Let A(G) be the adjacency matrix of a graph G. The rows of A(G) corresponding to a vertex v of G, denoted by ... Adjacency matrix, Mathematics Subject Classification: 05C99 ..... B. Kolman, R. Busby, S. C. Ross (2002), DiscreteMathematical ... Math. Soc., 4, 691â702. Harishchandra S. Ramane and Asha B. Ganagi (2013),.
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International Journal of Current Engineering and Technology
Research Article
Hamming Index of Class of Graphs Harishchandra S. Ramanea* and Asha B. Ganagia a
Department of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum - 590008, India
Abstract Let A(G) be the adjacency matrix of a graph G. The rows of A(G) corresponding to a vertex v of G, denoted by s(v) is the string which belongs to Z 2n , a set of n-tuples. The Hamming distance between the vertices u and v is the number of positions in which s(u) and s(v) differ. The Hamming index of a graph G is the sum of the Hamming distances between all pairs of vertices of G. In this paper we obtain the Hamming index of certain class of graphs. Keywords: Hamming distance, Hamming index. Adjacency matrix, Mathematics Subject Classification: 05C99
1. Introduction 1
Let Z2 ={0, 1} and (Z2 +) be the additive group where + denotes addition modulo 2. For any positive integer n, (n factors) = {( )} ) Thus every element of is an n-tuple ( written as where every is either 0 or 1 and is called a string or word. The number of 1’s in is called the weight of x and is denoned by wt(x). Let and be the elements of Then the sum x + y is computed by adding the corresponding components of x and y under addition modulo2. That is if and if The Hamming distance ( ) between the strings and is the number of i’s such that ,1 Thus ( ) = Number of positions in which x and y differ = wt(x ) Example: Let x = 01001 & y = 11010 and x + y = 10011, Therefore
(
)
wt(x
)=3
A graph G is called a Hamming graph [3,6 ] if each vertex ( ) can be labeled by a string of a fixed length s(v) ( ( ) ( )) ( ) for all such that ( ) ) is the length of shortest path joining u and where ( v in G. Hamming graphs are known as an interesting graph family in connection with the error-correcting codes and association schemes. For more details one can refer [S.
Bang et al,2008, V. Chepoi,1988, W. Imrich et al,1997, S. Klavzar et al, 2005, H. S. Ramane et al, 2013, R. Squer et al, 2001 ]. Motivated by the work on Hamming graphs in this we introduce and study the Hamming Index of a graph in association with its adjacency. 2. Preliminaries Let G be an undirected graph without loops and multiple + , ( ) * + edges. Let ( ) * be the vertex set and edge set of G respectively. The vertices adjacent to the vertex v are called the neighbours of v. The degree of a vertex v, denoted by degv is the number of neighbours of v. The adjacency matrix of G is a square matrix [ ] of order n where =0
if the vertex
( )
is adjacent to
otherwise.
Rows of A(G) represents the strings which belongs to . The Row of A (G) corresponding to the vertex ( )is the string denoted by s(v). Definition 2.1: Hamming distance between the vertices u ( ) is defined as ( ) and v of G denoted by ( ( ) ( )) Definition 2.2: The Hamming Index[10] of a graph G, denoted by H(G) is defined as, ( )
∑
(
) 1≤ i < j ≤ n
*Corresponding author: Harishchandra S. Ramane
