Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 47, No. 1 (2025) pp. 169-180

R. Ponraj¹

VECTOR BASIS S-CORDIAL LABELING

OF FRIENDSHIP GRAPH, FAN GRAPH,

AND LILLY GRAPH

and

R. Jeya²

Abstract: Let

basis . Let  :V(G)  S be a map. For each xy assign the label

 x,y  , where  x,y  denotes the inner product of x and y. We say that

G

be a (p,q) graph. Let

V

be an inner product space with

S



is

a

vector basis S-cordial labeling if |_x _y|  1 and

|_i  _j|  1 where

_x

denotes the number of vertices labeled with the

vector x and

_i

denotes the number of edges labeled with the scalar i. A

graph with a vector basis S-cordial labeling is called a vector basis

S-cordial graph. In this paper, we investigate the vector basis S-cordial

labeling of certain graphs like friendship graph, fan graph, lilly graph,

bistar

graph,

crown

graph

and

armed

crown

graph

where

S = {(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)} is a basis inR⁴

.

Keywords: Friendship Graph, Fan Graph, Lilly Graph, Bistar Graph, and

Crown Graph.

Mathematics Subject Classification (2020) No.: 05C78.

1. Introduction

In this paper, we consider only finite, simple and undirected graph

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R. PONRAJ AND R. JEYA

G  (V(G)

,

E(G)) where V(G) and E(G) respectively, denote the vertex set and

edge set of G. Note that p  |V(G)| and q  |E(G)|denote the number of vertices

and edges of G respectively. The idea of graph labeling was first introduced by Rosa

in 1967 [16]. Vertex odd graceful labeling has studied in [5]. Baskar Babujee and

Shobana [3] have examined the prime and prime cordial labeling for some

special graphs. Radio geometric mean labeling of some star like graphs have

investigated in [8]. Parmar [18] proved that for the wheel, fan and friendship graphs

are edge vertex prime.

The join G₁G₂[6] of two graphs G₁and G₂is defined as the graph whose

vertex set is V(G₁)V(G₂) and the edge set consists of these edges which are in G₁

and in G₂and the edges contained by joining each vertex of G₁to each vertex of G₂

The fan graph F_n[18] is a graph that is constructed by joining all vertices of a path

to a further vertex, called center. That is,F_n K₁ P_n. Amutha and Uma Devi

.

P

n

[1] have explored the super graceful labeling for some families of fan graphs.

Barasara [2] proved that the comb is an edge and total edge product cordial. For a

dynamic survey on graph labeling, we refer to Gallian [6].

The friendship graph C₃(n)[6] can be constructed by joining n copies of the

cycle graph C₃with a common vertex, which becomes a universal vertex for the

graph. The corona G₁G₂[6] of two graphs G₁and G₂is obtained by taking one

copy of G₁and |V(G₁)| copies of G₂and joining each vertex of the i^thcopy of

G₂to the i^thvertex of G₁

.

The concept of cordial labeling was first introduced by I. Cahit [4]. Mitra and

Bhoumik [11] have introduced the tribonacci cordial labeling of graphs. Parthiban

and Sharma proved that the Lilly graph is a prime cordial graph in [13]. The Lilly

graph I_n,n  2 [13] can be constructed by two star graphs 2K_1,n,n  2 joining two

paths 2P ,n  2 with sharing a common vertex. That is,I_n 2K

 2P_n. For the

n

1,n

terminologies and different notations of graph theory, we refer the book of Harary [7]

and of algebra; we refer the book of Herstein [9]. Sum divisor cordial labeling of

theta graph was examined by Sugumaran and Rajesh in [17]. Difference cordial

labeling for plus and hanging pyramid graphs have studied in [12]. Prajapati and

A. Vantiya have proved that the triangular snake, double triangular snake,

quadrilateral snake, double quadrilateral snake are SD-prime cordial in [14]. Kaneria

VECTOR BASIS S-CORDIAL LABELING OF FRIENDSHIP GRAPH

…

171

et al. [10] have investigated the balanced mean cordial labeling and graph operations.

