Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

1928 
https://internationalpubls.com 

Odd Hamming Distance Labeling of Some Path Related Graphs 

 

E.Esakkiammal 1, K.Thirusangu2 and S.Seethalakshmi 3 

1,2 Department of Mathematics, S.I.V.E.T.  College, Gowrivakkam, Chennai,India. 

3Department of Mathematics, R.V. Govt. Arts, Chengalpattu,Chennai,India.      

1esakkiammal2682@gmail.com,2kthirusangu@gmail.com,3seetha0687@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract 

Binary data strings of equal length are compared using the metric called hamming 

distance. It is the number of bit positions in which the two binary strings differ. The 

hamming distance between two binary strings m and n of equal length is denoted by 

hd(m,n). We introduced the concept of Hamming distance labeling and odd Hamming 

distance labeling. In this paper, it is shown that Path graph, Star graph, One point union 

of Path graphs, Coconut tree, are Odd hamming distance labeled graphs and obtained 

their Odd hamming distance number. Both Hamming and Odd Hamming distance 

labeling are used to send secret messages in Cryptography. 

Keywords: Hamming Distance, Odd Hamming Distance Labeling, Path graph, Star 

graph, One-point union for Path of graphs, Coconut tree graph. 

 

1. Introduction  

Let G = (V, E) be a graph with vertex set V and edge set E. A Path graph 𝑃𝑚 , 𝑚 ≥ 1  is an alternating 

sequence of vertices and edges, beginning and ending with vertices in which each edge is incident with 

two vertices immediately preceding and following it. Edges and vertices appear only once in a path. A 

Path graph of length m has m+1 vertices and m edges[2].The complete bipartite graph of the form  𝐾1,𝑛  

is a Star graph with n+1 vertices and it is denoted by 𝑆𝑛, 𝑛 ≥ 1[3].The One point union of Path graph 

𝑃𝑚
𝑛 𝑛,𝑚 ≥ 2, is obtain by replacing each edge of a Star graph  𝐾1,𝑛 by Path graph 𝑃𝑚, where n is the 

number of pendant edges in Star graph and m is the length of the Path graph[6].A Coconut tree CT(n,m), 

𝑛 ≥ 2,𝑚 ≥ 1 is the graph obtained from the Path 𝑃𝑚 by appending n new pendant edges at an end 

vertex of 𝑃𝑚
[7]. 

 Graph labeling is a function defined on the vertex set or edge set subject to certain conditions enforced 

on the number of vertices p or on the number of edges q or on both p and q[1]. The concept of graph 

labeling was introduced in the year 1967 by Rosa and it was further developed by Graham and Sloane 

in 1980[4]. We introduced the concept of Hamming distance labeling and proved that some Path related 

graphs are Hamming distance graphs [8]. In this paper, prove the existence of Odd hamming distance 

labeling of some path related Graphs. Here the notation [𝑥]2 denotes the binary conversion of the 

number 𝑥. 

 

 

mailto:esakkiammal2682@gmail.com
mailto:kthirusangu@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

1929 
https://internationalpubls.com 

2. Odd Hamming Distance Labelling of some Graphs 

2.1. Definition  

Let G = (V, E) be a graph. A function  𝑓: 𝑉 → 𝑁 ∪ {0}  is said to be an Odd hamming distance labeling  

if  there  exist  an   induced function 𝑓∗ ∶ 𝐸 → {1,3,5, … , n} such that for every 𝑢𝑣 ∈ 𝐸, 𝑓∗(𝑢𝑣) =

ℎ𝑑([𝑓(𝑢)]2, [𝑓(𝑣)]2) satisfying the following conditions: 

(i) For every vertex 𝑣 𝜖 𝑉, the set of all edges incident with 𝑣 receive distinct odd labels. 

(ii) For every edge 𝑒 = 𝑢𝑣, the adjacent vertices 𝑢 and 𝑣 receive distinct labels. 

