Example
2.5: In Figure 2 shows the pentagonal pyramidal graceful labeling of
𝐾1,4 ∪ 𝐾1,4
Fig 2. 𝐾1,4 ∪ 𝐾1,4-Star Graph
Theorem 2.6: 𝐾1,r ∪ 𝐵n ,s is a
pentagonal pyramidal graceful labeling the entire graph r ≥ 3 and n, s ≥
1
Proof: Assume G be a 𝐾1,r ∪ 𝐵n
,s graph for all r ≥ 3 and n, s ≥ 1.
Consider V(G) = {𝑢 , 𝑢p , 𝑣, 𝑣q , 𝑤,
𝑤𝑘 : 1 ≤ p ≤ r , 1 ≤ q ≤ n and 1 ≤ k ≤ s} and
E(G) = { 𝑢𝑢p , 𝑣𝑣q , 𝑣𝑤,
𝑤𝑤𝑘 : 1 ≤ p ≤ r, 1 ≤ q ≤ n and 1 ≤ 𝑘 ≤ s}
Here G contains r + n + s + 3 vertices with r+ n + s + 1 edges.
Let c = r + n + s + 1.
Define f : V(G) → {0,1,…, Bc} as follows
f (u) = 0
f (𝑢p ) = Bp , 1 ≤ p ≤ r
f (v) = f (𝑢𝑛−1) – 1.
f (𝑣q ) = B𝑛+q+1- f (v) , 1 ≤ q ≤ n
f (𝑤) = B𝑛+1- f (𝑣),
f (𝑤𝑘) = Bn+r+1+k - f (𝑤) , 1 ≤ 𝑘 ≤ s
It is evident that f is injective and
f prompts a bijective function f * : E(G) →{1,6,…,
Bc} as
f * ( 𝑢𝑢p ) = B𝑖 , 1 ≤ p ≤ r
f * ( 𝑣𝑣q ) = B𝑛+1+q , 1 ≤ q ≤ n
f * (𝑣𝑤) = B𝑛+1
f * ( 𝑤𝑤𝑘) = Br+n+1+k , 1 ≤ 𝑘 ≤ s
As a result, the edge labels are 1,6,…, Bc .
Thus f is a pentagonal pyramidal graceful labeling of G.
Therefore, G = 𝐾1,r ∪ 𝐵n ,s is a
pentagonal pyramidal graceful labeling graph.