Power-3 Heronian odd Mean Labeling of Graphs

Abstract

In this article, we discuss Power-3 Heronian odd Mean Labeling for some families of graphs.A function is said to be Power-3 Heronian odd mean labeling of a graph G with q edges, if f is a bijective function from the vertices of G to the set{1,3,5,.......2p-1}such that when each edges uv is assigned the label. The resulting edge labels are distinct numbers.

Introduction

I. INTRODUCTION

In this paper, the graphs are taken as simple, finite and undirected. V(G) represents the vertex set and E(G) represents Edge set. A graph labeling is an assignment of integers to its vertices or edges subject to some certain conditions. A vertex labeling is a function of V to a set of labels. A graph with such a vertex labeling function is defined as Vertex – labeled graph.An edge labeling is a function of E to a set of labels and a graph with such a function is called an edge labeled graph. In this article path,triangular snake,caterpillar are discussed Power-3 Heronian odd Mean Labeling Of Graphs.

All Graphs in this paper are finite and undirected.  The symbols V(G) and E(G) denote the vertex set and edge set of a graph G.  The cardinality of the vertex set is called the order of G denoted by p.  The cardinality of the edge set is called the size of G denoted by q edges is called a (p,q) graph.  A graph labeling  is an assignment of integers to the vertices or edges.  Bloom and Hsu[2] extended the notion of graceful labeling to directed graphs.  Graceful signed graphs f(uv) is the difference between f(v) and f(v), that is f(uv) = f(v) – f(u). Shalini, Paul Dhayabaran [14] introduced the concept A Study on Root Mean Square Labelings in Graphs. Shalini, Paul Dhayabaran [13] defined An Absolute Differences of Cubic and Square Difference Labeling. Shalini, Gowri, Paul Dhayabaran [15] discussed An Absolute Differences of Cubic and Square Difference Labeling For Some Families of Graphs. Shalini, Sri Harini, Paul Dhayabaran [19] introduced Sum of an Absolute Differences of Cubic And Square Difference Labeling For Cycle Related Graphs. Shalini, Gowri, Paul Dhayabaran [16] studied An Absolute Differences of Cubic and Square Difference Labeling for Some Shadow and Planar Graphs. Shalini, Subha, Paul Dhayabaran [20] investigated A Study on Disconnected Graphs for an Absolute Difference Labeling. Shalini, Subha, Paul Dhayabaran [22] discussed A Study on Disconnected Graphs for Sum of an Absolute Difference of Cubic and Square Difference  Labeling. Shalini, Sri Harini, Paul Dhayabaran [21] extended Sum of an Absolute Differences of Cubic And Square Difference Labeling For Path Related Graphs. Shalini.P, S.A.Meena[25] introduced “Lehmer -4 mean labelling of graphs”.

II. BASIC DEFINITIONS

1. Definition 2.1

In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges)

2. Definition 2.2

Caterpillar is attained by removing the pendant vertices of a path from the tree. It has vertices and edges.

3. Definition 2.3

A Triangular snake Tm is attained by attaching every pair of vertices of a path to another new vertex. (i,e.,) we can replace each edge of a path Pn by a cyclic graph C3.Generally, it has vertices and edges.

4. Definition 2.4

A graph G is said to be power-3 Heroine odd Mean Labeling graph, if it admits power-3 Heroine odd Mean labeling.

Conclusion

In this article, we proved some families of graphs which admits Power-3 Heronian odd Mean Labeling .Therefore, Path, Triangular snake, Caterpillar are Power-3 Heronian Odd Mean Labeling.

References

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Copyright

Copyright © 2022 Dr. P Shalini, S. Tamizharasi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.