205 |Proceedings of National Conference on ‘Women in Science & Engineering’ (NCWSE 2013), SDMCET Dharwad
Harishchandra S. Ramane et al
International Journal of Current Engineering and Technology, Special Issue1 (Sept 2013)
Hamming index is a graph invariant. Ex. :
H (G1 UG 2 )
H
v1
1i j n1
d
(u i , v j )
H
n1
d
n1 i j n1 n2
(u i , v j )
n1 n2
H
i 1 j n1 1
d
(u i , v j )
= n1
v2
deg( u ) deg( u (
v3
)
(
Fig. 1: Graph G For a graph G of Fig. 1, the adjacency matrix is 1 0 0 0 1 1 1 0 1 1 1 0
)
,
Let G1(n1, m1) and G2(n2, m2) be two graphs then the adjacency matrix of join of G1 and G2 is,
( )
)
(
)
(
)
(
)
(
)
(
)
Therefore H(G) = 4+1+1+3+3+2 = 14 ( G1 G 2 )
3. Hamming Index of certain class of graphs Definition: Consider two graphs G1(n1, m1) and G2(n2, m2. Then the graph whose vertex set n1Un2 and the edge set m1Um2 is called the union of G1 and G2 and is denoted by G1U G2. Theorem 3.1: Let G1(n1, m1) and G2(n2, m2) be two graphs then Hamming index of union of G1 and G2 is, H (G1 UG 2 ) H (G1 ) H (G 2 ) 2(n1 m2 n2 m1 )
Proof: Let G1(n1, m1) and G2(n2, m2) be two graphs then then the adjacency matrix of union of G1 and G2 is, ( G1 )
A(G ) 1 J
A(G 2 ) J
where A(G2 ) and A(G2 ) are adjacency matrices of G1 and G2 respectively and 1 1 J . 1
1 . . . 1 1 . . . 1 . . . 1 . . .1
Therefore,
H
1i j n1
( n1
)
d
(
n1 n2
{n i 1 j n1 1
1
H
(u i , v j )
H
(u i , v j )
d 1i j n1 n2
H (G1 G 2 )
] (G2)
where ( G1 ) and ( G2 ) are adjacency matrices of G1 and G2 respectively, O is the null matrix. Therefore,
)
Proof:
(
[
j
H (G1 G 2 ) H (G1 ) H (G 2 ) n1n2 (n1 n2 ) 2n1m2 2n2 m1
and the strings are ( ) ( ) ( )
( G1 UG 2 )
i
i 1 j n1 1
= H (G1 UG 2 ) H (G1 ) H (G 2 ) 2(n1 m2 n2 m1 ) Definition: Consider two graphs G1(n1, m1) and G2(n2, m2). The join of G1 and G2, denoted by G1 + G2, is G1UG2 along with each vertex of G1 joined to every vertex of G2 by an edge. Theorem 3.2: Let G1(n1, m1) and G2(n2, m2) be two graphs then Hamming index of G1 + G2 is,
v4
0 1 A(G ) 0 0
n1 n2
(u i , v j )
)
n1
d
n1 i j n1 n2
n1 n2
H
i 1 j n1 1
d
(u i , v j )
=
n2 [deg( u i ) deg( u j )]}
H (G1 G 2 ) H (G1 ) H (G 2 ) n1n2 (n1 n2 ) 2n1m2 2n2 m1 206 | Proceedings of National Conference on ‘Women in Science & Engineering’ (NCWSE 2013), SDMCET Dharwad
Harishchandra S. Ramane et al
International Journal of Current Engineering and Technology, Special Issue1 (Sept 2013)
Definition: Let G be a graph with vertices v1, v2,…vn . A graph G+ is obtained from G by inserting new vertices u1, u2, . . . un, and joining ui to vi by an edge, i=1, 2, . . . n
H (C ) 2
Theorem 3.3: Let Cn be cycle on n vertices then Hamming
H (C n ) 6n 2 6n
C n is,
index of
n
Proof: Let Cn be cycle on n vertices then the adjacency matrix of
C n is
A( C ) n I
I
O
H
H (C n )
1i j 2 n
H
1i j n
2 H
u , vCn
d
d
2n(n 2) 6n 2 6n 4n 5 n 2 H (C n ) =
(u i , v j )
(u, v)
d
n 1i j 2 n
d
Let Kn be complete graph on n vertices then the adjacency
2n
i 1 j n 1
H
n
d n 1i j 2 n
H
d n 1i j 2 n
(u i , v j )
2n
H
i 1 j n 1
n
(u i , v j )
d
n
(K )
(u i , v j )
i 1 j n 1
d
Therefore,
(u i , v j ) -
(u i , v j )
H
d
H i 1 j n 1
(u i , v j )
2n
H
i 1 j n 1
d
(u i , v j )
for pair of (u i , v j ) adjacent pairs
(u i , v j )
(u i , v j )
2 H
for pair of (u i , v j ) non adjacent pairs
d
(u i , v j ) hamming distances between n - adjacent pairs 4(n)
d
