Cmjv03i02p0145 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

On the Coefficients of the Vertex - Cover Polynomials of Wheels AYYAKUTTY VIJAYAN1 and BERNONDES STEPHEN JOHN2 1

Assistant Professor, Department of Mathematics, Nesamony Memorial Christian College, Marthandam Kanyakumari District, Tamil Nadu, 629165, India. 2 Assistant Professor, Department of Mathematics Annai Velankanni College, Tholayavattom Kanyakumari District, Tamil Nadu, 629157, India. (Received on: February 27, 2012) ABSTRACT The vertex cover polynomial of a graph G of order n has been already introduced in3. It is defined as The polynomial. | V(G) |

C (G, x) =

C (G, i) xi Σ i = β (G)

where c (G, i) is the number of vertex covering sets of G of size i and β (G) is the covering number of G. We obtain some properties of the coefficients of the vertex cover polynomial of the wheels. We obtain some recurrence relations for the coefficients of the vertex cover polynomials of wheels we establish some relation between the coefficients of the vertex cover polynomials of wheels and paths ; wheels and cycles. Keywords: Vertex covering sets : vertex cover number, vertex cover polynomial.

INTRODUCTION Let G = (V, E) be a simple graph. For any vertex v 0 V, the open neighborhood of v is the set N (v) = { u 0 V/ uv 0 E} and the closed neighborhood of v is the set N [v] = N (v) χ {v}. For a set S φV, the open neighborhood of S is N (s) = U N (v) and the closed v∈S

neighborhood of S is N [S] = N (S) χ S .

A set S φV is a vertex covering of G if every edge uv 0 E is adjacent to atleast one vertex in S. The vertex covering number β (G) is the minimum cardinality of the minimum vertex covering sets in G. A vertex covering set with cardinality β (G) is called a β set. Let C (G, i) be the family of vertex covering sets with cardinality i and let c (G, i) = | C (G, i)|. The | V(G) |

polynomial C (G, x) =

c (G, i) xi Σ i = β (G)

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Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

defined as the vertex cover polynomial of G. In3, many properties of the vertex cover polynomial have been studied and derived the vertex cover polynomials for some standard graphs. In this paper, we derived some properties of the coefficients of the vertex cover polynomials of wheels. Also, we obtain some relationship between the coefficients of vertex cover polynomials and paths ; wheels and cycles. The number of edges incident to the vertex v of a graph G is called the degree of the vertex v in G. It is denoted by deg (v). δ (G) and ∆ (G) are the minimum and

obtained as C (Pn , x) =

 n −1  i =    2  n

C (Wn, x) = xn – 1 +

 n + 1 i =    2 

n i+1  i  n − i

maximum of the degree of the vertices in G respectively. 2. VERTEX COVER POLYNOMIAL Definition 2.1 Let C (G, i) be the family of all vertex covering sets of G with cardinality i and let c (G, i) = | C (G, i)|. The vertex cover polynomial of G is defined as | V(G) |

C (G, x) =

Σ c (G, i) i = β (G)

xi.

In3 the vertex cover polynomial of paths, cycles and wheels are

 i  x ; C (Cn, x) = 

n i =   2

n  i  i   x i  n − i 

n − 1 i − 1 i  x . i − 1  n − i 

Theorem 2.2 The coefficients of the vertex cover polynomials the wheels have the following properties: i)

c (W2n + 1, n + 2) = n2

ii)

c  W2n + 1 ,  

 

iii) iv) v) vi)

, n∃2

 2n + 1     = 2 , n ∃ 2 2  

c (wn, n – 2) = c (wn – 1, n – 3) + c (w n – 2, n – 3) – 1 , n > 4 c (Wn, n – 2) = c (Wn – 1, n – 2) + c (Wn – 1, n – 3) – 2 , n > 4 c (Wn, n – 1) = c (Wn – 1, n – 2) + c (Wn – 2, n – 2), n > 4 c (Wn, n – 3) = c (Wn – 1, n – 4) + c (Wn – 2, n – 4) , n > 4 Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

where

and

i − 1   if i ≠ n – 1 i − 1 n − i n − 1  i − 1 c (Wn, i) =   + 1 ; if i = n – 1 i − 1  n − i

c (Wn, i) =

n − 1

Proof : i)

c (W2n + 1 , n + 2)

2n + 1 − 1  n + 2 − 1    n + 2 − 1  2n + 1 − n + 2  2n  n + 1  =   n + 1  n − 1   n + 1 2n =   n + 1  2  =

= n2.



