 Open Access
 Authors : K.Ranga Devi, Dr.D.Bharathi
 Paper ID : IJERTV13IS090082
 Volume & Issue : Volume 13, Issue 09 (September 2024)
 Published (First Online): 08102024
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Binary Cyclic Codes in Extending Circular Cliques
K.Ranga Devi Research Scholar,
Department of Mathematics, S.V.University, Tirupati 517502, A.P, India.
ABSTRACT: Let G be a coloring graph with circular chromatic number ()= {/d: ,d, gcd(k,d)=1 and dd}, ,d are prime circular cliques. If the two circular cliques ,d at distance such that some (,d
)precolouring of the two cliques is nonextendible. In
this section, we examine extending circular colourings of
,d , is the path of length 1 with vertex set
{1,2,,}. In view of the homomorphism G admits a (k,d)colouring if and only if, there is a homomorphism f:,d. there exist a uniquely extendible homomorphisms between circular cliques.
KEY WORDS: Edge coloring, Vertex coloring, Circular chromatic number, Homomorphism, Binary cyclic codes.

INTRODUCTION:
Graph coloring theory has a central position in discrete mathematics for its own interest as well as for the large variety of applications, dating back to the famous fourcolor problem stated by Guthrie in 1852 Zhu[9].
Define a(k,d)colouring of a graph G is an assignment :(){0,1,2,,1} such that for
(), d()()d, d is any positive integer. The circular complete graph or circular clique ,d has vertices {0,1,,1} and edges
{:dd}. Thus ,1 is simply the (classical) complete graph on vertices. Graph
coloring is the procedure of assignment of colors to
each vertex of a graph G such that no adjacent vertices get same color.
The minimum for which admits a coloring is called the chromatic number of and denoted by ().
There are now many papers on colouring extensions. The introduction of [3] provides a nice overview on coloring. We focus on the situation where the precoloured vertices
Dr.D.Bharathi Professor, Department of Mathematics,
S.V.University, Tirupati 517502, A.P, India.
induce a collection of cliques. Let be a graph with circular chromatic number ()=/d [8]is isomorphic to the circular clique ,d. Suppose the vertices of have been precoloured with a (,d)colouring. In [2] Albertson and Moore study the problem of extending a (+1)colouring of a colourable graph where the precoloured components are cliques. They also study the problem when the precoloured components are general subgraphs. In the latter case the penalty for having general subgraphs is a larger number of colours may be required for the extension. In this spirit we now turn attention to extending a (,d)colouring of a (,d)colourable graph where the precoloured components are circular cliques.
We now consider extending (classical) colourings where the precoloured components are ,d. The general problem of extending colourings where the precoloured components are not cliques is considered in [2]. In our work the assumption that the precoloured components are circular cliques

PRELIMINARIES:
DEFINITION 2.1: An undirected graph is a type of graph where the edges have no specified direction assigned to the them..
DEFINITION 2.2: A binary code is cyclic code if it is a linear [n, k] code and if for every codeword (c1, c2, . . . , cn) C we also have that (cn, c1, . . . , cn1) is again a codeword in C.

Vertex coloring is a concept in graph theory that refers to assigning colors to the vertices of a graph in such a way that no two adjacent vertices have the same color..

In graph theory, Edge coloring of a graph is an assignment of colors to the edges of the graph so that no two adjacent edges have the same color
G
such that no adjacent vertices get same color.
DEFINITION 2.3: Graph coloring is the procedure of assignment of colors to each vertex of a graph
DEFINITION 2.4: The chromatic number of a graph is the minimal number of colours needed to colour the vertices in such a way that no two adjacent vertices have the same colour.


