Journal of Prime Research in Mathematics

# Vertex-magic total labelings of disconnected graph

Slamin
Mathematics Education Study Program, FKIP, Universitas Jember, Jalan Kalimantan
37 Jember 68121 Indonesia.
Visiting Professor at School of Mathematical Sciences, GC University, 68-B New Muslim
Town, Lahore, Pakistan.
A.C. Prihandoko
Mathematics Education Study Program, FKIP, Universitas Jember, Jalan Kalimantan
37 Jember 68121 Indonesia.
T. B.
Setiawan
Mathematics Education Study Program, FKIP, Universitas Jember, Jalan Kalimantan
37 Jember 68121 Indonesia.
F. Rosita
Mathematics Education Study Program, FKIP, Universitas Jember, Jalan Kalimantan
37 Jember 68121 Indonesia.
B. Shaleh
Mathematics Education Study Program, FKIP, Universitas Jember, Jalan Kalimantan
37 Jember 68121 Indonesia.

$$^{1}$$Corresponding Author: slamin@unej.ac.id

### Abstract

Let G be a graph with vertex set $$V = V (G)$$ and edge set $$E = E(G)$$ and let $$e = |E(G)|$$ and $$v = |V (G)|$$. A one-to-one map $$λ$$ from $$V ∪ E$$ onto the integers $$\{1, 2, …, v + e\}$$ is called vertex magic total labeling if there is a constant $$k$$ so that for every vertex $$x$$, $$λ+\sum λ(xy)=k$$. where the sum is over all vertices $$y$$ adjacent to $$x$$. Let us call the sum of labels at vertex x the weight $$w_{λ}(x)$$ of the vertex under labeling $$λ$$; we require $$w_{λ}(x) = k$$ for all $$x$$. The constant $$k$$ is called the magic constant for $$λ$$. In this paper, we present the vertex magic total labelings of disconnected graph, in particular, two copies of isomorphic generalized Petersen graphs $$2P(n, m)$$, disjoint union of two non-isomorphic suns $$S_{m} ∪ S_{n}$$ and t copies of isomorphic suns $$tS_{n}$$.

#### Keywords:

Vertex magic total labeling, disconnected graph, generalized Petersen graph, sun.