 2n + 1     = c (W2n + 1, n + 1) 2       2n + 1  Q   = n + 1  2    2n + 1 − 1  n + 1 − 1    = n + 1 − 1  2n + 1 − n + 1  2n  n  =   n  n 

ii) c  W2n + 1 , 

= 2 iii)

R.H.S = c (Wn – 1, n – 3) + c (Wn – 2, n – 3) – 1

n −1 −1  n − 3 − 1  n − 2 −1   + n − 3 − 1  n − 1 − n − 3  n − 3−1

 n − 3 − 1   + 1 − 1  n−2 − n − 3   

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147

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Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

n − 2 n − 4 n − 3 n − 4   +   n − 4  2  n − 4  1  n − 2 (n − 4)(n − 5) = . + (n − 3) n − 4 2 n 2 − 5n + 4 =

= =

2 (n − 1) (n − 4) n −1

(n − 3)(n − 4)

n − 3 n −1

2 n − 3 =   n − 3  2  n − 1  n − 2 − 1 =   n − 3  n − n − 2 

(A)

= c (Wn, n – 2) = L.H.S. Therefore, c (Wn, n – 2) = c (Wn – 1 , n – 3) + c (Wn – 2 , n – 3) – 1 (iv)

R.H.S = c (Wn – 1, n – 2) + c (wn – 1, n – 3) – 2

 n − 3 − 1   [ .  n − 3 − 1  n − 1 − n − 3  n − 2 n − 4 – 2 .  = n – 1 + n − 4  2  (n − 2)(n − 5) = (n – 1) +

n −1 − 1

c (Wn, n – 1) = n]

= n–3 +

2 2 (n − 3) + (n − 2) (n − 5) 2

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

n 2 − 5n + 4 2

= c (Wn, n –2) = L.H.S

[

R.H.S. = c (Wn – 1, n – 2) + = = =

vi)

R.H.S = c (Wn – 1 , n – 4) + c (Wn – 2, n – 4)

by (A) in (iii) ]

c (Wn – 2, n – 2) = n – 1 + 1 n (Wn, n – 1) [ c (Wn, n) = 1 ; c (Wn, n – 1) = n] L.H.S

n −1− 1  n − 4 − 1  n − 2 − 1  n − 4 − 1    +   n − 4 − 1  n − 1 − n − 4  n − 4 − 1  n − 2 − n − 4  n − 2 n − 5 n − 3 n − 5   +   n − 5  3  n − 5  2  n − 2 (n − 5) (n − 6) (n − 7) n − 5

(n − 3) (n − 5) (n − 6) (n − 5)

 (n − 2) (n − 7) + 3(n − 3)   3!  

= (n – 6)  = = =

n − 6 3! n − 6 3! n − 1 3! n − 1

[n2 – 6n + 5] [ (n – 5) (n – 1) ] [ (n – 5) (n – 6) ]

 (n − 4) (n − 5) (n − 6)  .   n − 4  3!  n − 1 n − 4 =   n − 4  3 

149

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Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

 n − 3 − 1   n − 3 − 1  n − n − 3 

n − 1

= c (Wn, n – 3) = L.H.S. Theorem : 2.3 The coefficients of the vertex cover polynomials of the wheels satisfy the following identity : ∞

∞

c (Wi, n) = 2

i = 5

c (Wi, n – 1) – 1.

i= 4

Proof : When i < n or i > 2n – 1 , c (Wi , n) = 0 2n −1

Therefore the above relation reduces to

2n −3

c (Wi, n) = 2

i= n

c (Wi, n – 1) – 1.