RESULT AND DISCUSSION:
We find Vertex chromatic number, edges chromatic number, Degree and Dimensions of the generator matrix.
A(k,d)colouring of a graph G is an assignment :(){0,1,2,,1} such that
for (), d()()d, d is any positive integer. The circular complete graph or circular clique ,d has vertices {0,1,,1} and edges {:dd}. Thus ,1 is simply the (classical) complete graph on vertices. The circular complete graphs play the role in circular colourings as do the complete graphs in classical colourings. Adopting the homomorphism point of view, see [4], [5], admits a (,d)colouring if and only if, there is a homomorphism :,d. Recall, a homomorphism : is a
mapping :()() such
that () implies ()()(). We
write to indicate the existence of a
homomorphism. It turns out that ,d,d if and only if /d/d. Thus, given a graph , if ,d, then ,d for any /d/d is surjective. Suppose (k2d), d is positive integer and k is prime number with gcd(k,d)=1, the circular chromatic numbers includes all chromatic numbers ()=() as well as odd holes see the below figures.
The circular chromatic number of a graph is defined as ()=Inf{/d : ,d and gcd(k,d)=1}.
In [4], Bondy and Hell show the infimum may be replaced by a minimum. The proof depends on the fact that optimum colourings must be surjective. The surjective mappings play a key role in our constructions of nonextendible families.
Example 3.1.1: The circular chromatic number of a graph is defined as ()=inf{5/1 : 5,1 and gcd(5,1)=1}.
The adjacency matrix of X is
1 
0 
0 
0 
1 
1 
1 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
1 
1 
0 
[0 
0 
0 
1 
1] 
Vertex coloring graph Edge coloring graph (Fig 1.1)
The polynomial represented by X is k(x)=1+x4
In above Figure 1.1, the vertex chromatic number ()= 3 and Edge chromatic number is 3.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 4, dimension of the code is 5 and has no error correcting codes. (5/1)=3
Example 3.1.2: The circular chromatic number of a graph is defined as ()=inf{5/2 : 5,2 and gcd(5,2)=1}.
The adjacency matrix of X is
0 
0 
1 
1 
0 
0 
0 
0 
1 
1 
1 
0 
0 
0 
1 
1 
1 
0 
0 
0 
[0 
1 
1 
0 
0] 
Vertex coloring graph Edge coloring graph (Fig 1.2)
The polynomial represented by X is k(x)=x2+x3
In above Figure 1.2, the vertex chromatic number ()=5 and Edge chromatic number is 5.
Hence X corresponds to the cyclic code C =<x> . Since te degree of the generator polynomial k(x) is 3, dimension of the code is 5 and has no error correcting codes. (5/2)=5
Example 3.1.3: The circular chromatic number of a graph is defined as ()=inf{5/3 : 5,3 and gcd(5,3)=1 }.
The adjacency matrix of X is
0 
0 
1 
1 
0 
0 
0 
0 
1 
1 
1 
0 
0 
0 
1 
1 
1 
0 
0 
0 
[0 
1 
1 
0 
0] 
Vertex coloring graph Edge coloring graph (Fig 1.3)
The polynomial represented by X is k(x)= x2+x3
In above Figure 1.3, the vertex chromatic number ()= 5 and Edge chromatic number is 5.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 3, dimension of the code is 5 and has no error correcting codes. (5/3)=5
()=inf{ 5/1, 5/2 and 5/3} is 5/1=3
Example 3.2.1: The circular chromatic number of a graph is defined as ()=inf{7/1: 7,1 and gcd(7,1)=1 }.
0 
1 
0 
0 
0 
0 
1 
1 
0 
1 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
1 
0 
1 