We have introduced new labeling called vector basis S-cordial labeling in

[15] and investigated the vector basis vector basis {(1,1,1,1), (1,1,1,0), (1,1,0,0),

(1,0,0,0)}-cordial labeling behavior of some standard graphs like path, cycle, comb,

star and complete graph. In this paper, we investigate the vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling of certain graphs like friendship graph,

fan graph, lilly graph, bistar graph, crown graph and armed crown graph.

2. Vector basis S-cordial labeling

Definition 2.1: Let G be a (p,q) graph. Let V be an inner product space

with basis S. Let  :V(G)  S be a map. For each xy assign the label x,y 

,

where  x,y  denotes the inner product of x and y. We say that



is a vector basis

S-cordial labeling if |_x _y|  1 and |_i  _j|  1 where

_x

denotes the number

of vertices labeled with the vector x and

_i

denotes the number of edges labeled with

the scalar i. A graph with a vector basis S-cordial labeling is called a vector basis

S- cordial graph.

Theorem 2.2: [15] The set S = {(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)} is a

basis for R⁴over R.

Theorem 2.3: [15] A graph G is vector basis {(1,0),(0,1)}-cordial if and only

if G is a cordial graph.

Theorem 2.4: [15] The path

P

is a vector basis {(1,1,1,1), (1,1,1,0),

n

(1,1,0,0), (1,0,0,0)}-cordial graph for all n  1

.

Theorem 2.5: [15] The cycle C_nis a vector basis {(1,1,1,1), (1,1,1,0),

(1,1,0,0), (1,0,0,0)}-cordial if and only if n  1,2,3 (mod4)

.

In this paper, we consider the inner product space ꢀꢁ and the standard

product  x,y   x₁y₁ x₂y₂   x_ny_nwhere x  (x₁,x₂, ,x_n)

y  (y₁,y₂, ,y_n) x_iy_i R

inner

,

.

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R. PONRAJ AND R. JEYA

3. Main Results

In this section, we consider the basis S = {(1,1,1,1), (1,1,1,0), (1,1,0,0),

(1,0,0,0)}.

Theorem 3.1: The friendship graph C₃(n) is a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial if and only if n  0,1,2 (mod4)

.

Proof: The friendship graph C₃(n) is a planar, undirected graph with 2n  1

vertices and 3n edges. Let V(C₃(n))  {u,u_i|1  i  2n} andE(C₃(n))  {uu_2i1

uu_2i,u_2i1u_2i|1  i  n} respectively be the vertex and edge sets of C₃(n).Then

,

|V(C₃(n))|p2n1 and |E(C₃(n))| q  3n . There are four case arises.

Case (i): n  0 (mod 4)

Letn  4k . Then, p  2n  1  8k  1. Next, we assign the vector (1,1,1,1)

to the vertex u. Assign the vector (1,1,1,1) to the vertices u₁,u₂, ,u_2k. We assign

the vector (1,1,1,0) to the next vertices u_2k1

,

u

_2k2, ,u_4k. Then assign the vector

_4k2, ,u_6k. Also assign the vector (1,0,0,0)

_6k2, ,u_8k

(1,1,0,0) to the next vertices u_4k1

to the remaining vertices u_6k1

,

u

,

u

.

Case (ii): n  1 (mod 4)

Let n  4k  1. Then, p  8k  3. Now, we assign the vector (1,1,1,1) to the

vertex u. So assign the vector (1,1,1,1) to the vertices u₁,u₂, ,u_2k. Assign the

vector (1,1,1,0) to the next vertices u_2k1

,

u_2k2,,u_4k. We assign the vector

(1,1,0,0) to the next vertices u_4k1_4k2, ,u_6k

,

u

.

