Pelabelan total (a,d)-H-Antiajaib (Super) pada Graf

Suatu graf G =(V(G),E(G)) dikatakan mempunyai selimut-(Hl H2, Hk) jika setiap sisi di G menjadi sisi paling sedikit dari satu subgraf Hi, 1 SiS k. Jika untuk setiap i, Hi isomorfis dengan suatu graf H, maka G dikatakan mempunyai selimut-H. Pelabelan total (a,d)-H-anti ajaib dari graf G adalah fungsi bijektif f: V(G) U E(G) -t (1,2, . W(G)I + IE(G)!} sedemikian hingga himpunan bobot untuk setiap subgraf H yang isomorfis dengan H adalah a + (a + d) +... + (a + (t -l)d), untuk suatu bilangan bulat positif a dan d, dim ana t adalah banyaknya subgraf pada G yang isomorfis dengan H. Iika f: V(G) -t (1,2, .,W(G)I} maka G dikatakan mempunyai pelabelan total (a,d)-H-anti ajaib super. Penelitian in; mengkaji pelabelan total {a,d}-H-anti ajaib super pad a graf tangga (PnxP2) untuk H= Ca, Cs dan pelabelan total (a,d}-Cs-anti ajaib super pada graf prisma. Hasilnya adalah, jika graf tangga (PnxP2) mempunyai pelabelan tolal (a,d}-Ca-anti ajaib super maka nilai d S 36 dan d s 48 jika mempunyai pelabelan total {a,d}-Ca-anti ajaib super. Pelabelan total (a,d}-Co-anti ajaib super pada graf tangga diperoleh untuk 1 S d S 22 dan d = 24,27,30. Pelabelan total (a,d)CBanti ajaib super pada graf tangga diperoleh untuk d =3, 4, 6, 13, 14, 15, 16,21,22, 23,24, 29, 3D, 32, 40. Sedangkan untuk graf prisma belum diperoleh pola pelabelan anti ajaib supernya

Fractional Local Metric Dimension of Comb Product Graphs

 يعرف الرسم البياني المتصل G مع قمة الرأس (V (G ومجموعة الحافة (E (G، (حي الحل المحلي)   لذرتين متجاورتين u، v بواسطة   دالة الحل المحلية fi لـ  G هي دالة ذات قيمة حقيقية  بحيث يكون   لكل رأسين متجاورين البُعد المتري المحلي الجزئي لـ  الرسم البياني G يشير إلى ، وهو معرّف بواسطة  وهي دالة حل محلية لـ G}. إحدى العمليات في الرسم البياني هي الرسوم البيانية لمنتج Comb. الرسوم البيانية لمنتج Comb لـ G و  H يشار إليه  بواسطة  الهدف من هذا البحث هو تحديد البعد المتري المحلي الجزئي لـ  ، وذلك  لان  الرسم البياني G هو رسم بياني متصل والرسم البياني H هو رسم بياني كامل   نحصل  من   علىThe local resolving neighborhood  of a pair of vertices  for  and  is if there is a vertex  in a connected graph  where the distance from  to  is not equal to the distance from  to , or defined by . A local resolving function  of  is a real valued function   such that  for  and . The local fractional metric dimension of graph  denoted by , defined by  In this research, the author discusses about the local fractional metric dimension of comb product are two graphs, namely graph  and graph , where graph  is a connected graphs and graph  is a complate graph  and denoted by  We ge

On Commutative Characterization of Graph Operation with Respect to Metric Dimension

Let  G be a connected graph with vertex set V(G) and W={w1, w2, ..., wm} ⊆ V(G). A representation of a vertex v âˆˆ V(G) with respect to W is an ordered m-tuple r(v|W)=(d(v,w1),d(v,w2),...,d(v,wm)) where d(v,w) is the distance between vertices v and w. The set W is called a resolving set for G if every vertex of G has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim (G), is the number of vertices in a basis of G. In general, the comb product and the corona product are non-commutative operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient  conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension

