^{1}

^{2}

^{2}

^{1}

^{3}

^{4}

^{5}

^{1}

^{2}

^{3}

^{4}

^{5}

A total

We consider a simple and finite graph

Furthermore, labeling known as a vertex irregular total

A

The concept of vertex irregular total

Some results related to vertex irregular reflexive labeling have been studied by several researchers. Tanna et al. [

The following lemma and theorem will be used as a base construction of analysing the reflexive vertex strength of any graph with pendant vertex, namely, sunlet graph, helm graph, subdivided star graph, and broom graph.

For any graph

Proof. Let

It completes the proof.

Let

Given that a graph

Let

Define an injection

If

If

If

If

The vertex weight of point

From Table

Labeling of vertex and edge on pendant vertices.

0 | 0 | 0 | 0 | 2 | 2 | 4 | 4 | |||||

1 | 2 | 3 | ||||||||||

1 | 2 | 3 |

The vertex set apart from pendant vertices

Based on Case

It concludes the proof.

Let

Moreover, to determine the label of vertices

It concludes the proof.

For an illustration, see Figure

Define

Assign the labels of vertices and edges of pendants

Observe that the vertex weight on each pendant will be

The vertex weights of point (4),(7) are

When

Apart from pendants, assign the label of vertices

Observe, all pendant vertices and otherwise show a different vertex weight.

STOP.

The illustration of labeling on

Let

Let

Furthermore, we will show the upper bound of vertex irregular reflexive

Based on the above injection, the overall vertex weight sets are

It is easy to see that the above elements of set are all different. It concludes the proof.

Let

For the illustration of the vertex irregular reflexive,

Furthermore, we will show the upper bound of vertex irregular reflexive

For

For otherwise

Based on the above injection, the overall vertex weight sets of the subdivided star

It is easy to see that the above elements of the set are all different. It concludes the proof.

The illustration of labeling on

Let

Let

The Broom graph

Furthermore, we will show that

Define an injection

Based on the above injection, the overall vertex weight sets of

Moreover, to determine label of vertices

It is easy to see that the above elements of set

Given that the vertex weight

Assign the labels of vertices

When on point (ii), the label of edges is more than

STOP.

In this paper, we have studied the construction of the reflexive vertex

Determine an upper bound of reflexive vertex strength of any graph to find the gap between lower bound and upper bound, and continue to determine the exact values for reflexive vertex strength of any other special graphs

Determine the construction of the reflexive vertex

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The authors gratefully acknowledge from the support of CGANT Research Group—the University of Jember, Indonesia of year 2020.