Gaussian Twin Neighborhood Prime Labeling on Fan Digraphs

Abstract

Gaussian integers are complex numbers of the form \gamma=x+iy where x and y are integers and i^2=-1. The set of Gaussian integers is usually denoted by \mathbb{Z}[i]. A Gaussian integer \gamma=a+ib\in\mathbb{Z}[i] is prime if and only if either \gamma=\pm(1\pm i),N(\gamma)= a^2+b^2 is a prime integer congruent to 1(mod4), or \gamma=p+0i or =0+pi where p is a prime integer and |p|\equiv3(mod4). Let D=(V,A) be a digraph with |V|=n. An injective function f:V(D)\rightarrow\left[\gamma_n\right] is said to be a Gaussian twin neighborhood prime labeling of D, if it is both Gaussian in and out neighborhood prime labeling. A digraph which admits a Gaussian twin neighborhood prime labeling is called a Gaussian twin neighborhood prime digraph. In this paper, we introduce some definitions of fan digraphs. Further, we establish the Gaussian twin neighborhood prime labeling in fan digraphs using Gaussian integers.

Keywords

• gaussian, integers, neighborhood, prime, labeling, fan, double, digraphs