Further, assign the vector

(1,1,1,0) to the vertexu_6k1. Assign the vector (1,0,0,0) to the vertex u_6k2. Then

assign the vector (1,1,0,0) to the vertex u_6k3. Finally, assign the vector (1,0,0,0) to

the remaining 2k  1 vertices u_6k4

,

u

_6k5, ,u_8k3

.

VECTOR BASIS S-CORDIAL LABELING OF FRIENDSHIP GRAPH

…

173

Case (iii): n  2 (mod 4)

Let n  4k  2. Then, p  8k  5. Also, we assign the vector (1,1,1,1) to the

vertex u. Assign the vector (1,1,1,1) to the vertices u₁,u₂, ,u_2k1. We assign the

vector (1,1,1,0) to the next vertices u_2k2

,

u_2k3,,u_4k2. Then assign the vector

(1,1,0,0) to the vertices _4k3,u_4k4, ,u_6k3

u

.

Moreover, assign the vector

(1,0,0,0) to the remaining 2k vertices

u

_6k4,u_6k5, ,u_8k4

.

Case (iv): n  3 (mod 4)

Let n  4k  3. Then p  8k  7 and q  12k  9. If we assign vector

(1,1,1,1) to the vertex u and we have to assign (1,1,1,1) to the 2k  1 vertices, then

2k1

2

₄

 2k  1

 2k  1 k  1  3k  2, a contradiction. But ₄ 3k  1 or

₄ k  1 according as the vertex u receive the vector (1,1,1,1) or not. This is a

contradiction since the size of C₃(n) is 12k  9

.

Clearly the above labeling pattern provides a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling for the friendship graph C₃(n) if

n  0,1,2 (mod 4)

.

Theorem 3.2: The fan graph F_nis a vector basis {(1,1,1,1), (1,1,1,0),

(1,1,0,0), (1,0,0,0)}-cordial if and only if n  0 (mod 4)

.

Proof: Let V(F_n)  {u,u_i|1  i  n} and E(F_n)  {uu_i|1  i  n}

{u_iu_i1|1  i  n  1} respectively be the vertex and edge sets of F_n. Then

V(F_n)  p  n  1 and E (F_n)  q  2n  1. There are four cases arises.

Case (i): n  0 (mod 4)

Let n  4k . Then, p  4k  1. Next, we assign the vector (1,1,1,1) to the

vertex u. Assign the vector (1,1,1,1) to the first k verticesu₁,u₂, ,u_k. Then, assign

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R. PONRAJ AND R. JEYA

the vector (1,1,1,0) to the next

k

vertices u_k1,u_k2, ,u_2k. Also, assign the vector

_2k1,u_2k2, ,u_3k. Moreover, assign the vector

_3k1,u_3k2, ,u_4k

(1,1,0,0) to the next vertices

k

u

(1,0,0,0) to the remaining k vertices

u

.

Case (ii): n  1(mod 4)

Let n  4k  1. Then p  n  1  4k  2  (k  1) (k  1) k  k

and

q  2n  1  8k  1  (2k  1)  2k  2k  2k . Clearly, ₄ 2k  1 or ₄ k

according as the vertex u receive the vector (1,1,1,1) or not. This is a contradiction

since the size of F_nis 8k  1

.

Case (iii): n  2 (mod 4)

Let

q  8k  3  (2k  1) (2k  1) (2k  1) 2k

according as the vertex u receive the vector (1,1,1,1) or not. We get a contradiction

since the size of F_nis 8k  3

n  4k  2

.

Then p  4k  3  (k  1) (k  1) (k  1) k and

.

Thus, ₄ 2k  1 or ₄ k

.

Case (iv): n  3 (mod 4)

Let n  4k  3. Then p  4k  4  (k  1) (k  1) (k  1) (k  1) and

q  8k  5  (2k  2)  (2k  1)  (2k  1) (2k  1). Hence, ₄ 2k  1 or

₄

 k

according as the vertex u receive the vector (1,1,1,1) or not. We get a contradiction

since the size of F_nis 8k  5

.

