Example 2.2: A pentagonal pyramidal graceful labeling of star graph 𝐾1,8 is shown in Fig. 1.
Fig 1. 𝐾1,8-Star Graph
Definition 2.3: Let \(G_{1\ }=\ (V_{1},\ E_{1})\) and\(G_{2\ }=\ (V_{2},\ E_{2})\) be two graphs. Then the disjoint union of two graphs is \(G_{3\ }\) such that\(G_{3\ }=\ G_{1}\ \cup\ G_{2}\) whose vertex set\(V_{3\ }=\ V_{1}\ \cup\ V_{2}\) and the edge set\(E_{3\ }=\ E_{1}\ \cup\ E_{2}\)
Theorem 2.4: 𝐾1,m ∪ 𝐾1 ,s is a pentagonal pyramidal graceful labeling graph for all values of m, n satisfying m, s > 2.
Proof: Let G be a 𝐾1, a ∪ 𝐾1 ,bgraph for all a, b > 2.
Let V(G) = {𝑢 , 𝑢p , 𝑣, 𝑣q : 1 ≤ p ≤ a , 1 ≤ q ≤ b} and
E(G) = {𝑢𝑢p , 𝑣𝑣q : 1 ≤ p ≤ a , 1 ≤ q ≤ b}
Here G contains a + b + 2 vertices with a + b edges.
Let c = a + b
Define f : V(G) → {0,1,…, Bc} as follows
f (u) = 0
f (𝑢p ) = Bp , 1 ≤ p ≤ a
f (v) = f (𝑢𝑛−1) – 1.
f (𝑣q ) = B𝑛+q - f (v) , 1 ≤ q ≤ b.
Clearly, f is injective and f induces a bijective function f *: E(G) →{1,6,… Bc} as
f * ( 𝑢𝑢p ) = Bp , 1 ≤ p ≤ a
f * ( 𝑣𝑣q ) = B𝑛+q , 1 ≤ q ≤ b
Hence the edge labels are 1,6,18,… Bc.
Thus f is a pentagonal pyramidal graceful labeling of G.
Therefore, G = 𝐾1,a ∪ 𝐾1 ,b is a pentagonal pyramidal graceful labeling graph.