A graph which admits Odd hamming distance labeling is called Odd hamming distance graph.The Odd 

hamming distance number of a graph G is the least positive integer n such that  2𝑛 − 1 ≥ 𝑘, where 𝑘 

= max {𝑓(𝑣)/𝑣 ∈ 𝑉} and it is denoted by ηℎ𝑑
′ (G).  

2.2.1. Algorithm: Odd hamming distance labeling of 𝐏𝐦 graph 

Procedure: Vertex labeling of Pm graph, m ≥ 1 

Input: Path graph Pm 

V ← {vi  /0 ≤ i ≤ m}  

v0  ← 0; 

for i = 1 to m  do 

     vi  ←     {

1     if i ≡ 1(mod4)

6      if i ≡ 2(mod4)

2      if i ≡ 3(mod4)

5      if i ≡ 0(mod4)

; 

end for 

end procedure 

Output: The  labeled vertices of Path graph Pm. 

2.3.2.Theorem The Path graph 𝑃𝑚, 𝑚 ≥ 1 is an Odd hamming distance graph and the Odd hamming 

distance number is ηℎ𝑑
′ (𝑃𝑚) =  {

1    𝑖𝑓 𝑚 = 1
3    𝑖𝑓 𝑚 > 1

 . 

Proof: Let us consider the Path graph Pm with vertex set V = {vi/0 ≤ i ≤ m} and edge set  E =

{vivi+1 / 0 ≤ i ≤ m− 1}. Define a function 𝑓: V → N ∪ {0} such that   𝑓(𝑣𝑖) ≠ 𝑓(𝑣𝑖+1), 0 ≤ i ≤ m −

1 as given in the above algorithm 3.3.1. hence the adjacent vertices receive distinct labels.The edge 

labels are obtained as follows: 

f ∗(v0v1) = hd([f(v0)]2, [f(v1)]2)  = hd([0]2, [1]2)   = hd(00000, 00001) =1   

For 1 ≤ i ≤ m − 1 

Case (i): When  i ≡ 1(mod 4);f ∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) =  hd([1]2, [6]2) = 3 

Case (ii): When  i ≡ 2(mod4);f ∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) =  hd([6]2, [2]2) = 1  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

1930 
https://internationalpubls.com 

Case (iii): When  i ≡ 3(mod 4);f ∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) =  hd([2]2, [5]2) = 3 

Case (iv): When  i ≡ 0(mod4);f ∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) =  hd([5]2, [1]2) = 1  

From all the above cases, all the adjacent edges receive distinct odd labels. Hence the Path graph Pm 

admits Odd hamming distance labeling and the Odd hamming distance number is  ηℎ𝑑
′ (Pm) =

 {
1     𝑖𝑓 𝑚 = 1
3     𝑖𝑓 𝑚 > 1

. 

 

Figure 3.Odd Hamming Distance 𝑷𝟏𝟎 graph. 

3.3.3. Algorithm: Odd hamming distance labeling of 𝑺𝒏 graph. 

Procedure: Vertex labeling of 𝑆𝑛 graph 𝑛 ≥ 1. 

Input: Star graph 𝑆𝑛 

𝑉 ← {𝑣𝑖 /0 ≤ 𝑖 ≤ 𝑛} 

𝑣0  ← 0; 

for 𝑖 = 1 𝑡𝑜 𝑛  do 

 𝑣𝑖  ← 22𝑖−1 − 1; 

end for 

end procedure 

Output: The labeled vertices of star graph. 𝑆𝑛. 

3.3.4.Theorem The Star graph 𝑆𝑛, 𝑛 ≥ 1 is an Odd hamming distance graph  and the Odd hamming 

distance number is ηℎ𝑑
′ (𝑆𝑛) = 2𝑛 − 1,where n is the number of pendant edges. 

