2. Star Graph
Theorem 2.1: For all values of s, the star graph š¾1,s is a pentagonal pyramidal graceful.
Proof: Consider V(š¾1,s) = {ui: 1≤ i≤ s+1}.
Assume E(š¾1,s) = { un+1ui : 1≤ i≤ s}.
Define an injection Φ : V(š¾1,s)→{0,1,2,…, Bc } by
Φ(ui) = Bš‘– if 1≤ i≤ s and
Φ(un+1) = 0.
Thus Φ prompts a bijection Φp : E(š¾1,s) →{1,6,18,…Bc }.
Hence the star graph š¾1,s is a pentagonal pyramidal graceful for all values of s.