Extended Vertex Odd Mean Labeling in Certain Graphs

Abstract

Mean labeling was introduced and studied in some graphs by Somasundaram and Ponraj [8].
It is defined as an injective function f : V → { x : 0 ≤ x ≤ q} and the edge labels for each edge uv is allotted
from {{x : 1 ≤ x ≤ q} by the induced function f∗ as the mean value of f(u) and f(v) whenever f(u) + f(v)
is even and [f(u) + f(v) + 1]/2 wheneverf(u) + f(v) is odd. The idea of odd mean labeling was introduced
by K. Manickam and M. Marudai [3]. It is a mean graph with vertex set V = {x : 0 ≤ x ≤ 2q − 1} and edge
set E = {2x − 1 : 1 ≤ x ≤ q}. N. Revathi introduced Vertex Odd Mean and Even Mean Labeling and proved
that Umbrella graph, Mangolian tent and K1 + Cn graphs admit these labeling [5].
Motivated by the above studies, here in this paper, we prove the existence of extended vertex odd mean
labeling in the duplicate graphs of path, comb, twig, star and bistar graphs.