Proof: Let us consider the Star graph 𝑆𝑛 with vertex set 𝑉 = {𝑣𝑖 /0 ≤ 𝑖 ≤ 𝑛} and edge set     𝐸 =

{𝑣0𝑣𝑖 / 1 ≤ 𝑖 ≤ 𝑛}. Define a function 𝑓: 𝑉 ← 𝑁 ∪ {0}  such that 𝑓(𝑣0 ) ≠ 𝑓(𝑣𝑖 ), 1 ≤ 𝑖 ≤ 𝑛  as given 

in the above algorithm 3.3.3. Here all the adjacent vertices receive distinct labels.The edge labels are 

obtained as follows:  

For 1 ≤ 𝑖 ≤ 𝑛; 

𝑓∗(𝑣0𝑣𝑖) = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣𝑖)]2)  = ℎ𝑑([0]2, [2
2𝑖−1 − 1]

2
)  = 2𝑖 − 1.  

For each 𝑖, the corresponding vertex label and edge label are given in the following table: 

𝑖 1 2 3 4 5 6 7 ……

… 

N 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 9s (2025) 

  

1931 
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𝑓(𝑣𝑖) 1 7 31 127 511 2047 8191 ……

… 

22𝑛−1

− 1 

e = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣𝑖)]2) 1 3 5 7 9 11 13 ..……. 2n − 1 

 

where 𝑓(𝑣0) = 0.From the above table it is clear that all the adjacent edges receive distinct odd labels. 

Hence the star graph 𝑆𝑛 admits Odd hamming distance labeling and the Odd hamming distance number 

is   ηℎ𝑑
′ (𝑆𝑛) = 2𝑛 − 1 for any 𝑛 ≥ 1. 

3.3.5. Algorithm: Odd hamming distance labeling of 𝑷𝒎
𝒏  graph 

Procedure: Vertex labeling of 𝑃𝑚
𝑛, graph 𝑚, 𝑛 ≥ 2. 

Input: One point union of Path graph 𝑃𝑚
𝑛. 

𝑉 ← {𝑣𝑖𝑗  /1 ≤ 𝑖 ≤ 𝑛, 0 ≤ 𝑗 ≤ 𝑚; 𝑣10 = 𝑣20 = 𝑣30 = ⋯ = 𝑣𝑛0}  

𝑣0  ← 0; 𝑣0  = 𝑣10 = 𝑣20 = 𝑣30 = ⋯ = 𝑣𝑛0 

for 𝑗 = 1 𝑡𝑜 𝑚  do 

𝑣1𝑗  ←

{
 

 
1 if  𝑗 ≡ 1(𝑚𝑜𝑑4)

6 if  𝑗 ≡ 2(𝑚𝑜𝑑4)

 2 if  𝑗 ≡ 3(𝑚𝑜𝑑4)

5 if  𝑗 ≡ 0(𝑚𝑜𝑑4)

;   𝑣2𝑗  ←

{
 

 
7  if  𝑗 ≡ 1(𝑚𝑜𝑑4)

3 if  𝑗 ≡ 2(𝑚𝑜𝑑4)

4 if  𝑗 ≡ 3(𝑚𝑜𝑑4)

0 if  𝑗 ≡ 0(𝑚𝑜𝑑4)

; 

end for 

for 𝑖 = 3 𝑡𝑜 𝑛 do 

     𝑣𝑖1 ← 22𝑖−1 − 1; 𝑣𝑖2  ← 22𝑖−2 − 1; 

     𝑣𝑖3  ← 22𝑖−5 − 1; 𝑣𝑖4  ← 22𝑖−6 − 1; 

     for 𝑗 = 5 𝑡𝑜 𝑚 do 

      if 𝑖 = 3  

             𝑣𝑖𝑗 ← (𝑣(𝑖−1)(𝑗−4)); 

       else 

  𝑣𝑖𝑗 ← (𝑣(𝑖−2)(𝑗−2)); 

       end if 

      end for 

end for 

end procedure 

Output:  The labeled vertices of one point union of path graph 𝑃𝑚
𝑛. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

1932 
https://internationalpubls.com 

3.3.6.Theorem The one point union of path graphs 𝑃𝑚
𝑛 , 𝑚, 𝑛 ≥ 2 is an Odd hamming distance graph 

and the Odd hamming distance number ηℎ𝑑
′ (𝑃𝑚

𝑛) = 2𝑛 − 1.  