The Dominant Metric Dimension of Corona Product Graphs

               البعد المتري والمجموعة المسيطرة هما مفهوم نظرية الرسم البياني الذي يمكن تطويره من حيث المفهوم وتطبيقه في عمليات الرسم البياني. ان  حل الرقم المهيمن هو أحد المفاهيم في نظرية الرسم البياني التي تجمع بين هذين المفهومين . في هذه الورقة ، يتم تقديم تعريف حل الرقم المسيطر مرة أخرى كمصطلح البعد المتري السائد. حيث تهدف هذه الورقة إلى إيجاد البعد المتري السائد لبعض الرسوم البيانية الخاصة ورسومات حاصل ضرب الاكليل للرسوم البيانية المتصلة ، ولبعض الرسوم البيانية الخاصة. يُشار إلى البعد المتري السائد لـ ويتم الإشارة إلى البعد المتري السائد في الرسم البياني للمنتج الإكليل G و H بواسطة. The metric dimension and dominating set are the concept of graph theory that can be developed in terms of the concept and its application in graph operations. One of some concepts in graph theory that combine these two concepts is resolving dominating number. In this paper, the definition of resolving dominating number is presented again as the term dominant metric dimension. The aims of this paper are to find the dominant metric dimension of some special graphs and corona product graphs of the connected graphs  and , for some special graphs  . The dominant metric dimension of  is denoted by  and the dominant metric dimension of corona product graph G and H is denoted by

Embedding Cycle Graphs Complements

A graph is embeddable on a surface if it can be drawn on that surface without any edges intersect. The cycle graphs can always be embedded on the plane and the torus, but this is not occurred for their complements. We prove that the maximum order of cycle graphs such that their complements still can be embedded on the plane is 6. But, the maximum order of cycle graphs such that their complements still can be embedded on the torus is 9. Also, the crossing number of complements of cycle graphs which can’t be embedded on the plane with minimum order will be presented

Pelabelan total (a,d)-H-Antiajaib (Super) pada Graf

Suatu graf G =(V(G),E(G)) dikatakan mempunyai selimut-(Hl H2, Hk) jika setiap sisi di G menjadi sisi paling sedikit dari satu subgraf Hi, 1 SiS k. Jika untuk setiap i, Hi isomorfis dengan suatu graf H, maka G dikatakan mempunyai selimut-H. Pelabelan total (a,d)-H-anti ajaib dari graf G adalah fungsi bijektif f: V(G) U E(G) -t (1,2, . W(G)I + IE(G)!} sedemikian hingga himpunan bobot untuk setiap subgraf H yang isomorfis dengan H adalah a + (a + d) +... + (a + (t -l)d), untuk suatu bilangan bulat positif a dan d, dim ana t adalah banyaknya subgraf pada G yang isomorfis dengan H. Iika f: V(G) -t (1,2, .,W(G)I} maka G dikatakan mempunyai pelabelan total (a,d)-H-anti ajaib super. Penelitian in; mengkaji pelabelan total {a,d}-H-anti ajaib super pad a graf tangga (PnxP2) untuk H= Ca, Cs dan pelabelan total (a,d}-Cs-anti ajaib super pada graf prisma. Hasilnya adalah, jika graf tangga (PnxP2) mempunyai pelabelan tolal (a,d}-Ca-anti ajaib super maka nilai d S 36 dan d s 48 jika mempunyai pelabelan total {a,d}-Ca-anti ajaib super. Pelabelan total (a,d}-Co-anti ajaib super pada graf tangga diperoleh untuk 1 S d S 22 dan d = 24,27,30. Pelabelan total (a,d)CBanti ajaib super pada graf tangga diperoleh untuk d =3, 4, 6, 13, 14, 15, 16,21,22, 23,24, 29, 3D, 32, 40. Sedangkan untuk graf prisma belum diperoleh pola pelabelan anti ajaib supernya

A central local metric dimension on acyclic and grid graph

The local metric dimension is one of many topics in graph theory with several applications. One of its applications is a new model for assigning codes to customers in delivery services. Let $G$ be a connected graph and $V(G)$ be a vertex set of $G$. For an ordered set $W = \{ x_1, x_2, \ldots, x_k\} \subseteq V(G)$, the representation of a vertex $x$ with respect to $W$ is $r_G(x|W) = \{(d(x, x_1), d(x, x_2), \ldots, d(x, x_k) \}$. The set $W$ is said to be a local metric set of $G$ if $r(x|W)\neq r(y|W)$ for every pair of adjacent vertices $x$ and $y$ in $G$. The eccentricity of a vertex $x$ is the maximum distance between $x$ and all other vertices in $G$. Among all vertices in $G$, the smallest eccentricity is called the radius of $G$ and a vertex whose eccentricity equals the radius is called a central vertex of $G$. In this paper, we developed a new concept, so-called the central local metric dimension by combining the concept of local metric dimension with the central vertex of a graph. The set $W$ is a central local metric set if $W$ is a local metric set and contains all central vertices of $G$. The minimum cardinality of a central local metric set is called a central local metric dimension of $G$. In the main result, we introduce the definition of the central local metric dimension of a graph and some properties, then construct the central local metric dimensions for trees and establish results for the grid graph