Ratio Mathematica                                                                  Volume 44, 2022 

 

 
 

 

 

Polygonal Graceful Labeling of Some Simple 

Graphs 

 
A. Rama Lakshmi 1 

M.P. Syed Ali Nisaya2  

 

 

Abstract 

Let 𝐺 = (𝑉, 𝐸) be a graph with 𝑝
 
vertices and 𝑞

 
edges.  Let 𝑉(𝐺) and𝐸(𝐺)

 
be the vertex 

set and edge set of 𝐺respectively.  A polygonal graceful labeling of a graph 𝐺 is an 
injective function 𝜂: 𝑉(𝐺) → 𝑍+, where 𝑍+ is a set of all non-negative integers that 
induces a bijection 𝜂*: 𝐸(𝐺) → {𝑃𝑠(1),𝑃𝑠(2), . . . , 𝑃𝑠(𝑞)}, where 𝑃𝑠(𝑞) is the 𝑞

𝑡ℎ polygonal 

number such that
 
𝜂*(𝑢𝑣) = |𝜂(𝑢) − 𝜂(𝑣)| for every edge 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺).  A graph 

which admits such labeling is called a polygonal graceful graph. For 𝑠 = 3, the above 
labeling gives triangular graceful labeling. For 𝑠 = 4, the above labeling gives tetragonal 
graceful labeling and so on. In this paper, polygonal graceful labeling is introduced and 

polygonal graceful labeling of some simple graphs is studied.
 
 

 

Keywords:  Polygonal numbers, Polygonal graceful labeling, Polygonal graceful graph. 

 

AMS Classification: 05C123 

 

 

 

 

 

 

 
1Research Scholar (Part Time-Internal), Department of Mathematics, The M.D.T Hindu college, Affiliated 

to Manonmaniam Sundaranar University, Tirunelveli–627010, Tamil Nadu, India. 

Mail Id: guruji.ae@gmail.com 
2 Assistant Professor, Department of Mathematics, The M.D.T Hindu college, Affiliated to Manonmaniam 

Sundaranar University, Tirunelveli–627010, Tamil Nadu, India. Mail Id: 

syedalinisaya@mdthinducollege.org 
3Received on June 9th, 2022. Accepted on Sep 5st, 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.883. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement 

9

mailto:guruji.ae@gmail.com
mailto:syedalinisaya@mdthinducollege.org


A. Rama Lakshmi and M.P. Syed Ali Nisaya  

 

1. Introduction 

We shall consider a simple, undirected and finite graph 𝐺 = (𝑉, 𝐸) on 𝑝 = |𝑉|vertices 
and 𝑞 = |𝐸|edges.  For all standard terminology, notations and basic definitions, we 
follow Harary [2] and for number theory, we follow Apostal [1].  A graph labeling is an 

assignment of integers to the vertices or the edges or both subject to certain conditions.  

If the domain of the mapping is the set of vertices (edges/both) then the labeling is called 

a vertex (edge/ total) labeling.  Rosa [6] introduced –valuation of a graph.  Golomb [4] 

called it as graceful labeling.  For a detailed survey of graph labeling, one can refer Gallian 

[3].  Ramesh and Syed Ali Nisaya [5] introduced some more polygonal graceful labeling 

of path.  Here, we shall recall some definitions which are used in this paper. 

 

2. Preliminaries  

Definition 2.1.  The star graph 𝐾1,𝑛of order 𝑛 + 1 is a tree on 𝑛 edges with one vertex 
having degree 𝑛 and other vertices having degree 1. 
 

Definition 2.2.  A graph 𝑆(𝐺)
 
which can be obtained from a given graph by breaking up 

each edge into one (or) more segment by inserting intermediate vertices between its two 

ends is called subdivision graph.  It is denoted by 𝑆(𝐺). 
 

Definition 2.3.  A path 𝑃𝑛is obtained by joining𝑢𝑖to the consecutive vertices𝑢𝑖+1 for 1 ≤
𝑖 ≤ 𝑛 − 1. 
 

