Communications on Applied Nonlinear Analysis 

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Topological Cordial Labeling of some Graphs 

Dr. S. Selestin Lina 

Assistant Professor, Department of Mathematics, Nanjil Catholic College of Arts and 

Science, Kaliyakkavilai, Kanyakumari District, Affiliated to Manonmaniam 

Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. 

E-mail: selestinlina@gmail.com 

  

 

Article History: 

Received: 02-01-2025 

Revised: 25-02-2025 

Accepted: 20-03-2025 

Abstract  

B.D. Acharya [3] introduced the notion of set – valuation as set analogue of 

number valuation as introduced by A. Rosa [5].  Let G be a graph and X, a 

non-empty set. Define an injective function f:V(G)→2^X such that {f(V(G))}  

is a topology on X. If the induced function f^* on E(G) is defined by 

 f^* (uv)={■(1  if f(u)∩f(v)  is not an empty set and singleton set@0  

otherwise                                                                              )┤     for every 

uv∈E(G) such that 

 |e_f (0)-e_f (1)|≤1 where e_f (0)= number of edges labeled with 0 and〖 e〗

_f (1)= number of edges labeled with 1 then f is a topological cordial labeling 

and a graph which admits such a labeling is called topological cordial graph. 

In this paper we proved Dodecahedral graph, Paley graph and some 

constructed graphs are topological cordial graph.  

Key words:-Dodecahedral graph, Paley graph and topological cordial graph. 

 

Introduction  

             The graphs treated in this paper are simple. For standard terminology and notations 

we follow F. Harary [4]. Given a graph 𝐺 = (𝑉, 𝐸), we can relate it to different topological 

structures. The relation between topology and graph theory is undergone many investigations. 

In 1983 Acharya [3] established another link between graph theory and point – set topology. 

He defined a set – indexer as follows: Let 𝐺 =  (𝑉, 𝐸) be a graph, X any non – empty set and 

2𝑋   denote the set of all subsets of  𝑋. A set – indexer of 𝐺 is an injective set valued function 

𝑓 ∶  𝑉(𝐺) → 2𝑋  such that the induced function  𝑓∗ ∶  𝐸(𝐺) →  2𝑋 − {𝜙} defined by 

𝑓∗(𝑣1𝑣2 )  =  𝑓 (𝑣1) ∆ 𝑓 (𝑣2) for every 𝑣1𝑣2 ∈ 𝐸(𝐺) is also injective , where ∆  denotes the 

symmetric difference of sets. A graph 𝐺 =  (𝑉, 𝐸) is said to be a bitopological graph if there 

exist a set indexer 𝑓: 𝑉(𝐺)  → 2𝑋  such that 𝑓(𝑉) and 𝑓∗(𝐸) ∪ {𝜙} are both topologies on the 

corresponding ground set. Let 𝐺 be a graph and 𝑋, a non-empty set. Define an injective 

mailto:1selestinlina@gmail.com


Communications on Applied Nonlinear Analysis 

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function 𝑓: 𝑉(𝐺) → 2𝑋 such that {𝑓(𝑉(𝐺))} is a topology on 𝑋. If the induced function 𝑓∗ on 

𝐸(𝐺) is defined by  

𝑓∗(𝑢𝑣) = {
1  if 𝑓(𝑢) ∩ 𝑓(𝑣) is not an empty set and singleton set

0  otherwise                                                                              

  

for every 𝑢𝑣 ∈ 𝐸(𝐺) such that  |𝑒𝑓(0) − 𝑒𝑓(1)| ≤ 1 where 𝑒𝑓(0) = number of edges labeled 

with 0 and 𝑒𝑓(1) = number of edges labeled with 1 then 𝑓 is a topological cordial labeling 

and a graph which admits such a labeling is called topological cordial graph. This definition 

is defined and introduced in [8]. In this paper we proved Dodecahedral graph, Paley graph 

and some constructed graphs are topological cordial graph.  

1.Preliminaries 

Definition 1.1  The Dodecahedral graph is a 3-connected graph with 20 vertices and 30 

edges. 

Definition 1.2  A complete bipartite graph or biclique is a special kind of bipartite graph 

where every vertex of the first set is connected to every vertex of the second set. 