i = n −1

2n − 3

R.H.S. =

2 c (Wi, n – 1) – 1

i = n −1

= 2 [c (Wn – 1, n – 1) + c (Wn, n – 1) + (Wn + 1, n – 1) + . . . + c (W2n – 4, n – 1) + c (W2n – 3, n – 1) – 1.

n − 2 n − 2  n − 1  n − 2     +    + 1  n − 2  0  n − 2  n − n − 1    

= 2 

n + 1− 1  n − 1 − 1  2n − 4 − 1   + . . . + n − 1 −1  n + 1 − n −1  n − 1−1

 n − 1 − 1     2n − 4 − n − 1 

2n − 3 − 1  n − 1 − 1     − 1 n − 1 − 1  2n − 3 − n − 1    n − 2 n − 2 n − 1 n − 2 n  +   + 1 +   + ...+ n − 2  1  n − 2  2    0 

= 2   

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

2n − 5  n − 2  2n − 4   + n − 2 n −3 n − 2

 n − 2    – 1  n − 2  

151 (1)

 n − 2   n − 2   n − 1   n − 1 n (n − 3)   +...+  +    +    +    + 2   0    0   1     1   2n − 5  n − 2  2n − 4  n − 2   2n − 4  n − 2     +    +   + 2–1 n − 2  0   n − 2  0   n − 2  1  n − 2  (n − 2) (n + 1)  =  + [1 + n − 1] +    + . . . + (2n – 3)] + 2 + 1  0  2      n − 1 (n − 2) (n + 1) =  + n+ + . . . + (2n − 3) + 2 + 1   0  2    n − 1 n  n − 1 n + 1 (n − 1) (n − 2) =  + + +. . .+     0   1  n − 1 n − 1 2!    

=  

2n − 3  n − 1    + n − 1  n − 2   n − 1 n =  +    n −1  0 

2 (n − 1)  n − 1    + 1 n −1 n − 1    n − 1 n+ 1  n−1   +   n−1  2   1  2n − 3  n − 1  2n − 2   + n −1 n − 2 n −1

 n − 1 n − 1  n − 1   n   +  1 +  n − 1  0   n − 1  n+1− n n − 1  2n−2−1    n −1  2n − 2 − n 

+. . .+  n − 1   + 1 − n 1  

 n − 1    + . . .+  n +2 − n  n − 1  2 n − 1− 1  +   n −1  2n − 1 − n 

  n+1   +   n − 1

= c (Wn, n) + c (Wn + 1, n) + c (Wn + 2, n) + . . + (W2n – 2, n) + c (W2n – 1, n) Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

152

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012) 2n − 1

c (wi, n)

i= n ∞

Hence , we have,

∞

c (Wi, n) = 2

i = 5

c (Wi, n – 1) – 1.

i= 4

Theorem 2.4 The coefficients of the vertex cover polynomials of the wheels satisfies the following recurrence relation: n −1

n+1

n+2 i =    2 

c (Wn + 1, i) + 1 =

n i =   2

c (Wn – 1, i) +

 n + 1 i =    2 

c (Wn, i)

Proof: First we take n = 2k,

n  = k ; 2

Therefore, 

n+1 n+2   = k + 1 and   = k+1  2   2 

n −1

R.H.S. =

c (Wn – 1, i) +

n i =   2 2k − 1

c (Wn – 1, i) +

i= k

 n + 1 i =    2  2k

c (Wn, i)

i = k+1

= c (Wn – 1, k) + c (Wn – 1, k + 1) + c (Wn – 1, k + 2) + . . . + c (Wn – 1, 2k – 2) + c (Wn – 1, 2k – 1) + c (Wn, k + 1) + c (Wn, k + 2) + c (Wn, k + 3) + . . . + c (Wn, 2k – 1) + c (Wn, 2k) =