Proof: Let    us   consider    the     one    point    union    of   path   graphs   𝑃𝑚
𝑛   with   vertex   set 

𝑉 = {𝑣𝑖𝑗  /1 ≤ 𝑖 ≤ 𝑛, 0 ≤ 𝑗 ≤ 𝑚; 𝑣10 = 𝑣20 = 𝑣30 = ⋯ = 𝑣𝑛0}, let  𝑣0 = 𝑣𝑖0, 1 ≤ 𝑖 ≤ 𝑛 and edge 

set 𝐸 = {    𝑣𝑖𝑗𝑣𝑖(𝑗+1)     /   1 ≤ 𝑖 ≤ 𝑛,   0 ≤ 𝑗 ≤ 𝑚 − 1}. This graph has mn+1 vertices and mn edges. 

Define a function 𝑓:𝑉 → 𝑁 ∪ {0} such that 𝑓(𝑣𝑖𝑗) ≠ 𝑓(𝑣𝑖(𝑗+1)), 1 ≤ 𝑖 ≤ 𝑛 , 0 ≤ 𝑗 ≤ 𝑚 − 1 as given 

in the above algorithm 3.3.6. hence the adjacent vertices receive distinct labels.The edge labels are 

obtained as follows: 

For 1 ≤ 𝑖 ≤ 𝑛, 

ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣𝑖1)]2) = 2𝑖 − 1. Which is given in the following table, here 𝑓(𝑣0) = 0. 

𝑖 1 2 3 4 5 6 7 ……

… 

N 

𝑓(𝑣𝑖1) 1 7 31 127 511 2047 8191 ……

… 

22𝑛−1

− 1 

e 

= ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣𝑖1)]2) 

1 3 5 7 9 11 13 ……

… 

2𝑛 − 1 

 

For  1 ≤ 𝑖 ≤ 𝑛 

Case (i): when 𝑗 ≡ 1(𝑚𝑜𝑑4) 

𝑓∗(𝑣11𝑣12) = ℎ𝑑([𝑓(𝑣11)]2, [𝑓(𝑣12))]2) =  hd([1]2, [6]2) = 3. 

𝑓∗(𝑣21𝑣22) = ℎ𝑑([𝑓(𝑣21)]2, [𝑓(𝑣22))]2) =  hd([7]2, [3]2) = 1 

𝑖 3 4 5 6 7 ……… n 

𝑓(𝑣i1) 31 127 511 2047 8191 ………. 22𝑛−1

− 1 

𝑓(𝑣𝑖2) 15 63 255 1023 4095 ………. 22𝑛−2

− 1 

e 

= ℎ𝑑([𝑓(𝑣i1)]2, [𝑓(𝑣𝑖2)]2) 

1 1 1 1 1 1 1 

 

For  1 ≤ 𝑖 ≤ 𝑛, 5 ≤ 𝑗 ≤ 𝑚 

𝑓∗(𝑣𝑖𝑗𝑣𝑖(𝑗+1)) = ℎ𝑑([𝑓(𝑣𝑖𝑗)]2, [𝑓(𝑣𝑖(𝑗+1)))]2
) = 1. 



Communications on Applied Nonlinear Analysis 

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1933 
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𝑖 1 2 3 4 5 … n 

𝑓(𝑣ij) 1 7 7=𝑣(3−1)(j−4) 4=

𝑣(4−2)(j−2) 

1=𝑣(5−2)(j−2) … 𝑣(n−2)(j−2) 

𝑓(𝑣𝑖(𝑗+1)) 6 3 3=𝑣(3−1)(j−3) 0=

𝑣(4−2)(j−1) 

0=𝑣(5−2)(j−1) … 𝑣(n−2)(j−1) 

e 

= ℎ𝑑([𝑓(𝑣ij)]2,
[𝑓(𝑣𝑖(𝑗+1))]2

) 

3 1 1 1 1  1 

 

Case (ii): when 𝑗 ≡ 2(𝑚𝑜𝑑4) 

𝑓∗(𝑣12𝑣13) = ℎ𝑑([𝑓(𝑣12)]2, [𝑓(𝑣13))]2) =  hd([6]2, [2]2) = 1. 