Definition 1.3   The double star 𝑆(𝑛, 𝑚), where 𝑛 ≥ 𝑚 ≥ 0, is the graph consisting of the 

union of two stars 𝐾1,𝑛 𝑎𝑛𝑑 𝐾1,𝑚 together with a line joining their centers. 

Definition 1.4 The Paley graph of order q with q a prime power is a graph on q nodes with 

two nodes adjacent if their difference is a square in the finite field GF(q). This graph is 

undirected when q=1 (mod 4). Simple Paley graphs therefore exist for orders 5, 9, 13, 17, 

25,…..     

2.Topological Cordial Labeling 

Definition 2.1  Let 𝐺 be a graph and 𝑋, a non-empty set. Define an injective function 

𝑓: 𝑉(𝐺) → 2𝑋 such that {𝑓(𝑉(𝐺))} is a topology on 𝑋. If the induced function 𝑓∗ on 𝐸(𝐺) is 

defined by  

𝑓∗(𝑢𝑣) = {
1  if 𝑓(𝑢) ∩ 𝑓(𝑣) is not an empty set and singleton set

0  otherwise                                                                              

  



Communications on Applied Nonlinear Analysis 

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for every 𝑢𝑣 ∈ 𝐸(𝐺) such that  |𝑒𝑓(0) − 𝑒𝑓(1)| ≤ 1 where 𝑒𝑓(0) = number of edges labeled 

with 0 and 𝑒𝑓(1) = number of edges labeled with 1 then 𝑓 is a topological cordial labeling 

and a graph which admits such a labeling is called topological cordial graph. 

2.Topological Cordial Labeling of named graphs 

Theorem 2.1. Dodecahedral graph is topological cordial graph. 

Proof: Let 𝐺 be Dodecahedral graph with 20 vertices and 30 edges. Let 𝑉(𝐺) = {𝑣𝑖/1 ≤ 𝑖 ≤

5} ∪  {𝑢𝑖/1 ≤ 𝑖 ≤ 10} ∪ {𝑤𝑖/1 ≤ 𝑖 ≤ 5}   and 𝐸(𝐺) = {𝑣𝑖𝑣𝑖+1/1 ≤ 𝑖 ≤ 𝑖 +

1, 𝑤ℎ𝑒𝑟𝑒 𝑣𝑖+1 = 𝑣𝑖} ∪ { 𝑢𝑖𝑢𝑖+1/1 ≤ 𝑖 ≤ 𝑖 + 1, 𝑤ℎ𝑒𝑟𝑒 𝑢𝑖+1 = 𝑢𝑖} ∪ {𝑤𝑖𝑤𝑖+1/1 ≤ 𝑖 ≤ 𝑖 +

1, 𝑤ℎ𝑒𝑟𝑒 𝑤𝑖+1 = 𝑤𝑖}  ∪ {𝑣𝑖𝑢2𝑖−2 /2 ≤ 𝑖 ≤ 4} ∪ {𝑤𝑖𝑢2𝑖−1/1 ≤ 𝑖 ≤ 5} ∪ {𝑣1𝑢10}  . Let 𝑋 =

{1,2, … ,20}. Now, define 𝑓: 𝑉(𝐺) → 2𝑋  by 

𝑓(𝑣1) = 𝜙, 𝑓(𝑣2) = {1}, 𝑓(𝑣3) = {2}, 𝑓(𝑣4) = {1,2,3}, 𝑓(𝑣5) = {4}, 𝑓(𝑣6) = {3,4} , 

𝑓(𝑣7) = {1,2}, 𝑓(𝑣8) = {2,3}, 𝑓(𝑣9) = {3}, 𝑓(𝑣10) = {1,4}, 𝑓(𝑣11) = {1,3,4}, 𝑓(𝑣12) =

{2,3,4}, 𝑓(𝑣13) = {1,2,3,4}, 𝑓(𝑣14) = {1,3}, 𝑓(𝑣15) = {2,4}, 𝑓(𝑣16) = {1,2,4}, 𝑓(𝑣17) =

{1,2,3,4,5}, 𝑓(𝑣18) = {1,2, … .6}, 𝑓(𝑣19) = {1,2, … . .7}, 𝑓(𝑣20) = 𝑋  

Then the vertex labels are distinct and {𝑓(𝑉(𝐺))} is a topology on 𝑋. 

