Some New Families of Mean Graphs

S K Vaidya,Lekha Bijukumar

doi:10.5539/jmr.v2n3p169

Abstract

We contribute some new results for mean labeling of graphs. It has been proved that the graphs obtained by the composition of paths Pm and P2 denoted by Pm[P2], the square of path Pn and the middle graph of path Pn admit mean labeling. We also investigate mean labeling for some cycle related graphs.

Highlights

We begin with simple,finite,connected and undirected graph G = (V(G), E(G)) with p vertices and q edges
Theorem-2.14: The middle graph of a path Pn denoted by M(Pn) admits mean labeling
Proof: Let v1,v2,. . . vn be the vertices and e1,e2,. . . , en−1 be the edges of path Pn.The middle graph of a path Pn denoted by M(Pn) is a graph with V[M(Pn)] = V(Pn) E(Pn) and two vertices are adjacent in M(Pn) if and only if they are adjacent in Pn or one is a vertex and other is an incident edge to that vertex in Pn

Summary

Introduction

2. Main Results Theorem-2.1: The composition of paths Pm and P2 denoted as Pm[P2] admits mean labeling except for m = 2. For i = m; f (ui, v1) = f (ui−1, v1) + 6 f (ui, v2) = f (ui−1, v2) + 3 In view of the above defined labeling pattern the graph under consideration admits mean labeling. Theorem-2.4:The graph obtained by duplication of an arbitrary vertex by a new edge in cycle Cn admits mean labeling.

Results

Conclusion