Pelabelan Selimut Ajaib Super Pada Graf Lintasan (original) (raw)

Abstract

Let 𝐺 = (𝑉, 𝐸) be a simple graph. An edge covering of 𝐺 is a family of subgraphs 𝐻1 , … , π»π‘˜ such that each edge of 𝐸(𝐺) belongs to at least one of the subgraphs 𝐻𝑖 , 1 ≀ 𝑖 ≀ π‘˜. If every 𝐻𝑖 is isomorphic to a given graph 𝐻, then the graph 𝐺 admits an 𝐻 βˆ’ covering. Let 𝐺 be a containing a covering 𝐻, and 𝑓 the bijectif function 𝑓: (𝑉 βˆͺ 𝐸) β†’ {1,2,3, … , |𝑉| + |𝐸|} is said an 𝐻 βˆ’magic labeling of 𝐺 if for every subgraph 𝐻 β€² = (𝑉 β€² ,𝐸 β€² ) of 𝐺 isomorphic to 𝐻, is obtained that βˆ‘ 𝑓(𝑉) + βˆ‘ 𝑓(𝐸) π‘’βˆˆπΈ(𝐻′ π‘£βˆˆπ‘‰(𝐻 ) β€² ) is constant. 𝐺 is said to be 𝐻 βˆ’super magic if 𝑓(𝑉) = {1, 2, 3, … , |𝑉|}. In this case, the graph 𝐺 which can be labeled with 𝐻-magic is called the covering graph 𝐻 βˆ’magic. The sum of all vertex labels and all edge labels on the covering 𝐻 βˆ’ super magic then obtained constant magic is denoted by βˆ‘ 𝑓(𝐻). The duplication graph 2 of graph 𝐷2 (𝐺) is a graph obtained from two copies of graph 𝐺, called 𝐺 and 𝐺 β€² , with connecting each respectively vertex 𝑣 in 𝐺 with the vertexs immedia...

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References (18)

  1. Gambar 3 : Pelabelan Titik dan Sisi Selimut 𝐷 2 (𝑃 3 ) pada Graf 𝐷 2 (𝑃 6 ) pada Teorema 1 Gambar 4 : Pelabelan Titik dan Sisi Selimut 𝐷 2 (𝑃 3 ) pada Graf 𝐷 2 (𝑃 6 ) β€² pada Teorema 2
  2. V. KESIMPULAN Berdasarkan hasil penelitian yang telah dilakukan, dapat disimpulkan bahwa duplikasi dari graf lintasan (𝐷 2 (𝑃 𝑛 )) memiliki pelabelan selimut ajaib super berbentuk 𝐷 2 (𝑃 π‘š ) untuk 𝑛 β‰₯ 4 dan 3 ≀ π‘š ≀ 𝑛 -1 dengan konstanta ajaib untuk semua selimut adalah βˆ‘ 𝑓(𝐷 2 (𝑃 π‘š ) (𝑠
  3. = βˆ‘ 𝑓(𝐷 2 (𝑃 π‘š ) (𝑠+1) ).
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