Definition 2.4.  A coconut tree 𝐶𝑇(𝑛, 𝑚)
 
is a graph obtained from the path 𝑃𝑛  by 

appending m new pendant edges at an end vertex. 

 

Definition 2.5.  Let 𝑃2 be a path on two vertices and let 𝑢 and 𝑣 be the vertices of 𝑃2.  
From 𝑢, there are m pendant vertices say 𝑢1, 𝑢2, . . . , 𝑢𝑚and from 𝑣,  there are 𝑛 pendant 
vertices say𝑣1, 𝑣2, . . . , 𝑣𝑛.  The resulting graph is a bistar𝐵𝑚,𝑛.  
 

Definition 2.6. A graceful labeling of a graph 𝐺 is an injective function 𝑓: 𝑉(𝐺) →
{0,1,2, . . . , 𝑞} that induces a bijection 𝑓*: 𝐸(𝐺) → {1,2, . . . , 𝑞} of the edges of  𝐺 defined 
by

 
𝑓*(𝑒) = |𝑓(𝑢) − 𝑓(𝑣)|,∀𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺).  The graph which admits such a labeling is 

called a graceful graph.  

 

Definition 2.7.  A polygonal number is a number represented as dots (or) pebbles 

arranged in the shape of regular polygon. If 𝑠
 
is the number of sides in a polygon, the 

formula for the 𝑛𝑡ℎ𝑠–gonal number 𝑃𝑠(𝑛) is 𝑃𝑠
(𝑛) =

(𝑠−2)(𝑛)2−(𝑠−4)(𝑛)

2
. For 𝑠 = 3

 
gives 

triangular numbers.  For 𝑠 = 4
  
gives tetragonal numbers and so on. 

  

3. Main results 

Definition 3.1:    Let 𝐺 = (𝑉, 𝐸) be a graph with 𝑝
 
vertices and 𝑞

 
edges.  Let 𝑉(𝐺) 

and𝐸(𝐺)
 
be the vertex set and edge set of 𝐺respectively.  A polygonal graceful labeling 

10



Polygonal Graceful Labeling of Some Simple Graphs 

 

 

of a graph 𝐺 is an injective function 𝜂: 𝑉(𝐺) → 𝑍+, where 𝑍+ is a set of all non-negative 
integers that induces a bijection 𝜂*: 𝐸(𝐺) → {𝑃𝑠(1), 𝑃𝑠(2), . . . , 𝑃𝑠(𝑞)}, where 𝑃𝑠(𝑞) is the 
𝑞𝑡ℎ polygonal number such that

 
𝜂*(𝑢𝑣) = |𝜂(𝑢) − 𝜂(𝑣)| for every edge 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺).  

A graph which admits such labeling is called a polygonal graceful graph. For 𝑠 = 3, the 
above labeling gives triangular graceful labeling. For 𝑠 = 4, the above labeling gives 
tetragonal graceful labeling and so on. 

 

Example 3.2: The hexagonal graceful labeling of 𝐾1,5is given in figure (1) 

 
Figure (1) 

 

Theorem 3.3:  The star graph 𝐾1,𝑛 admits polygonal graceful labeling. 

Proof: Consider the star graph 𝐾1,𝑛. Let 𝑉(𝑘1,𝑛) = {𝑣, 𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛} 
and𝐸(𝑘1,𝑛) =

{𝑣𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛}.   
Then 𝐾1,𝑛 has 𝑛 + 1 vertices and 𝑛 edges. Define 𝜂: 𝑉(𝐾1,𝑛) → {0,1,2, . . . , 𝑃𝑠(𝑛)} as 
follows. 

𝜂(𝑣) = 0 
 

 

 

= 𝑃𝑠(𝑖),1 ≤ 𝑖 ≤ 𝑛 
Clearly 𝜂 is injective and the induced edge labels are the first 𝑛 polygonal numbers. Hence 
the star graph 𝐾1,𝑛 admits polygonal graceful graph. 
 