Clearly the above labeling method provides a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling for the fan graph F_nif n  0

(mod 4)

.

Example 3.3: The

following Figure

1 illustrates

the

vector basis

{(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling fan graph F₄

.

VECTOR BASIS S-CORDIAL LABELING OF FRIENDSHIP GRAPH

…

175

Figure 1

Vector basis {(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling of F₄

.

Theorem 3.4: The Lilly graph I_nis a vector basis {(1,1,1,1), (1,1,1,0),

(1,1,0,0), (1,0,0,0)}-cordial graph for all n  2

.

Proof:

Consider

the

Lilly

graph

I_n, n  2

.

Let

V(I_n)  {u,u_i,v_i|1  i  n  1}



{x_i, y_i|1  i  n}

and

E(I_n)  {ux_i, uy_i|1  i  n} {uv₁,uu₁,u_iu_i1,,v_iv_i1|1  i  n  2} respectively

be the vertex and edge sets of I_n. Then and

.

V(I_n)  p  4n  1

E(I_n)  q  4n  2

First, we assign the vector (1,1,1,1) to the vertex u. Assign the vector

(1,1,1,1) to the vertices u₁,u₂, ,u_n1. Then, assign the vector (1,1,1,0) to the

verticesv₁,v₂, ,v_n1. Also, assign the vector (1,1,0,0) to the vertices x₁, x₂, , x_n

. Moreover, assign the vector (1,0,0,0) to the vertices y₁, y₂, , y_n

.

Hence, the above labeling technique provides a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}- cordial labeling for the Lilly graph I_n

.

Example 3.5: The following Figure 2 illustrates the vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling of Lilly graph I₄

.

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R. PONRAJ AND R. JEYA

Figure 2

Vector basis {(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling of I₄

.

Theorem 3.6: The crown graph C_n K₁is a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial if and only if n is odd.

Proof: Consider the crown graph C_n K₁. Let C_nbe the cycle

u₁u₂u_nu₁. Let

and E(C_n K₁)  E(C)

V(C_n K₁) V(C_n) {v_i|1  i  n}

{u_iv_i|1  i  n} respectively be the vertex and edge sets of C_n K₁. Then

|V(C_n K₁)|  p  2n and |E(C_n K₁)|  q  2n . We have considered the two

cases.

Case (i): p  0 (mod 4)

Let p  4k . To get the edge label 4, the vector (1,1,1,1) should be assigned

to the consecutive vertices of C_n K₁. As the size of C_n K₁is 2n  p  4k , the

maximum edges with label 4 is k  1, a contradiction arises.

Case (ii): p  2 (mod 4)

Let p  4k  2. Then, assign the vector in the following order

u₁, u₂,  u_n, v₁, v₂,  v_n. We assign the vector (1,1,1,1) to the first k  1 vertices

u₁, u₂, , u_k1. Also, assign the vector (1,1,1,0) to the next k vertices

VECTOR BASIS S-CORDIAL LABELING OF FRIENDSHIP GRAPH

…

177

u_k2, u_k3, , u_n. We assign the vector (1,1,0,0) to the k  1

vertices

v₁, v₂, , v_k1. Moreover, assign the vector (1,0,0,0) to the next k vertices

v_k2, v_k3, , v_n

.

Thus, the above labeling technique provides a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling for the crown graph C_n K₁

.

Theorem 3.7: The armed crown graph AC_nis a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial if and only if n  1,2,3 (mod 4)

.

Proof: The armed crown graph AC_nis the graph obtained from the cycle

u₁u₂u_nu₁with V(ACn) V(C_n) {v_i, w_i|1  i  n} and E(AC_n)  E(C_n) 

{u_iv_i, v_iw_i|1  i  n}respectively be the vertex and edge sets of AC_n. Then

V(AC_n)  p  3n and E(AC_n)  q  3n . We have considered the four cases.