𝑓∗(𝑣22𝑣23) = ℎ𝑑([𝑓(𝑣22)]2, [𝑓(𝑣23))]2) =  hd([3]2, [4]2) = 3. 

𝑖 3 4 5 6 7 ……… n 

𝑓(𝑣i2) 15 63 255 1023 4095 ………. 22𝑛−2 − 1 

𝑓(𝑣𝑖3) 1 7 31 127 511 ………. 22𝑛−5 − 1 

e 

= ℎ𝑑([𝑓(𝑣i2)]2, [𝑓(𝑣𝑖3)]2) 

3 3 3 3 3 3 3 

 

For  1 ≤ 𝑖 ≤ 𝑛, 6 ≤ 𝑗 ≤ 𝑚 

𝑓∗(𝑣𝑖𝑗𝑣𝑖(𝑗+1)) = ℎ𝑑([𝑓(𝑣𝑖𝑗)]2, [𝑓(𝑣𝑖(𝑗+1)))]2
) = 3. 

𝑖 1 2 3 4 5 … n 

𝑓(𝑣ij) 6 3 3=𝑣(3−1)(j−4) 0=

𝑣(4−2)(j−2) 

0=

𝑣(5−2)(j−2) 

… 𝑣(n−2)(j−2) 

𝑓(𝑣𝑖(𝑗+1)) 2 4 4=𝑣(3−1)(j−3) 7=

𝑣(4−2)(j−1) 

7=

𝑣(5−2)(j−1) 

… 𝑣(n−2)(j−1) 

e=

 ℎ𝑑([𝑓(𝑣ij)]2
, [𝑓(𝑣𝑖(𝑗+1))]2

) 

1 3 3 3 3 3 3 

Case (iii): when 𝑗 ≡ 3(𝑚𝑜𝑑4) 



Communications on Applied Nonlinear Analysis 

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For  1 ≤ 𝑖 ≤ 𝑛 

𝑖 1 2 3 4 5 … N 

𝑓(𝑣i3) 2 4 1 7=𝑣(4−2)(j−2) 31=𝑣(5−2)(j−2) … 𝑣(n−2)(j−2) 

𝑓(𝑣𝑖4) 5 0 0 3=𝑣(4−2)(j−1) 15=𝑣(5−2)(j−1) … 𝑣(n−2)(j−1) 

e 

= ℎ𝑑([𝑓(𝑣i1)]2, [𝑓(𝑣𝑖2)]2) 

3 1 1 1 1 1 1 

 

For  1 ≤ 𝑖 ≤ 𝑛, 7 ≤ 𝑗 ≤ 𝑚 

𝑓∗(𝑣𝑖𝑗𝑣𝑖(𝑗+1)) = ℎ𝑑([𝑓(𝑣𝑖𝑗)]2, [𝑓(𝑣𝑖(𝑗+1)))]2
) = 1. 

𝑖 1 2 3 4 5 … n 

𝑓(𝑣ij) 2 4 4=𝑣(3−1)(j−4) 7=𝑣(4−2)(j−2) 7=

𝑣(5−2)(j−2) 

… 𝑣(n−2)(j−2) 

𝑓(𝑣𝑖(𝑗+1)) 5 0 0=𝑣(3−1)(j−3) 3=𝑣(4−2)(j−1) 3=

𝑣(5−2)(j−1) 

… 𝑣(n−2)(j−1) 

e 

= ℎ𝑑([𝑓(𝑣ij)]2,
[𝑓(𝑣𝑖(𝑗+1))]2

) 

3 1 1 1 1 1 1 

 

Case (iv): when 𝑗 ≡ 0(𝑚𝑜𝑑4) 

𝑖 1 2 3 4 5 … n 

𝑓(𝑣i4) 5 0           0 3

= 𝑣(4−2)(j−2) 

15

= 𝑣(5−2)(j−2) 

… 𝑣(n−2)(j−2) 

𝑓(𝑣𝑖5) 3 7 7

= 𝑣(3−1)(j−3) 

