PELABELAN ANTI AJAIB JARAK PADA SUATU GRAF PETERSEN DIPERUMUM

Salah satu jenis pelabelan pada graf adalah pelabelan jarak yang merupakan pelabelan graf berdasarkan jarak antara titik-titiknya. Pelabelan jarak ini disebut distance magic labeling (pelabelan ajaib jarak) jika setiap titik mempunyai bobot pelabelan jarak yang sama. Pelabelan jarak ini disebut distance antimagic labeling (pelabelan anti ajaib jarak) jika setiap titik mempunyai bobot pelabelan jarak yang berbeda. Yang membentuk suatu deret. Tulisan ini membahas tentang pelabelan anti ajaib jarak pada graf petersen diperumum yaitu G= P(n, m) dengan n ≥ 3, 1 ≤ m <n/2 suatu graf teratur berderajat 3 yang mempunyai 2n titik dan 3n sisi. Lebih lanjut, tulisan ini juga membahas tentang pelabelan (a,d)-anti ajaib jarak -{1} pada suatu graf petersen diperumum.

Orthogonal labeling

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let ∆</span><span>G </span><span>be the maximum degree of a simple connected graph </span><span>G</span><span>(</span><span>V,E</span><span>). An injective mapping </span><span>P </span><span>: </span><span>V </span><span>→ </span><span>R</span><span>∆</span><span>G </span><span>is said to be an orthogonal labeling of </span><span>G </span><span>if </span><span>uv,uw </span><span>∈ </span><span>E </span><span>implying (</span><span>P</span><span>(</span><span>v</span><span>) </span><span>− </span><span>P</span><span>(</span><span>u</span><span>)) </span><span>· </span><span>(</span><span>P</span><span>(</span><span>w</span><span>) </span><span>− </span><span>P</span><span>(</span><span>u</span><span>)) = 0, where </span><span>· </span><span>is the usual dot product defined in Euclidean space. A graph </span><span>G </span><span>which has an orthogonal labeling is called an orthogonal graph. This labeling is motivated by the existence of several labelings defined by some algebraic structure, i.e. harmonious labeling and group distance magic labeling. In this paper we study some preliminary results on orthogonal labeling. One of the early result is the fact that cycle graph with even vertices are orthogonal, while ones with odd vertices are not. The main results in this paper state that any graph containing </span><span>K</span><span>3 </span><span>as its subgraph is non-orthogonal and that a graph </span><span>G</span><span>′ </span><span>obtained from adding a pendant to a vertex in orthogonal graph </span><span>G </span><span>is orthogonal. In the end of the paper we state the corollary that any tree is orthogonal.<br /> </span></p></div></div></div>