Example 3.4:  The pentagonal graceful labeling of 𝐾1,7 is shown in the following figure 
(2) 

. 

Figure (2) 

 

Theorem 3.5: 𝑆(𝐾1,𝑛), the subdivision of the star 𝐾1,𝑛 admits polygonal graceful labeling. 
Proof:  Consider the subdivision of the star graph 𝑆(𝐾1,𝑛).  Let 𝑉(𝑆(𝐾1,𝑛)) =
{𝑣, 𝑣𝑖, 𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛} and 𝐸(𝑆(𝐾1,𝑛)) = {𝑣𝑣𝑖, 𝑣𝑖𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛}.  Thus 𝑆(𝐾1,𝑛) has 2𝑛 +

1 vertices and 2𝑛 edges. Define 𝜂: 𝑉(𝑆(𝐾1,𝑛)) → {0,1,2, . . . , 𝑃𝑠(2𝑛)} 
as follows 

 𝜂(𝑣) = 0 

ni
isis

v
i


−−−

= 1,
2

)4()2(
)(

2



11



A. Rama Lakshmi and M.P. Syed Ali Nisaya  

 

𝜂(𝑣𝑖) =
(𝑠 − 2)(2𝑛 − 𝑖 + 1)2 − (𝑠 − 4)(2𝑛 − 𝑖 + 1)

2
, 1 ≤ 𝑖 ≤ 𝑛 

= 𝑃𝑠(2𝑛 − 𝑖 + 1),1 ≤ 𝑖 ≤ 𝑛
 

𝜂(𝑢𝑖) = 𝜂(𝑣𝑖) +
(𝑠 − 2)(𝑛 − 𝑖 + 1)2 − (𝑠 − 4)(𝑛 − 𝑖 + 1)

2
, 1 ≤ 𝑖 ≤ 𝑛 

= 𝜂(𝑣𝑖) + 𝑃𝑠(𝑛 − 𝑖 + 1), 1 ≤ 𝑖 ≤ 𝑛
 Clearly 𝜂 is injective and the induced edge labels are the first 2𝑛 polygonal numbers. 

Hence 𝑆(𝐾1,𝑛) admits polygonal graceful graph. 
 

Example 3.6:  The triangular graceful labeling of 𝑆(𝐾1,3) is shown in the following figure 
(3). 

 
Figure (3) 

 

Theorem 3.7:  The path of length 𝑛 is a polygonal graceful graph for all 𝑛 ≥ 3. 
Proof.  Let 𝐺 = (𝑉, 𝐸) be the path of length 𝑛 with the vertex set 𝑉 = {𝑣1, 𝑣2, . . . , 𝑣𝑛}and 
the edge set 𝐸 = {𝑣𝑖𝑣𝑖+1/1 ≤ 𝑖 ≤ 𝑛 − 1}.  Then 𝐺 has 𝑛 vertices and 𝑛 − 1 edges. Define 
𝜂: 𝑉(𝐺) → {0,1,2, . . . , 𝑃𝑠(𝑛 − 1)} as follows. 

𝜂(𝑣2𝑖−1) = (𝑖 − 1)((𝑠 − 2)𝑛 − (𝑠 − 2)𝑖 + 1),𝑖 = 1,2, . . , ⌊
𝑛 + 1

2
⌋
 

𝜂(𝑣2𝑖) =
1

2
(𝑛 − 1)[(𝑠 − 2)𝑛 − 2𝑠 + 6] − [(𝑠 − 2)𝑛 − (𝑠 − 2)𝑖 − (𝑠 − 3)](𝑖 − 1),𝑖

= 1,2, . . , ⌊
𝑛

2
⌋ 

Clearly 𝜂 is injective and the induced edge labels are the first 𝑛 − 1 polygonal numbers. 
Hence the path of length 𝑛 is a polygonal graceful graph for all 𝑛 ≥ 3. 
 

Example 3.8: The heptagonal graceful labeling of path of length 6 is given in figure (4). 
 