4

= 𝑣(4−2)(j−1) 

1 = 𝑣(5−2)(j−1) … 𝑣(n−2)(j−1) 

 𝑒 =

ℎ𝑑([𝑓(𝑣i2)]2, [𝑓(𝑣𝑖3)]2) 

3 1 1 1 1 1 1 

 

For  1 ≤ 𝑖 ≤ 𝑛, 8 ≤ 𝑗 ≤ 𝑚 

𝑓∗(𝑣𝑖𝑗𝑣𝑖(𝑗+1)) = ℎ𝑑([𝑓(𝑣𝑖𝑗)]2, [𝑓(𝑣𝑖(𝑗+1)))]2
) = 3. 



Communications on Applied Nonlinear Analysis 

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1935 
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𝑖 1 2 3 4 5 … n 

𝑓(𝑣ij) 5 0 0=𝑣(3−1)(j−4) 3=𝑣(4−2)(j−2) 3=𝑣(5−2)(j−2) … 𝑣(n−2)(j−2) 

𝑓(𝑣𝑖(𝑗+1)) 1 7 7=𝑣(3−1)(j−3) 4=𝑣(4−2)(j−1) 4=𝑣(5−2)(j−1) … 𝑣(n−2)(j−1) 

e=

 ℎ𝑑([𝑓(𝑣ij)]2,
[𝑓(𝑣𝑖(𝑗+1))]2

) 

3 1 1 1 1 1 1 

 

From all the above cases, all the adjacent edges receive distinct odd labels. Hence the one point union 

of Path graphs 𝑃𝑚
𝑛, admits Odd hamming distance labeling and the Odd hamming distance number  

ηℎ𝑑
′ (𝑃𝑚

𝑛) is 2𝑛 − 1. 

 

Figure 5 Odd Hamming Distance 𝑷𝟓
𝟖 graph 

3.3.7. Algorithm: Odd hamming distance labeling of CT(n,m) graph 

Procedure: Vertex labeling of Coconut tree CT(n,m) 

Input: Coconut tree graph CT(n,m). 

V ← {{ui /0 ≤ i ≤  m} ∪ {vj /1 ≤ j ≤ n}} 

u0  ← 0; 

for i = 1 to m do 

ui  ← {

1      if  i ≡ 1(mod4)

6     if  i ≡ 2(mod4)

2     if  i ≡ 3(mod4)

5     if  i ≡ 0(mod4)

 

end fo 

for j = 1 to n do 

vi  ← {

6     if  i ≡ 1(mod4)

2     if  i ≡ 2(mod4)

5     if  i ≡ 3(mod4)

1     if  i ≡ 0(mod4)

 



Communications on Applied Nonlinear Analysis 

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1936 
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end for 

for j = 2 to n do 

                 vj ← 22j+1 − um + 1) 

end for 

end procedure 

Output: The labeled vertices of Coconut tree graph CT(n,m). 

3.3.8.Theorem The Coconut tree graph CT(n,m) is an Odd hamming distance graph and the Odd 

hamming distance number is ηℎ𝑑
′ (CT(n,m) ) = 2𝑛 + 1. 

Proof: Let us consider the coconut tree graph CT(n,m) with vertex set                    𝑉 =

{{𝑢𝑖 /0 ≤ 𝑖 ≤  𝑚} ∪ {𝑣𝑗 /1 ≤ 𝑗 ≤ 𝑛}} and edge set 𝐸 = {{𝑢𝑖𝑢𝑖+1 / 0 ≤ 𝑖 ≤ 𝑚 − 1} ∪ {𝑢𝑚𝑣𝑗 / 1 ≤

𝑗 ≤ 𝑛}}.This graph has m+n+1 vertices and m+n edges. Define a function 𝑓: 𝑉 → 𝑁 ∪ {0} such that       

𝑓(𝑢) ≠ 𝑓(𝑣) for any two adjacent vertices 𝑢 and 𝑣 as given in the above algorithm 3.3.7.Hence all the 

adjacent vertices receive distinct labels.The edge labels are obtained as follows: 

f ∗(𝑢0𝑢1) = hd([f(𝑢0)]2, [f(𝑢1)]2) = hd([0]2, [1]2) =1. 

