A Lipschitz labeling of a graph G with vertex set V(G) with |V(G)| = m and edge set E(G) with |E(G)| = n ≥ 1 is a function f: V(G) ∪ E(G) → {1, 2, . . ., m + n} which satisfy the following conditions.

(i) f is bijective.

(ii) f (v) ∈ {1, 2, . . ., m} for all v ∈ V(G)

(iii) f (x) ∈ {m + 1, m + 2, . . ., m + n} for all x ∈ E(G)

(iv) There exists a positive integer L such that

| f (x) − f (y)| ≤ L|xf − yf |

for all x, y ∈ E(G) and xf = min {f (u), f (v)} for x = uv.

A graph which admits a Lipschitz labeling is called a Lipschitz graph. In this work, we focus on examining the existence of Lipschitz labeling for a fork graph with two prongs, an h graph, a perfect binary tree and a flag graph.