On $\mathbb{Z}_k-$vertex-magic labeling of prime graph $PG(\mathbb{Z}_n)$

On $Z_{k} -$ vertex-magic labeling of prime graph $P G (Z_{n})$

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematics and Sciences, University of Brawijaya, Malang, Indonesia

Abstract

Let

G = (V (G), E (G))

be a graph,

(A, +)

be an Abelian group with identity

0_{A}

, and

(R, +, \cdot)

be a ring. The

A

-vertex-magic labeling of

G

is a mapping from

V (G)

A - {0_{A}}

such that the total labels of every adjacent vertex with

u

are equal for every

u

V (G)

. The prime graph over

R

, denoted by

P G (R)

, is a graph with

V (P G (R)) = R

such that

u v

is an edge if and only if

u R v = {0_{R}}

v R u = {0_{R}}

, for every vertex

u \neq v

. In this article, we discuss the

Z_{k}

-vertex-magic labeling of the prime graph over ther ring

Z_{n}

. We study some literature to develop the properties of

Z_{k}

-vertex-magic labeling of

P G (R)

. We investigate some classes of prime graphs over ring

Z_{n}

for

n = p, n = p^{2},

and

n = p q

, with

p \neq q

primes.

Keywords