1 
0 
0 
0 
0 
1 
0] 
The adjacency matrix of X is
0
[.
Vertex coloring graph Edge coloring graph
(Fig 1.4)
The polynomialrepresented by X is k(x)= x+x6
In above Figure 1.4, the vertex chromatic number ()=3 and Edge chromatic number is 3.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 6, dimension of the code is 7 and has no error correcting codes. (7/1)=3
Example 3.2.2: The circular chromatic number of a graph is defined as ()=inf{7/2: 7,2 and gcd(7,2)=1 }.
The adjacency matrix of X is
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
1 
1 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
1 
1 
0 
0 
1 
0 
0 
0 
[0 
1 
0 
0 
1 
0 
0] 
Vertex coloring graph Edge coloring graph (Fig 1.5)
The polynomial represented by X is k(x)= x2+x5 In above Figure 1.3, the vertex chromatic number ()= 4 and Edge chromatic number is 7.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 5, dimension of the code is 7 and has no error correcting codes. (7/2)=4
Example 3.2.3: The circular chromatic number of a graph is defined as ()=inf{7/3: 7,3 and gcd(7,3)=1 }.
The adjacency matrix of X is
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
0 
0 
0 
0 
1 
1 
1 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
[0 
0 
1 
1 
0 
0 
0] 
Vertex coloring graph Edge coloring graph
(Fig 1.6)
The polynomial represented by X is k(x)= x3+x4
In above Figure 1.3, the vertex chromatic number ()= 4 and Edge chromatic number is 7.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 4, dimension of the code is 7 and has no error correcting codes. (7/3)=4
()=inf{ 7/1, 7/2 and 7/3} is 7/1=3
Example 3.3.1:The circular chromatic number of a graph is defined as ()=inf{11/1: 11,1 and gcd(11,1)=1 }.
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 

1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0] 
The adjacency matrix of X is
0
[The polynomial represented by X is k(x)=x+x10
In above Figure 1.7, the vertex chrmatic number ()= 3 and Edge chromatic number is 3.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 10, dimension of the code is 11 and has no error correcting codes. (11/1)=3
Example 3.3.2:The circular chromatic number of a graph is defined as ()=inf{11/2 : 11,2 and gcd(11,2)=1 }.
The adjacency matrix of X is
0 
0 
1 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
[0 
1 
0 
0 
0 
0 
0 
0 
1 
0 
0] 
.
Vertex coloring graph Edge coloringraph
(Fig 1.8)
The polynomial represented by X is k(x)=x2 +x9
In above Figure 1.8, the vertex chromatic number ()= 4 and Edge chromatic number is 6.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 9, dimension of the code is 11 and has no error correcting codes. (11/2)=4
Example 3.3.3:The circular chromatic number of a graph is defined as ()=inf{11/3 : 11,3 and gcd(11,3)=1 }.
The adjacency matrix of X is
Vertex coloring graph Edge coloring graph
(Fig 1.7)
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
[0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0] 
The polynomial represented by X is k(x)= x4 +x7 In above Figure 1.10, the vertex chromatic number ()= 4 and Edge chromatic number is 6.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 7, dimension of the code is 11 and has no error correcting codes. (11/4)=4
Example 3.3.5:The circular chromatic number of a graph defined as ()=inf{11/5 : 11,5 and gcd(11,5)=1}
The adjacency matrix of X is
Vertex coloring graph Edge coloring graph
(Fig 1.9)
The polynomial represented by X is k(x)= x3 +x7
In above Figure 1.9, the vertex chromatic number ()= 4, edge chromatic number is 6.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 7, dimension of the code is 11 and has no error correcting codes. (11/3)=4
Example3.3.4:The circular chromatic number of a graph is defined as ()=inf{11/4 : 11,4 and gcd(11,4)=1}
The adjacency matrix of X is
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
p>0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
[0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0] 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
0 
0 
[0 
0 
0 
0 
1 
1 
0 
0 
0 . 
0 
0] 
Vertex coloring graph Edge coloring graph (Fig 1.11)
The polynomial represented by X is k(x)= x5 +x6
In above Figure 1.11, the vertex chromatic number ()= 4 and Edge
chromatic number is 6.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 6, dimension of the code is 11 and has no error correcting codes. (11/5)=4
c()= inf{11/1, 11/2, 11/3, 11/4 and 11/5} is 11/1=3
Vertex coloring graph Edge coloring graph (Fig 1.10)
.
We observe that the above graphs , the two circular cliques ,d at distance such that some (,d) precolouring of the two cliques (Vertex chromatic number and Edge chromatic
number d ) are nonextendible. And the dimensions of generator matrix are same. Also the circular chromatic numbers are