The induced function 𝑓∗ on 𝐸(𝐺) is defined as follows:  

𝑓∗(𝑢𝑣) = {
1 if 𝑓(𝑢) ∩ 𝑓(𝑣) is not an empty set and singleton set

0 otherwise                                                                             

 

for every 𝑢𝑣 ∈ 𝐸(𝐺). Then, |𝑒𝑓(0) − 𝑒𝑓(1)| = 15 − 15 = 0 ≤ 1 where 𝑒𝑓(0) = number of 

edges labeled with 0 and 𝑒𝑓(1) = number of edges labeled with 1. Hence 𝑓 is a topological 

cordial labeling. Thus 𝐺 is topological cordial graph. 

Illustration 2.1.  Dodecahedral graph is topological cordial graph. 



Communications on Applied Nonlinear Analysis 

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Fig. 2.1 

 

Theorem 2.2  A Paley graph of order 13 is a topological cordial graph. 

Proof: Let 𝐺 be a Paley graph of order 13. Thus it has 13 vertices and 39 edges. Let 𝑉(𝐺) =

{𝑣𝑖/1 ≤ 𝑖 ≤ 13} an 𝐸(𝐺) = {𝑣𝑖𝑣𝑖+1/1 ≤ 𝑖 ≤ 13 where 𝑣13 = 𝑣1} ∪ {𝑣𝑖𝑣𝑖+3/1 ≤ 𝑖 ≤ 10} ∪

{𝑣1𝑣𝑖+4/1 ≤ 𝑖 ≤ 9} ∪ {𝑣𝑖𝑣𝑖+10/1 ≤ 𝑖 ≤ 3} ∪ {𝑣𝑖𝑣𝑖+9/1 ≤ 𝑖 ≤ 4}. Let 𝑋 = {1,2, … ,13}. 

Define 𝑓: 𝑉(𝐺) → 2𝑋  by  𝑓(𝑣1) = 𝜙, 𝑓(𝑣𝑖+1) = {1,2, . . . , 𝑖}, 1 ≤ 𝑖 ≤ 3,  𝑓(𝑣𝑖) =

{1,2, … , 𝑖 + 1}, 8 ≤ 𝑖 ≤ 12, 𝑓(13) = 𝑋. Then the vertex labels are distinct and {𝑓(𝑉(𝐺))} is 

a topology on 𝑋. The induced function 𝑓∗ on 𝐸(𝐺) is defined as follows: 

𝑓∗(𝑢𝑣) = {
1  if 𝑓(𝑢) ∩ 𝑓(𝑣) is not an empty set and singleton set

0  otherwise                                                                              

 

for every  𝑢𝑣 ∈ 𝐸(𝐺). Then, |𝑒𝑓(0) − 𝑒𝑓(1)| ≤ 1 where 𝑒𝑓(0) = number of edges labeled 

with 0 and  𝑒𝑓(1) = number of edges labeled with 1. Hence 𝑓 is a topological cordial 

labeling. Thus 𝐺 is topological cordial graph. 

Illustration 2.2 A Paley graph of order 13 is a topological cordial graph. 

 

 

 

 



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ig 2.2 

3.Topological Cordial Labeling of generalized graphs 

Theorem 3.1. The graph 𝐾(𝑚, 𝑛) is topological cordial graph, if 3 ≤ 𝑛 ≤ 6. 

Proof: Let 𝐾(𝑚, 𝑛) be a complete bipartite graph. 

Let 𝑉(𝐾) = {𝑣𝑖/1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑤𝑗/1 ≤ 𝑗 ≤ 𝑛} and 𝐸(𝐾) = {𝑣𝑖𝑤𝑗/1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑛, 

where   3 ≤ 𝑛 ≤ 6}  Let 𝑋 = {1,2,3, … , 𝑛 − 1}, Define 𝑓: 𝑉(𝐾) → 2𝑋 . We label the vertices 

and edges satisfying the condition of topology. Therefore the vertex labels are distinct and 

{𝑓(𝑉(𝐾))}  is a topology on 𝑋.The induced function 𝑓∗ on 𝐸(𝐾) is defined as follows:  

𝑓∗(𝑢𝑣) = {
1 if 𝑓(𝑢) ∩ 𝑓(𝑣) is not an empty set and singleton set

 0 otherwise                                                                               

 

for every  𝑢𝑣 ∈ 𝐸(𝐾). Then, |𝑒𝑓(0) − 𝑒𝑓(1)| ≤ 1 where 𝑒𝑓(0) = number of edges labeled 

with 0 and  𝑒𝑓(1) = number of edges labeled with 1. Hence 𝑓 is a topological cordial 

labeling. Thus 𝐾 is topological cordial graph. 

