-eternal domination model. The size of the smallest m-eternal dominating set is called the m-eternal domination number and is denoted by γm∞G. In this paper, we find γm∞Pn,1 and γm∞Pn,3 for n≡0 mod 4. We also find upper bounds for γm∞Pn,2 and γm∞Pn,3 when n is arbitrary

بعض المؤشرات وكثيرات الحدود للجداء الديكارتي للبيان الثنائي التجزئة التام K_(1,n) مع حلقة C_m

     لقد استخدمت المؤشرات وكثيرات الحدود في الكيمياء ولها العديد من التطبيقات الكيميائية الهامة مثل التنبؤ بسلوك بعض المركبات الكيميائية وحساب بعض الثوابت الحرارية. في هذا البحث نقوم بحساب  كثيرة حدود للجداء الديكارتي للبيان الثنائي التجزئة التام مع حلقة  من أجل أية قيمة لـ  ونحسب من خلال  كثيرة حدود بعض المؤشرات التي تعتمد عليها مثل مؤشر راندك العام ومؤشر راندك العكسي ومؤشر زغرب الأول والثاني ومؤشر زغرب المعدل الثاني ومؤشر التقسيم المتناظر ومؤشر زغرب الموسّع ومؤشر المجموع العكسي من أجل  ونحسب بعض المؤشرات وكثيرات الحدود التي تعتمد على التعريف مثل مؤشر فرط زغرب ومؤشر زغرب المضاعف الأول والثاني والمؤشر المنسي وكثيرة حدود زغرب الأولى والثانية  وكثيرة الحدود ذي الدرجة العكسية والمؤشر ذو الدرجة العكسية ومؤشر زغرب العام الأول والثاني ومؤشر لورديك وكثيرة الحدود المتناسقة والمؤشر المتناسق ومؤشر اتصال الرابطة الذرية من أجل

Chromatic Schultz and Gutman Polynomials of Jahangir Graphs J2,m and J3,m

Topological polynomial and indices based on the distance between the vertices of a connected graph are widely used in the chemistry to establish relation between the structure and the properties of molecules. In a similar way, chromatic versions of certain topological indices and the related polynomial have also been discussed in the recent literature. In this paper, we present the chromatic Schultz and Gutman polynomials and the expanded form of the Hosoya polynomial and chromatic Schultz and Gutman polynomials, and then we derive these polynomials for special cases of Jahangir graphs

Hosoya, Schultz, and Gutman Polynomials of Generalized Petersen Graphs Pn,1 and Pn,2

The graph theory has wide important applications in various other types of sciences. In chemical graph theory, we have many topological polynomials for a graph G through which we can compute many topological indices. Topological indices are numerical values and descriptors which are used to quantify the physiochemical properties and bioactivities of the chemical graph. In this paper, we compute Hosoya polynomial, hyper-Wiener index, Tratch–Stankevitch–Zefirov index, Harary index, Schultz polynomial, Gutman polynomial, Schultz index, and Gutman index of generalized Petersen graphs Pn,1 and Pn,2

On Eternal Domination of Generalized Js,m

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex, and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m-eternal domination model. The size of the smallest m-eternal dominating set is called the m-eternal domination number and is denoted by γm∞G. In this paper, we find the domination number of Jahangir graph Js,m for s≡1,2 mod 3, and the m-eternal domination numbers of Js,m for s,m are arbitraries

Game Chromatic Number of Generalized Petersen Graphs and Jahangir Graphs

Let G=V,E be a graph, and two players Alice and Bob alternate turns coloring the vertices of the graph G a proper coloring where no two adjacent vertices are signed with the same color. Alice's goal is to color the set of vertices using the minimum number of colors, which is called game chromatic number and is denoted by χgG, while Bob's goal is to prevent Alice's goal. In this paper, we investigate the game chromatic number χgG of Generalized Petersen Graphs GPn,k for k≥3 and arbitrary n, n-Crossed Prism Graph, and Jahangir Graph Jn,m

A Systematic Review to Uncover a Universal Protocol for Accuracy Assessment of 3-Dimensional Virtually Planned Orthognathic Surgery

The aim of this study was to systematically review methods used for assessing the accuracy of 3-dimensional virtually planned orthognathic surgery in an attempt to reach an objective assessment protocol that could be universally used.status: publishe