()= inf{ 5/1, 5/2 and 5/3} is 5/1=3

()=inf{ 7/1, 7/2 and 7/3} is 7/1=3

()= inf{11/1, 11/2, 11/3, 11/4 and 11/5} is 11/1=3.
HOMOMORPHISM OF A CIRCULAR GRAPHS:
A kcoloring, for some integer k, is an assignment of one of k colors to each vertex of a graph G such that the endpoints of each edge get different colors. The kcolorings of G correspond exactly to homomorphisms from G to the complete graph Kk.[3] Indeed, the vertices of Kk correspond to the k colors, and two colors are adjacent as vertices of Kk if and only if they are different. Hence a function defines a homomorphism to Kk if and only if it maps adjacent vertices of G to different colors (i.e., it is a kcoloring). In particular, G is kcolorable if and only if it is Kkcolorable.[3]
If there are two
homomorphisms G H and H Kk, then their composition G Kk is also a homomorphism.[1] In other words, if a graph H can be colored with k colors, and there is a homomorphism from G to H, then G can also be kcolored. Therefore, G H implies (G) (H), where denotes the chromatic number of a graph (the least k for which it is kcolorable).[4]
DIRECT PRODUCT GRAPHS: The direct product
G Ã— H of graphs G and H is the graph with the vertex set V (G) Ã— V (H), two vertices (x, y) and (v, w) being adjacent in G Ã— H if and only if xv E(G) and yw E(H).
PREPOSITION3.3: Let : be a homomorphism and let V(). The homomorphism is uniquely extendible at if whenever : is a homomorphism with g()=() for all , then ()=(). If is uniquely extendible at for all V(), we simply say is uniquely extendible[1].
PREPOSITION 3.4: Let and be graphs. The extension product X has as its vertex set
()Ã—() with (1,p)(2,p) an edge if 12() and either pp() or p=p. The direct product of with a reflexive copy (a loop on each vertex) of .
Proof: The direct product of G and H is the graph, denoted as GÃ—H, whose vertex is V (G)Ã—V (H), and for which vertices (g, h) and(g, h) are adjacent precisely if ggE(G) and hhE(H).
Thus, V (GÃ—H) = {(g, h) : gV (G) and hV (H)},
E(GÃ—H) = {(g, h)(g, h) : ggE(G) and
hhE(H)}.
Other names for the direct product that have appeared in the literature are tensor product, Kronecker product, cardinal product, relational product, cross product, conjunction, weak direct product, or categorical product.
A product GÃ—H has a loop at (g, h)if and only if both G and H have loops at g and h, respectively.
Moreover, if G has no loop at g, then the Hlayer H(g,h) is disconnected; whereas if G has a loop at g, then H(g,h) is isomorphic to H.
Suppose (g, h) and (g, h) are vertices of a direct product GÃ—H and n is an integer for which G has a g, g walk of length n and H has an h, h walk of length
n. Then GÃ—H has a walk of length n from (g, h) to (g, h). The smallest such n (if it exists) equals d((g, h), (g, h)). If no such n exists, then d((g, h), (g, h)) =
.
Example 3.4.1:Let and be graphs.The extension product :2,2 P2Ã—P2 has as its vertex set
(2)Ã—(2) with (1,p)(2,p) an edge
if 12() and either pp() or p=p.
Vertex and Edge coloring graphs of G2 and G3
Let :2,2 P2Ã—P2 the generatoer of a matrix is [1 1]
1 1
P2Ã—P2 Vertex and Edge graphs
(Fig 1.12)
The polynomial represented by X is k(x)=1+x
In above Figure 1.12, the vertex chromatic number
()= :2,2 P2Ã—P2 is 4, edge chromatic number is 2.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 1, dimension of the code is 2 and has no error correcting codes.
Example 3.4.2:Let and be graphs.The extension product P2Ã—P3 has as its vertex set (2)Ã—(3) with (1,p)(2,p) an edge if 12() and
either pp() or p=p.
Let :2,3 P2Ã—P3 , the generator of a matrix is [1 1 1]
1 1 1
P2Ã—P3 Vertex and Edge graphs (Fig 1.13)
The polynomial represented by X is k(x)=1 + x+x2
In above Figure 1.13, the vertex chromatic number ()= :2,3 P2Ã—P3 is 4, edge chromatic number is 4.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 2, dimension of the code is 2 and has no error correcting codes.
Example 3.4.3 :Let and be graphs.The extension product :2,5 P2Ã—P5 has as its vetex set
(2)Ã—(5) with (1,p)(2,p) an edge if 12() and either pp() or p=p.
Let :2,5 P2Ã—P5, the generator of a matrix is [1 1 1 1 1]
1 1 1 1 1
P2Ã—P5 Vertex and Edge graphs
(Fig 1.14)
The polynomial represented by X is k(x)= 1 + x + x2+x3+x4
In above Figure 1.14, the vertex chromatic number ()= :2,5 P2Ã—P5 is 4, edge chromatic number is 6.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 4, dimension of the code is 2 and has no error correcting codes.
Example 3.4.4:Let and be graphs.The extension product :2,7 P2Ã—P7 has as its vertex set
(2)Ã—(7) with (1,p)(2,p) an edge if 12() and either pp() or p=p.
Let :2,7 P2Ã—P7, the generator of a matrix is [1 1 1 1 1 1 1]
1 1 1 1 1 1 1
P2Ã—P7 Vertex and Edge graphs (Fig 1.15)
The polynomial represented by X is k(x)= 1 + x + x2+x3+x4+x5+x6
In above Figure 1.15, the vertex chromatic number ()= :2,7 P2Ã—P7 is 4, edge chromatic number is 6.
Hence X corresponds to the cyclic code C =<x> . Since the degree of the generator polynomial k(x) is 6, dimension of the code is 2 and has no error correcting codes.
We observe that from the graphs , the product of two circular cliques ,d at distance such that some (,d)precolouring of the two cliques( Vertex chromatic number and Edge chromatic number d ) are nonextendible. And the dimensions of generator matrix are same.