Illustration 3.1  𝐾(5,5) is topological cordial graph. 



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Fig 3.1 

Theorem 3.2 A double star 𝑆(𝑛, 𝑚) is a topological cordial graph.  

Proof: Let 𝐺 be a double star 𝑆(𝑛, 𝑚). 

Let  𝑉(𝐺) = {𝑢0, 𝑢1, 𝑢2, … … . 𝑢𝑛} ∪ {𝑣0, 𝑣1, 𝑣2, … . . 𝑣𝑚} and  𝐸(𝐺)  = {𝑢0𝑣0} ∪ {𝑢0𝑢𝑖/1 ≤

𝑖 ≤ 𝑛} ∪ {𝑣0𝑣𝑖/1 ≤ 𝑖 ≤ 𝑚} . 

Then G has 𝑚 + 𝑛 + 2  𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑎𝑛𝑑 𝑛 + 𝑚 + 1 𝑒𝑑𝑔𝑒𝑠 Let 𝑋 = {1,2, … … 𝑛 + 𝑚 + 2} 

 Define 𝑓: 𝑉(𝐺) → 2𝑋 . We label the vertices and edges satisfying the condition of topology. 

Then the vertex labels are distinct and {𝑓(𝑉(𝐺))}  is a topology on 𝑋. The induced function 

𝑓∗ on 𝐸(𝐺)  is defined as follows:  

𝑓∗(𝑢𝑣) = {
1  if 𝑓(𝑢) ∩ 𝑓(𝑣) is not an empty set and singleton set

0  otherwise                                                                              

 

for every  𝑢𝑣 ∈ 𝐸(𝐺). Therefore, |𝑒𝑓(0) − 𝑒𝑓(1)| ≤ 1 where 𝑒𝑓(0) = number of edges 

labeled with 0 and 𝑒𝑓(1) = number of edges labeled with 1. Hence 𝑓 is a topological cordial 

labeling.  Thus 𝐺 is topological cordial graph. 

Illustration 3.2 A double star 𝑆(2,4) is a topological cordial graph. 



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Fig. 3.2 

4.Conclusion  

In this paper deals with topological cordial graphs. The aim of this paper is to make some 

progress to a better understanding of topological cordial labeling.  

References : 

[1] Acharya B.D., Set indexers of a graph and set – graceful graphs, Bull. Allahabad Math. 

Soc., 16 (2001), 1-23 

[2] Acharya B.D,  Germina K.A , Princy K.L and Rao S.B., Topologically set graceful 

graphs , Paper under revision. 

[3] Acharya B.D., Set valuations and their applications, MRI Lecture note in Applied 

Mathematics, No.2, Mehta Research Institute of Mathematics and Mathematical 

Physics, 1983. 

[4] Germina K.A , Bindhu K.Thomas., On Bitopological Graphs, International Journel of 

Algorithm, Computing and Mathematics,vol.4 No.1, Feb.2011. 

[5] Haraey F., Graph Theory, Addison Wesley, reading Massachusetts, 1969. 

[6] Joseph A Gallian 2015, ‘A Dynamic Survey of Graph Labeling’, The Electronic Journal 

of Combinatorics. 

[7] Rosa A., On certain valuations of the vertices of a graph , Gorden and Breach, New York 

and Dunod, Paris, 1967, Proceedings of the International Symposium in Rome. 

[8] Selestin Lina S, Asha S, ‘On Topological Cordial Graphs’, Journal of Science and 

Technology, vol.5, Jan-Feb 2020, 25-28. 

[9] Selestin Lina S, Asha S, ‘Topological cordial labeling of some graphs’, Malaya Journal 

of Matematik, Vol.9, No.1, 861-863,2021.