4. Comb Graph
Definition 4.1: Consider graph G. From the graph each edge is broken into precisely two segments. This is done by the insertion of intermediate vertices in between two ends. The new graph obtained is known as a graph of subdivisions, which is denoted as S(G).
Definition 4.2: 𝑛𝐚 is a graph possessing n copies of the graph G, which means, 𝑛𝐚 = ⋃i-1nGi where each 𝐚𝑖 = 𝐚.
Definition 4.3: Comb graphs is the graph that is formed by connecting a single pendant edge to every vertex of a path which is represented by 𝑃𝑛âĻ€ðū1
Theorem 4.4: The comb graph 𝑃𝑛âĻ€ðū1 is said to be a pentagonal pyramidal graceful graph for all 2â‰Īn.
Proof: Assume 𝐚 to be a comb graph 𝑃𝑛âĻ€ðū1.Then V(𝐚) = { ui, wi âˆķ ð‘Īℎ𝑒𝑟𝑒 1 â‰Ī 𝑖 â‰Ī n}
E(G) = {ð‘Ē𝑖ð‘Ē𝑖+1 : ð‘Īℎ𝑒𝑟𝑒 1 â‰Ī 𝑖 â‰Ī r − 1} ∊ {ð‘Ē𝑖ð‘Ī𝑖 âˆķ ð‘Īℎ𝑒𝑟𝑒 1 â‰Ī 𝑖 â‰Ī r}
Hence 𝐚 contains 2𝑛 vertices and 2𝑛 − 1 edge.
Let 𝑠 = 2𝑛 − 1.
Define ÎĶ âˆķ (𝐚) → {0,1,2, â€Ķ , B𝑠 } as follows.
ÎĶ(ð‘Ē1)= 0
ÎĶ(ð‘Ē𝑖) = ÎĶ(ð‘Ē𝑖−1) − B𝑠−(𝑖−2) 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 2 â‰Ī 𝑖 â‰Ī r
ÎĶ(ð‘Ē𝑖−1) + B𝑠−(𝑖−2) 𝑖𝑓 𝑖 𝑖𝑠 𝑒ð‘Ģ𝑒𝑛 2 â‰Ī 𝑖 â‰Ī r
ÎĶ(ð‘Ī1) = B2𝑠+1
ÎĶ(ð‘Ī𝑖) = ÎĶ(ð‘Ē𝑖) + B𝑠+(𝑖−1), 2 â‰Ī 𝑖 â‰Ī r.
ÎĶ is injective.
ÎĶ∗ the induced edge function defined from V(𝐚) → {B1, B2, â€Ķ, B2𝑛−1} is as given below.
ÎĶ∗ (ð‘Ē𝑖ð‘Ē𝑖+1 ) = B𝑛−𝑖 , 1 â‰Ī 𝑖 â‰Ī r – 1
ÎĶ(ð‘Ē1ð‘Ī1 ) = B2𝑠+1
ÎĶ(ð‘Ē𝑖ð‘Ī𝑖 ) = B𝑠+(𝑖−1), 2 â‰Ī 𝑖 â‰Ī r.
Clearly ÎĶ∗ is a bijection and ÎĶ∗((𝐚)) = {B1, B2, â€Ķ , B2𝑛−1}.
Hence 𝐚 permits pentagonal pyramidal graceful labeling.
Hence the comb 𝑃𝑛âĻ€ðū1 is a pentagonal pyramidal graceful graph for all 𝑛 â‰Ĩ 2.
Example 4.5: Figure 3 is a representation of pentagonal pyramidal graceful labeling of 𝑃5âĻ€ðū1