n−2  k − 1  k − 1  n − 1 − k

k k+1    n−2  n−2   +   +     n − 1 − k +1   n − 1 − k +2  k k + 1      2k − 3 2k − 2   n−2  n−2    +   + .... + 1 + 2k − 3  n − 1 − 2k − 2  2k − 2  n − 1 − 2k − 1  Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

k  n−1  n−1  k+1  n−1   +   + k  n − k + 1  k + 1  n − k + 2  k + 2

+...+ 1+

 2k − 2  n−1   + 2k − 2  n − 2k − 1  2k − 1

n−1

 2k − 3  n−2   +  2k − 3  2k − 2 1 

n−2

 k+2    n −k+3  

 2k − 1     n − 2k   

n − 2  k − 1 n−2  k  n−2   +   + k − 2 k − 1  k − 1  k k+1  

153

 k+1    + ...+ k − 3  

 2k − 2      0  

n−1  k  n−1  k+1  n−1   +   +  k − 1 k k + 1  k − 2  k + 2    2k − 2  n−1   +  2k − 2  2k − 1 1 

n−1

2k − 2 k − 1

n−2 k

k n−2   + 2 k+1  

 2k − 1    +2   0    k + 1   +...+  4   

 2k − 3  2k −   +  2k − 3  2k − 1  n − 1  k + 1 n−1   + + k + 1  3  k+2

n−2

 k+2    + ...+ k − 3  

 2k − 2  n−1  k    +     k − 1 2  k 0    k+2 n − 1  2k − 2    +...+    5    2k − 2 1     2k − 1  2k − 1    + 2 +  2k − 1  0 

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Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

n − 2  k  n − 1  k  n−2   +   +  k  k − 1    k  2   k + 1 

=2+ 

 k + 1 n − 1  k + 1   +    4   3  k + 1    

 2k − 2  2k − 2  n − 1  2k − 2   2k − 1  2k − 1    +   +   + 2 +.... +  2k − 1  0   2k − 2  0  2k − 2  1    (n − 2) (k − 1)  + n − 1 = 2 +  2   (n − 1) k (k − 1)   (n − 2) k (k − 1) (k − 2) +  +  + . . . + [1 + n – 1] 4! 3!    2k − 1  +  + 2  0     (2k − 2) (k − 1) + 2 (2k − 2)  = 2 +   + 2!    (2k − 2) k (k − 1) (k − 2) + 4 (2k − 1) k(k − 1)  4! 

= 2+

= 2 +

2k k

2k 2 2

2.k k 2

(2k 4 − 2k 2 )

 2k − 1   + 2 0  

+ . . + 2k +  

2k (k + 1) (k) (k − 1) 4!

 k + 1 2k   + k + 1  2  k + 2

 2k − 1   + 2 + . . . + 2k +   0   

k+2   +...+  4   

 2k − 1  2k  2k    +   + 2   0  2k − 1  2k 1   

 2k − 1   + . . . + 2k +   + 2  0   

  2k   2k − 1   Q =    0       0 

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

n k n  k + 1 n k+2   +   +   +...+ k  k  k + 1  k − 1  k + 2  k − 2 

155

 2k − 1  n  2k    +   + 2   0  2k − 1  2k 1   

k k k+2    n  n  n    +   +   k  n + 1 − k + 1  k + 1  n + 1 − k + 2  k + 2  n + 1 − k + 3 

+...+

 2k  2k − 1    n n  1 +   +   +1 2k − 1  n + 1 − 2k   2k  n + 1 − 2k+1   

= c (Wn + 1, k + 1) + c (Wn + 1, k + 2) = c (Wn + 1, k + 3) + . . . + c (Wn + 1, 2k) + c (Wn + 1, 2k + 1) + 1. 2k + 1

c (Wn + 1, i) + 1 = L.H.S

i = k+1

Similarly, for n = 2k + 1 the above result follows.

