3. Power of three- acyclic graph
Definition 3.1: A power of three- acyclic graph
Hr consists of acyclic graph Hr-1 for
every r ≥ 0 which is associated together with the one’s root being the
leftmost child of the other’s root, the power of three acyclic graphs is
denoted by Hr. H0 comprises a solitary
vertex. The power of three-acyclic graph Hr is an
arranged non-cyclic diagram characterized recursively.
Hr consists of the power of three- acyclic graph
Hr −1 which is associated with each other, ie., the
leftmost child of the other root. Note that the vertices in
Hr are 3r.
Theorem 3.2: Every power of three trees is a pentagonal
pyramidal graceful labeling graph.
Proof: Assume G to be a tree containing s vertices.
Assume V (G) = {vp : 1≤ p ≤ s} as the set of vertex G
and
E(G) = {𝑣p𝑣p+1:1 ≤ p ≤ s-1} as the set
of edge of G.
Hence G has s vertices and s-1 edges.
Let c = s-1.
Consider a function Φ: V(G) →{0,1,2,…, Bc}
defined as stated below.
Φ (𝑣1) = 0
Φ(𝑣2 ) = Bc
Φ (vp ) = Φ (𝑣p−1) –
Bc−(p−2) if p is odd and 3 ≤ p ≤ s.
= Φ (𝑣p−1) + Bc−(p−2) if p is even and 3
≤ p ≤ s
Let Φ * be the induced edge labeling of f
Then Φ (𝑣1𝑣2) = B
Φ*(vpvp+1) = Bc−(p−1) ;
2 ≤ p ≤ s-1.
The induced edge labels B1, B2,…,
Bc are separate, sequential pentagonal pyramidal
numbers.
Hence G, the graph is proved to be a pentagonal pyramidal graceful.
Example 3.3: Figure 2 shows the Pentagonal pyramidal graceful
labeling of power of three trees.