Square Difference Prime Labeling for Duplication of Graphs

Abstract: Let 𝑮(𝑽,𝑬) be a graph with 𝒑 vertices and 𝒒 edges. Let 𝒇∶ 𝑽 (𝑮) → {𝟎,𝟏,𝟐,...,𝒑−𝟏} be a bijection such that the induced function 𝒇∗∶ 𝑬(𝑮) → 𝑵 defined by 𝒇𝒔𝒒𝒅𝒑∗(𝒖𝒗)=|[𝒇(𝒖)]𝟐−[𝒇(𝒗)]𝟐| for every𝒖𝒗∈𝑬(𝑮).If 𝒇𝒔𝒒𝒅𝒑∗ is injective, then 𝒇𝒔𝒒𝒅𝒑∗ is called square difference labeling of 𝑮. A graph 𝑮 which admits square difference labeling is called square difference graph. The greatest common incidence number (gcin) of a vertex v of degree 𝒗 > 𝟏 is defined as the greatest common divisor (g.c.d) of the labels of the incident edges on 𝒗. A square difference labeling 𝒇 is said to be a square difference prime labeling if for each vertex v of degree > 1 then 𝒈𝒄𝒊𝒏(𝒗)= 𝟏.In this paper we investigate the square difference prime labelling of Petal graph and duplication of petal graph