Case (i): p  0 (mod 4)

Let p  4k . To get the edge label 4, the vector (1,1,1,1) should be assigned

to the consecutive vertices of the graph.AC_nAs the size of AC_nis 3n  p  4k

,

the maximum edges with label 4 is k  1 , a contradiction.

Case (ii): p  1(mod 4)

Let p  4k  1. Then, assign the vector in the following order u₁,u₂,  u_n

,

v₁,w₁,v₂,w₂,  v_n,w_n. We assign the vector (1,1,1,1) to the first k  1 vertices.

Next, assign the vector (1,1,1,0) to the next k vertices. Also assign the vector

(1,1,0,0) to the next k vertices. Further, assign the vector (1,0,0,0) to the remaining k

vertices.

Case (iii): p  2 (mod 4)

Let p  4k  2. Then, assign the vector in the following order u₁,u₂,  u_n

,

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R. PONRAJ AND R. JEYA

v₁,w₁,v₂,w₂,  v_n,w_n. Now, assign the vector (1,1,1,1) to the first k  1 vertices.

So assign the vector (1,1,1,0) to the next k vertices. Next, assign the vector (1,1,0,0)

to the next k  1 vertices. Finally, assign the vector (1,0,0,0) to the remaining k

vertices.

Case (iv): p  3 (mod 4)

Let p  4k  3. Then, assign the vector in the following order u₁,u₂,  u_n

,

v₁,w₁,v₂,w₂,  v_n,w_n. Also, assign the vector (1,1,1,1) to the first k  1 vertices.

Assign the vector (1,1,1,0) to the next k vertices. Then, assign the vector (1,1,0,0) to

the next k  1 vertices. Moreover, assign the vector (1,0,0,0) to the remaining k  1

vertices.

Therefore, the above labeling method provides a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling for the armed crown graphAC_n

.

Theorem 3.8: The bistar graph B_n,nis a vector basis {(1,1,1,1), (1,1,1,0),

(1,1,0,0), (1,0,0,0)}-cordial graph for all n.

Proof:

Let

V(B_n,n)  {u,u,u_i,v_i|1  i  n}

and

E(B_n,n)  {uv,uu_i,vv_i|1  i  n} respectively be the vertex and edge sets of B_n,n

.

n,n

Note that V(B

)  p  2n  2 and E(B_n,n)  q  2n  1. We have considered

the two cases.

Case (i): p  0 (mod 4)

Let p  4k . Next, we assign the vector (1,1,1,1) to the vertices u and v.

Assign the vector (1,1,1,1) to the vertices u₁,u₂, ,u_k2. Then, assign the vector

(1,1,1,0) to the vertices u_k1,u_k, ,u_2k2. Also, assign the vector (1,1,0,0) to the

next k vertices v₁,v₂, ,v_k. Moreover, assign the vector (1,0,0,0) to the remaining k

vertices.

VECTOR BASIS S-CORDIAL LABELING OF FRIENDSHIP GRAPH

…

179

Case (ii): p  2 (mod 4)

Let p  4k  2. Now, we assign the vector (1,1,1,1) to the vertices u and v.

Assign the vector (1,1,1,1) to the vertices u₁,u₂, ,u_k1. Next, assign the vector

(1,1,1,0) to the vertices u_k,u_k1, ,u_2k1. So, assign the vector (1,1,0,0) to the

next k vertices v₁,v₂, ,v_k. Further, assign the vector (1,0,0,0) to the remaining

vertices.

Clearly the above labeling method provides a vector basis {(1,1,1,1),

(1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling for the bistar graphB_n,n

.