For 1 ≤ 𝑖 ≤ 𝑚, where 𝑢𝑚+1 = 𝑣1 

Case (i): if 𝑖 ≡ 1(𝑚𝑜𝑑 4);  𝑓∗(𝑢𝑖𝑢𝑖+1) = ℎ𝑑([𝑓(𝑢𝑖)]2, [𝑓(𝑢𝑖+1)]2) =   hd([1]2, [6]2) = 3. 

Case (ii): if 𝑖 ≡ 2(𝑚𝑜𝑑 4); 𝑓∗(𝑢𝑖𝑢𝑖+1) = ℎ𝑑([𝑓(𝑢𝑖)]2, [𝑓(𝑢𝑖+1)]2) =  hd([6]2, [2]2) = 1. 

Case (iii): if 𝑖 ≡ 3(𝑚𝑜𝑑 4);  𝑓∗(𝑢𝑖𝑢𝑖+1) = ℎ𝑑([𝑓(𝑢𝑖)]2, [𝑓(𝑢𝑖+1)]2) =  hd([2]2, [5]2) = 3. 

Case (iv): if 𝑖 ≡ 0(𝑚𝑜𝑑 4);  𝑓∗(𝑢𝑖𝑢𝑖+1) = ℎ𝑑([𝑓(𝑢𝑖)]2, [𝑓(𝑢𝑖+1)]2) =  hd([5]2, [1]2) = 1. 

For 2 ≤ 𝑗 ≤ 𝑛 

Case (i): if 𝑚 ≡ 1(𝑚𝑜𝑑 4) 

𝑗 2 3 4 5 6 ……… n 

𝑓(𝑢m) 1 1 1 1 1 ………. 1 

𝑓(𝑣𝑗) 30 126 510 2046 8190 ………. 22j+1 − 2 

e = ℎ𝑑([𝑓(𝑢m)]2, [𝑓(𝑣𝑗)]2) 5 7 9 11 13 19 2n+1 

 

Case (ii): if 𝑚 ≡ 2(𝑚𝑜𝑑 4) 

𝑗 2 3 4 5 6 ……… n 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 9s (2025) 

  

1937 
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𝑓(𝑢m) 6 6 6 6 6 ………. 6 

𝑓(𝑣𝑗) 25 121 505 2041 8185 ………. 22j+1 − 7 

e 

= ℎ𝑑([𝑓(𝑢m)]2, [𝑓(𝑣𝑗)]2) 

5 7 9 11 13 19 2n+1 

 

Case (iii): if 𝑚 ≡ 3(𝑚𝑜𝑑 4) 

𝑗 2 3 4 5 6 ……… n 

𝑓(𝑢m) 2 2 2 2 2 ………. 2 

𝑓(𝑣𝑗) 29 125 509 2045 8189 ………. 22j+1 − 3 

e = ℎ𝑑([𝑓(𝑢m)]2, [𝑓(𝑣𝑗)]2) 5 7 9 11 13 19 2n+1 

 

Case (iv): if 𝑚 ≡ 0(𝑚𝑜𝑑 4) 

𝑗 2 3 4 5 6 ……… n 

𝑓(𝑢m) 5 5 5 5 5 ………. 5 

𝑓(𝑣𝑗) 26 122 506 2042 8186 ………. 22j+1 − 6 

e = ℎ𝑑([𝑓(𝑢m)]2, [𝑓(𝑣𝑗)]2) 5 7 9 11 13 19 2n+1 

 

From all the above cases, all the adjacent edges receive distinct odd labels. Hence the coconut tree 

graph CT(n,m) admits Odd hamming distance labeling and the Odd hamming distance number 

ηℎ𝑑
′ (CT(n,m)) is 2𝑛 + 1. 

3. Conclusion  

In this paper, the Odd hamming distance number of  some Path related graphs were obtained. 

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Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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1938 
https://internationalpubls.com 

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