(u i , v j ) hamming distance between (n - n) non - adjacent pairs
2n
d
d
H
d n 1i j 2 n
n
(u i , v j )
2n
H
i 1 j n 1
d
(u i , v j )
2n
H
i 1 j n 1
n
H
1i j 2 n
1i j n n
d
I O
where A( K n ) is the adjacency matrix of Kn, I is the Identity matrix of order n and O is the null matrix
2n
H
is
A( K ) n I
(u i , v j )
(u i , v j ) 2( n C 2 )
n
2n
K n
matrix of
ii) H
K n is,
Proof:
H ( K n )
H
i 1 j n 1
H ( K n ) H ( K n ) n 3 n 2
- - - - (1) Where, i)
H
Hamming index of
where A(Cn ) is the adjacency matrix of Cn, I is the Identity matrix of order n and O is the null matrix Therefore,
n
C2 2 n C2 4n 2(2n) 4(n 2 3n)
Theorem 3.4: Let Kn be complete graph on n vertices then
C n )
n
n
[H. S. Ramane et al, 2013] = 2n(4n-5
4n 5 H (C n ) H (C n ) n2
(
H (C n )
2
u , vCn
H
d n 1i j 2 n
n
(u i , v j )
2n
H
i 1 j n 1
d
(u i , v j )
--
Where, i)
H
d
(ui , v j ) 2( n C 2 )
ii) n
2n
H i 1 j n 1
n
d
(u i , v j )
2n
H
i 1 j n 1 n
+Hamming distance between [(n2-n)-2n]-pairs with non common neighbor= 2(2n) + 4(n2-3n)
(u, v)
- - - (1)
n 1i j 2 n
= Hamming distance between 2n pairs with common neighbour
d
d
H
i 1 j n 1
n
2n
H i 1 j n1
d
(u i , v j )
2n
d
(u i , v j )
for pair of (u i , v j ) adjacent pairs for pair of (u i , v j ) non adjacent pairs
(ui , v j ) hamming distances between n - adjacent pairs (n 1)(n)
Substituting in eqn(1), 207 | Proceedings of National Conference on ‘Women in Science & Engineering’ (NCWSE 2013), SDMCET Dharwad
Harishchandra S. Ramane et al
n
2n
H i 1 j n 1
International Journal of Current Engineering and Technology, Special Issue1 (Sept 2013)
(ui , v j ) hamming distance between (n - n) non - adjacent pairs = (n - 1)(n 2
d
Substituting in eqn(1), H ( K n )
H (K ) 2 C 2 n
n
n
2
H (Kn ) n n 3
C2 n(n 1) (n 1)(n2 n)
2
Acknowledgement Authors are thankful to Prof. P. R. Hampiholi for his suggestions. References S. Bang, E. R. van Dam, J. H. Koolen (2008), Spectral characterization of the Hamming graphs, Linear Algebra Appl., 429, 2678–2686 V. Chepoi (1988), d-connectivity and isometric subgraphs of Hamming graphs, Cybernetics, 1 (1988), 6–9.
2
W. Imrich, S. Klavˇzar (1993), A simple O (mn) algorithm for recognizing Hamming graphs, Bull. Inst. - n) Combin. Appl., 9, 45–56. W. Imrich, S. Klavˇzar (1996), On the complexity of recognizing Hamming graphs and related classes of graphs, European J. Combin., 17, 209–221. W. Imrich, S. Klavˇzar, Recognizing Hamming graphs in linear time and space, Inform. Process. Lett., 63 (1997), 91–95. S. Klavˇzar, I. Peterin (2005), Characterizing subgraphs of Hamming graphs, J. Graph Theory, 49, 302– 312. B. Kolman, R. Busby, S. C. Ross (2002), DiscreteMathematical Structures, Prentice Hall of India, New Delhi, M. Mollard (1991), Two characterizations of generalized hypercubes, Discrete Math., 93, 63–74. B. Park, Y. Sano (2011), The competition numbers of Hamming graphs with diameter at most three, J. Korean Math. Soc., 4, 691–702. Harishchandra S. Ramane and Asha B. Ganagi (2013), Hamming distance between vertices and Hamming index of graphs, SIAM Journal on Discrete Mathematic(preprint) R. Squier, B. Torrence (2001), A. Vogt, The number of edges in a subgraph of a Hamming graph, Appl. Math. Lett., 14,701– 705 D. B. West (2009), Introduction to Graph Theory, PHI Learning, New Delhi
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