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: ððš is a graph possessing n copies of the graph
G, which means, ððš =
âi-1nGi where each
ðšð = ðš.
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 ððâĻðū1
Theorem 4.4: The comb graph
ððâĻðū1 is said to be a pentagonal
pyramidal graceful graph for all 2âĪn.
Proof: Assume ðš to be a comb graph
ððâĻðū1.Then V(ðš) = { ui,
wi âķ ðĪâððð 1 âĪ ð âĪ n}
E(G) = {ðĒððĒð+1 : ðĪâððð 1 âĪ ð âĪ r â 1}
⊠{ðĒððĪð âķ ðĪâððð 1 âĪ ð âĪ r}
Hence ðš contains 2ð vertices and 2ð â 1 edge.
Let ð = 2ð â 1.
Define ÎĶ âķ (ðš) â {0,1,2, âĶ , Bð } as follows.
ÎĶ(ðĒ1)= 0
ÎĶ(ðĒð) = ÎĶ(ðĒðâ1) â
Bð â(ðâ2) ðð ð ðð ððð 2 âĪ ð âĪ r
ÎĶ(ðĒðâ1) + Bð â(ðâ2) ðð ð ðð ððĢðð 2 âĪ ð âĪ
r
ÎĶ(ðĪ1) = B2ð +1
ÎĶ(ðĪð) = ÎĶ(ðĒð) +
Bð +(ðâ1), 2 âĪ ð âĪ r.
ÎĶ is injective.
ÎĶâ the induced edge function defined from V(ðš) â
{B1, B2, âĶ,
B2ðâ1} is as given below.
ÎĶâ (ðĒððĒð+1 ) =
Bðâð , 1 âĪ ð âĪ r â 1
ÎĶ(ðĒ1ðĪ1 ) = B2ð +1
ÎĶ(ðĒððĪð ) = Bð +(ðâ1), 2 âĪ
ð âĪ r.
Clearly ÎĶâ is a bijection and ÎĶâ((ðš)) = {B1, B2, âĶ ,
B2ðâ1}.
Hence ðš permits pentagonal pyramidal graceful labeling.
Hence the comb ððâĻðū1 is a pentagonal
pyramidal graceful graph for all ð âĨ 2.
Example 4.5: Figure 3 is a representation of pentagonal
pyramidal graceful labeling of ð5âĻðū1