4. Conclusion

Vector basis {(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)}-cordial labeling

behavior of certain standard graphs like friendship graph, fan graph, lilly graph, bistar

graph, crown graph and armed crown graph have been investigated in this paper. The

investigation of different kinds of families of graphs for existence of vector basis

S-cordial labeling is an open problem.

REFERENCES

[

1] S. Amutha and M. Uma Devi (2019): Super graceful labeling for some families of fan

graphs, Journal Comput. Math. Sci., Vol. 10(8), pp. 1551-1562.

[2] C. M. Barasara (2018): Edge and total edge product cordial labeling of some new graphs,

Internat. Engin., Sci. Math., Vol. 7(2), pp. 263-273.

[3] J. Baskar Babujee and L. Shobana (2010): Prime and prime cordial labeling for some

special graphs, Int. J. Contemp. Math. Sciences, Vol. 5, pp. 2347-2356.

[4] I. Cahit (1987): Cordial Graphs: A weaker version of Graceful and Harmonious Graphs,

Ars Comb., Vol. 23(3), pp. 201-207.

[5] S. N. Daoud (2019): Vertex odd graceful labeling, Ars Combin., Vol. 142, pp. 65-87.

[6] A. Gallian (2024): A Dynamic Survey of Graph Labeling, Electronic Journal of

Combinatorics, Vol. 27, pp. 1-712.

[7] F. Harary (1988): Graph Theory, Narosa Publishing House, New Delhi.

180

R. PONRAJ AND R. JEYA

[8] V. Hemalatha, V. Mohanaselvi, and K. Amuthavalli (2017): Radio geometric mean

labeling of some star like graphs, J. Informatics Math. Sci., 9(3), pp. 969-977.

[9] I. N. Herstein (1991): Topics in Algebra, John Wiley & Sons, United States of America.

[10] V. J. Kaneria, M J Khoda, and H M Karavadiya (2016): Balanced mean cordial labeling

and graph operations, Int. J. Math. Appl., Vol. 4(3-A), pp. 181-184.

[11] S. Mitra and S. Bhoumik (2022): Tribonacci cordial labeling of graphs, Journal of

Applied Mathematics and Physics, Vol. 10(4), pp. 1394-1402.

[12] V. Mohan and A. Sugumaran (2019): Difference cordial labeling for plus and hanging

pyramid graphs, Internat. J. Res. Analytical Reviews, Vol. 6(1), pp. 1106-1114.

[13] A. Parthiban and Vishally Sharma (2021): Some results on prime cordial labeling of

Lilly graphs, Journal of Physics: Conference Series, 1831(012035), pp. 1-10.

[14] U. M. Prajapati and A. V. Vantiya (2019): SD-prime cordial labeling of some snake

graphs, JASC: J. Appl. Sci. Comput., Vol. VI(IV), pp. 1857-1868.

[15] R. Ponraj and R. Jeya: Vector basis S-cordial labeling of graphs (Submitted by the

Journal)

[16] A. Rosa (1967): On certain valuations of the vertices of a graph, Theory of Graphs,

(Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod

Paris, pp. 349-355.

[17] A. Sugumaran and K. Rajesh (2017): Sum divisor cordial labeling of theta graph, Annals

Pure Appl. Math., Vol. 14(2), pp. 313-320.

[18] Y. M. Parmar (2017): Edge vertex prime labeling for wheel, fan and friendship graph,

International Journal of Mathematics and Statistics Invention, Vol. 5(8), pp. 23-29.

1. Department of Mathematics,

(Received, December 23, 2024)

Sri Paramakalyani College,

Alwarkurichi - 627412, Tenkasi dt, Tamilnadu, India

E-mail: ponrajmaths@gmail.com

2. Research Scholar, Reg. No. 22222102092010,

Department of Mathematics,

Sri Paramakalyani College, Alwarkurichi-627412, Tamilnadu, India

(Affiliated to Manonmaniam Sundaranar University, Abhishekapatti,

Tirunelveli - 627012, India)

E-mail: jeya67205@gmail.com