^{1}

We prove that all uniform

Labeling of a graph is an assignment of labels (numbers) to its vertices or/and edges or faces, which satisfy some conditions. These are different from coloring problems since some properties and structures of numbers such as ordering, addition, and subtraction used here are not properties of colors. Graph labelings have several applications in many fields. They have found usage in various coding theory problems, including the design of good radar-type codes, synch-set codes, and convolutional codes with optimal autocorrelation properties. They facilitate the optimal nonstandard encodings of integers. They have also been applied to determine ambiguities in X-ray crystallographic analysis, to design of a communication network addressing system, to determine optimal circuit layouts, and to problems in additive number theory.

Graham and Sloane [

A

A uniform

To prove our results, we name the vertices of any

Order to name the vertices.

Uniform

In this section, we list a few existing labelings which are useful for the development of this paper. Here, we consider a graph

An injective function

This was introduced by Graham and Sloane [

A graph which has a harmonious labeling is called a harmonious graph.

An injective function

This was introduced by Chang et al. [

A graph which has a strongly

An injective function

This was introduced by Grace [

A graph which has a sequential labeling is called a sequential graph.

Every uniform

Let

To prove

Define a labeling

That is, the edges receive the labels

That is, the edges receive the labels

That is, the edges receive the labels

That is, the edges receive the labels

That is, the edges receive the labels

That is, the edges receive the labels

Hence,

Hence,

Every uniform

Let

Define a labeling

The odd vertices

That is, the edge labels are

That is,

Every uniform

Let

Define a labeling

The odd vertices

That is,

Every uniform

Let

Define a labeling

The author would like to thank the editors for their valuable comments.