Theorem 2.5 The coefficients of the vertex cover polynomials of the paths, wheels and cycles satisfy the following identities. ∞

i= 4

∞

c (Wi, n) =

∞

ii)

i= 4

c (Ci, n – 1) + 1

i= 3 ∞

c (Wn + 1, i) =

c (Cn, i) + 1

i= 3

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

156 iii)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

c (Cn, n – 2) + 1 = c (Pn, n – 2)

Proof : 1)

By Theorem 2.4 of [3] , we have c (Wn, x) = x C (Cn – 1, x) + xn – 1.

Therefore,

c (Wi, n) = c (Ci – 1, n – 1)

, i = n, n + 1, . . . , 2n − 1

and when

c (Wn, n – 1) = c (Cn – 1, n – 2) + 1 i < n or i > 2n – 1, c (Wi , n) = 0

when

i < n – 1 or i > 2n – 2, c (Ci , n – 1 ) = 0

[ by (A) ]

2n − 2

R.H.S =

i = n −1

c (Ci, n – 1) + 1

= c (Cn – 1 , n – 1) + c (Cn, n – 1) + c (Cn + 1, n – 1) + . . . + c (C2n – 3, n – 1) + . . . + c (C2n – 2, n – 1) + 1 = c (Cn – 1, n – 1) + [c (Cn, n – 1) + 1] + c (Cn + 1, n – 1) + . . . + c (C2n – 3, n – 1) + c (C2n – 2, n – 1) = c (Wn, n) + c (Wn + 1, n) + c (Wn + 2, n) + . . . + c (W2n – 2, n – 1) + c (W2n – 1, n – 1). 2n − 1

c (Wi, n)

i= n n −1

ii)

n i =  +1 2

c (Wn + 1, i) =

Σ n  c (Cn, i) + 1

i=   2

n  + 1 or i > n, C (Wn + 1 , i) = 0 2 n Also, when i <   or i > n – 1, c (Cn , i) = 0 2 When i < 

First we take, n = 2k Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012) 2k

Therefore ,

157

C (Wn +1, i) =

i = k +1

c (Cn, i) + 1

i= k

2k − 1

R.H.S =

(Cn, i) + 1 = c (Cn, k) + c (n, k + 1) + ...+ c (Cn, 2k – 2) + c (Cn, 2k – 1) + 1

i= k

= c (C2k, k) + c (C2k, k + 1) + . . . + c (C2k, 2k – 2 + [c (C2k, 2k – 1)) + 1] = c (W2k + 1, k + 1) + c (W2k + 1, k + 2) + . . .+ c(W2k + 1,2k – 1) + c (W2k + 1, 2k) = c (Wn +1, k + 1) + c (Wn + 1, k + 2) + . . . + c (Wn + 1, 2k) n

c(Wn + 1 , i)

i = k +1

(iii)

c (Cn, n – 2) + 1

 n − 2    + 1 n − 2  n − n − 2 

n − 2   + 1 n − 2  2 

n (n − 3) 2

+ 1

n 2 − 3n + 1 2 (n − 1) (n − 2) 2  n − 1   2 

=  

 n − 2 + 1   n − n−2 

=  

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

158

Ayyakutty Vijayan, et al., J. Comp. & Math. Sci. Vol.3 (2), 145-158 (2012)

= c (Pn, n – 2)

n  i   Q c (C , i) = n    i  n − i   i+1  c (Pn , i) =    n − i  

REFERENCES 1. Alikhani. S and Peng. Y.H. Introduction to Domination Polynomial of a Graph. arXiv : 09052241 v1 [math.co] 14 May (2009). 2. Alikhani. S. and Peng. Y.H. Domination Sets and Domination polynomials of cycles. Global Journal of Pure and Applied Mathematics, Vol. 4, No. 2, (2008).

3. Dong. F.M, Hendy M.D, Teo K.L. Little. C.H.C. The vertex – cover polynomial of a graph, Discrete Mathematics 250,71 – 78 (2002). 4. Douglas B. West, Introduction to Graph Theory. 5. Frucht. R and Harary. F, Corona of two graphs, A equations. Math. 4, 322-324 (1970). 6. Gary Chartrand and Ping Zhang; Introduction to Graph Theory.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)