Some results on vertex-edge neighborhood prime labeling

Abstract

Let G be a graph with vertex set V (G) and edge set E(G). For u ∈ V (G), NV (u) = {w ∈ V (G)|uw ∈ E(G)} and NE(u) = {e ∈ E(G)|e = uv, for some v ∈ V (G)}. A bijective function f : V (G) ∪ E(G) → {1, 2, 3, . . . , |V (G) ∪ E(G)|} is said to be a vertex-edge neighborhood prime labeling, if for u ∈ V (G) with deg(u) = 1, gcd {f(w), f(uw)|w ∈ NV (u)} = 1 ; for u ∈ V (G) with deg(u) > 1, gcd {f(w)|w ∈ NV (u)} = 1 and gcd {f(e)|e ∈ NE(u)} = 1. A graph which admits vertex-edge neighborhood prime labeling is called a vertex-edge neighborhood prime graph. In this paper we investigate vertex-edge neighborhood prime labeling for generalized web graph, generalized web graph without central vertex, splitting graph of path, splitting graph of star, graph obtained by switching of a vertex in path, graph obtained by switching of a vertex in cycle, middle graph of path.