 
Figure (4) 

 

Theorem 3.9: Coconut tree 𝐶𝑇(𝑛, 𝑚) is a polygonal graceful labeling∀𝑛 ≥ 1, 𝑚 ≥ 2. 

Proof: Consider the graph coconut tree 𝐶𝑇(𝑛, 𝑚). Let 𝑉(𝐶𝑇(𝑛, 𝑚)) = {𝑣, 𝑣𝑖, 𝑢𝑗: 1 ≤ 𝑖 ≤

𝑛, 1 ≤ 𝑗 ≤ 𝑚 − 1} and 𝐸(𝐶𝑇(𝑛, 𝑚)) = {𝑣𝑣𝑖, 𝑣𝑢1, 𝑢𝑗𝑢𝑗+1: 1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚 − 1} .  

Then 𝐶𝑇(𝑛, 𝑚) has 𝑚 + 𝑛 vertices and 𝑚 + 𝑛 − 1 edges. Define𝜂: 𝑉(𝐶𝑇(𝑛, 𝑚)) →
{0,1,2, . . . , 𝑃𝑠(𝑚 + 𝑛 − 1)} as follows. 

𝜂(𝑣) = 0
 

𝜂(𝑣𝑖) =
(𝑠 − 2)(𝑚 + 𝑛 − 𝑖)2 − (𝑠 − 4)(𝑚 + 𝑛 − 𝑖)

2
, 1 ≤ 𝑖 ≤ 𝑛 

12



Polygonal Graceful Labeling of Some Simple Graphs 

 

 

= 𝑃𝑠(𝑚 + 𝑛 − 𝑖),1 ≤ 𝑖 ≤ 𝑛
 

𝜂(𝑢1) =
(𝑠 − 2)(𝑚 − 1)2 − (𝑠 − 4)(𝑚 − 1)

2
 

= 𝑃𝑠(𝑚 − 1)  

𝜂(𝑢𝑗) = 𝜂(𝑢𝑗−1) +
(𝑠−2)(𝑚−𝑗)2−(𝑠−4)(𝑚−𝑗)

2
, 𝑖𝑓𝑗𝑖𝑠𝑜𝑑𝑑𝑎𝑛𝑑2 ≤ 𝑗 ≤ 𝑚 − 1 

𝜂(𝑢𝑗−1) −
(𝑠 − 2)(𝑚 − 𝑗)2 − (𝑠 − 4)(𝑚 − 𝑗)

2
, 𝑖𝑓𝑗𝑖𝑠𝑒𝑣𝑒𝑛𝑎𝑛𝑑2 ≤ 𝑗 ≤ 𝑚 − 1

 
 

 

Clearly 𝜂
 
is an injective function.  Let 𝜂 ⥂*

 
be the induced edge labeling of 𝜂. Then  

𝜂*(𝑣𝑣𝑖) =
(𝑠 − 2)(𝑛 + 𝑚 − 𝑖)2 − (𝑠 − 4)(𝑛 + 𝑚 − 𝑖)

2
, 1 ≤ 𝑖 ≤ 𝑛 

= 𝑃𝑠(𝑛 + 𝑚 − 𝑖),1 ≤ 𝑖 ≤ 𝑛  

𝜂*(𝑣𝑢1) =
(𝑠 − 2)(𝑚 − 1)2 − (𝑠 − 4)(𝑚 − 1)

2
 

= 𝑃𝑠(𝑚 − 1)  
  

𝜂*(𝑢𝑗𝑢𝑗+1) =
(𝑠 − 2)(𝑚 − 𝑗 − 1)2 − (𝑠 − 4)(𝑚 − 𝑗 − 1)

2
, 1 ≤ 𝑗 ≤ 𝑚 − 2 

= 𝑃𝑠(𝑚 − 𝑗 − 1),1 ≤ 𝑗 ≤ 𝑚 − 2 
Hence the induced edge labels are the first 𝑚 + 𝑛 − 1polygonal numbers. Hence 
𝐶𝑇(𝑛, 𝑚) is a polygonal graceful graph. 
 