()= :2,2 P2Ã—P2 is 4

()= :2,3 P2Ã—P3 is 4

()= :2,5 P2Ã—P5 is 4

()= :2,7 P2Ã—P7 is 4
IV.CONCLUSION:
We finish the paper with an extension result for (,d)colourings of ,d cliques in colourable graphs.

In above figures, the two circular cliques ,d at distance such that some (,d)precolouring of the two cliques( Vertex chromatic number and Edge chromatic number) are nonextendible. And the dimensions of generator matrix are same. The circular chromatic numbers are always same.

()= inf{ 5/1, 5/2 and 5/3} is 5/1=3

()=inf{ 7/1, 7/2 and 7/3} is 7/1=3

()= inf{11/1, 11/2, 11/3, 11/4 and 11/5} is 11/1=3, etc.


:k,dPn , if is uniquely extendible at for all
(), we simply say is uniquely extendible. The product of two circular cliques ,d at distance such that some (,d)precolouring of the two
cliques( Vertex chromatic number and Edge chromatic number d 2) are nonextendible. Andthe dimensions of generator matrix are same, but degree of the polynomials is increasing.

()= :2,2 P2Ã—P2 is 4

()= :2,3 P2Ã—P3 is 4

()= :2,5 P2Ã—P5 is 4

c()= :2,7 P2Ã—P7 is 4


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