Example 3.10. The octagonal graceful labeling of coconut tree 𝐶𝑇(5,6) is given in the 
following figure (5). 

 
Figure (5) 

 

Theorem 3.11. The bistar 𝐵𝑚,𝑛 admits polygonal graceful labeling. 

Proof.: Consider the graph 𝐵𝑚,𝑛. Let 𝑉(𝐵𝑚,𝑛) = {𝑢, 𝑣, 𝑢𝑖, 𝑣𝑗/1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛} and 

𝐸(𝐵𝑚,𝑛) = {𝑢𝑣, 𝑢𝑢𝑖, 𝑣𝑣𝑗/1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛}.  Then 𝐵𝑚,𝑛 has 𝑚 + 𝑛 + 2 vertices and 

𝑚 + 𝑛 + 1 edges. Define𝜂: 𝑉(𝐵𝑚,𝑛) → {0,1,2, . . . , 𝑃𝑠(𝑚 + 𝑛 + 1)} by 𝜂(𝑢) = 0 

12),()(

12),()()(

1

1

−−−

−−+=

−

−

mjandevenisjifjmPu

mjandoddisjifjmPuu

sj

sjj





13



A. Rama Lakshmi and M.P. Syed Ali Nisaya  

 

𝜂(𝑣) =
(𝑠 − 2)(𝑚 + 𝑛 + 1)2 − (𝑠 − 4)(𝑚 + 𝑛 + 1)

2
 

= 𝑃𝑠(𝑚 + 𝑛 + 1)𝜂(𝑢𝑖) =
(𝑠 − 2)(𝑚 + 𝑛 − 𝑖 + 1)2 − (𝑠 − 4)(𝑚 + 𝑛 − 𝑖 + 1)

2
, 1 ≤ 𝑖

≤ 𝑚 
= 𝑃𝑠(𝑚 + 𝑛 − 𝑖 + 1),1 ≤ 𝑖 ≤ 𝑚 

 
 

𝜂(𝑣𝑗) = 𝜂(𝑣) −
(𝑠 − 2)(𝑛 − 𝑗 + 2)2 − (𝑠 − 4)(𝑛 − 𝑗 + 2)

2
, 1 ≤ 𝑗 ≤ 𝑛 

= 𝜂(𝑣) − 𝑃𝑠(𝑛 − 𝑗 + 2),1 ≤ 𝑗 ≤ 𝑛
 Clearly 𝜂 is injective and the induced edge labels are the first 𝑚 + 𝑛 + 1polygonal 

numbers. Hence 𝐵𝑚,𝑛admits polygonal graceful graph. 
 

Example 3.12:  The hexagonal graceful labeling of 𝐵5,4 is given in the following figure 
(6). 

 
Figure (6) 

4. Conclusion 

In this paper, polygonal graceful labeling is introduced and polygonal graceful labeling 

of some simple graphs is studied. This work contributes several new results to the theory 

of graph labeling. 

 

References 
 

[1] M. Apostal, Introduction to Analytic Number Theory, Narosa Publishing House, 

Second Edition – (1991). 

[2] Frank Harary, Graph Theory, Narosa Publishing House – (2001).  

[3] J.A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of 

Combinatorics, 16(2013), #DS6. 

[4] S.W. Golomb, How to number a graph, Graph Theory and Computing, R.C. Read, 

Academic Press, New York (1972), 23-37. 

[5] D. S. T. Ramesh and M.P. Syed Ali Nisaya, Some More Polygonal Graceful Labeling 

of Path, International Journal of Imaging Science and Engineering, Vol. 6, No.1, January 

2014, 901-905.  

[6] A. Rosa, On Certain Valuations of the vertices of a graph, Theory of Graphs, (Proc. 

Internet Symposium, Rome, 1966, Gordon and Breach N.Y. and Dunod Paris (1967), 

349-355. 

14