Cmjv02i06p0888 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (6), 888-892 (2011)

Irregular Total Labeling of Grid Networks INDRA RAJASINGH1, BHARATI RAJAN1 and S.TERESA AROCKIAMARY2 1

Department of Mathematics, Loyola College, Chennai 600 034, India 2 Department of Mathematics, Stella Maris College, Chennai 600 086, India ABSTRACT Given a graph G (V, E) a labeling ∂: V∪E→ {1, 2… k} is called an edge irregular total k-labeling if for every pair of distinct edges uvand xy, ∂(u) + ∂(uv) + ∂(v) ≠∂(x) + ∂(xy) + ∂(y). The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength of G. In this paper we examine the grid network which is a well known interconnection network, and prove that it is total edge irregular. Keywords: Irregular Total Labeling, Interconnection Networks, Grid Networks, Labeling.

1. INTRODUCTION

One way for processors to communicate data is to use a shared memoryand shared variables. However this is unrealistic for large numbers of processors. A more realistic assumption is that each processor has its own private memory and data communication takes place using message passing through an interconnection network. The interconnection network plays a central role in determining the overall performance of a multicomputer system. An interconnection network may be modeled by a simple graph whose vertices represent components of the

network and whose edges represent physical communication links, where directed edges represent one-way communication links and undirected edges represent two-way communication links, and the incidence function specifies a way that components of the network are interconnected by links. Such a graph is called the topological structure of the interconnection network, or network topology. Some interconnection network topologies are designed and some borrow from nature. For example hypercubes, complete binary trees, butterflies and torus networks are some

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 888-892 (2011)

of the designed architectures. Grids, Hexagonal networks, Honeycomb networks and Diamond networks, for instance, bear resemblance to atomic or molecular lattice structures.They are called natural architectures. It is known that there exist three regular plane tessellations, composed of the same kind of regular polygons: triangular, square, and hexagonal. They are the basis for the designs of direct interconnection networks with highly competitive overall performance. Grid connected computers and tori are based on regular square tessellations, and are popular and well-known models for parallel processing.

example of such network system is the Grid Network. Grid networks use the resources of many computers connected within the network to solve large-scale computational problems. Grid-based networks (or grids for short), such as meshes and tori, have been the underlying topology for many multicomputers, and have been extensively studied in the past as a graph topology. We generate a sequence { xr } of numbers such that x1 = 2, xr = 2 xr −1 − 1 . In our study, we consider square matrices M[ xr × xr ] of order , xr , r ≥ 1. The number of vertices in M[ xr × xr )] is

2. GRID NETWORKS

xr 2 and number of edges in M[( xr × xr )]

For decades, numerous methods have been developed to maximize the use of networked computers for largescale computing, and several protocols have been developed to efficiently utilize the resources within a distributed computing system. All these developments in technology have led to the possibility of using wide area distributed computers for solving large-scale problems. Large scale computing networks can provide the ability to achieve higher throughput computing by taking advantage of many networked computers that simulates a virtual computer architecture environment where process execution is distributed among the computers in the network. An

is xr 2 r , r ≥ 1. We begin with a few known results on tes(G). Theorem 1 Every multigraphG = (V, E) without loops of order n, size m, and maximum degree 0<∆<

10 −3 m 8n

satisfies tes(G) =  E + 2  .  3 

Theorem 2 Every graph G = (V, E) of order n, minimum degree δ > 0, and maximum degree ∆ such that ∆ 10 − 3 n satisfies tes(G) =  m + 2 . < δ

 3   

Theorem 3 For every integer ∆ ≥ 1, there is some n(∆) such that every graph

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 888-892 (2011)

G = (V, E) without isolated vertices with order n ≥ n(∆), size m, and maximum degree at most ∆ satisfies tes(G)=  m + 2  . 



We now give an algorithm to prove that for the square grid networks , the lower bound for tes is obtained.

890

1. Let t =tes(M[ x r × x r ])-tes(M[ x r −1 × x r −1 ]). Label the vertex ( y i , y j ) , 2 ≤ i ≤ x r − x r −1 , 1 ≤ j ≤ x r −1 as l ( y i , y j ) = l ( xr − xr −1 +1, y j ) + ( xr − xr −1 +1- yi ) x r . l ( y1 , y j ) = l

Algorithm tes(M[ x r × x r ])

( xr − xr −1 +1, y j ) + ( xr − xr −1 -1) x r + t

Input :Square Matrix M[ xr × xr ].

mod x r . 2. Having labeled the vertices of x r − x r −1

Algorithm: 1. Label the vertices and edges of M[ x 2 × x 2 ]as shown in Figure 1.

The n-2, n-1 and nth rows of M[ x3 × x3 ] and 1, 2 and 3rd columns of M[ x3 × x3 ] constitute a 3× 3 grid A isomorphic to Go (see Figure 2). Thus there are xr − xr −1 number of columns and rows that are yet to be labeled. Let us label the vertices first and the edges later so that the consecutive sums are got as in Figure 2. Denote the vertex of the M[ x3 × x3 ] matrix that is to be labeled as ( y i , y j ) . 4

1 2 1

xr − xr −1 number of columnsto the right of Go about the 3rd column. Denote the vertex of M[5 × 5] matrix that is to be labeled as ( y j , yi ) Label the vertex ( y j , yi ) , xr −1 ≤ j ≤ xr ,

i = xr − 1 as l ( y j , yi ) = l ( xr − xr −1 +1, yi −1 ) + ( xr − xr −1 +1- yi − 2 ) xr and xr −1 ≤ j ≤ xr , i = xr l( y j , yi ) = l

for

( xr − xr −1 +1, yi −1 ) + ( xr − xr −1 -1) x r + t mod x r .

Go : 3

number of rows and xr − xr −1 + 1 number of columns about the 3rd column to the left and above of Gowe now go to label the left out xr number of rows and

Figure 1: Edge irregular k-labeling of M[3× 3] square grid when k = 5.

3. Now we label the vertices of the diagonal matrix to Go. Let us denote the vertex of M[5 × 5] matrix that is to be labeled as ( y j , yi ) Label the vertex ( y j , yi ) , j = xr −1 − 1 ,

i = xr − 1 as l( y j , yi ) = l ( xr − xr −1 +1, yi −1 ) +

( xr − xr −1 +1- yi − 2 ) x r and for

j = xr −1 − 2 , xr −1 ≤ i ≤ xr l( y j , yi ) = l

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 888-892 (2011)

( xr − xr −1 +1, yi −1 ) +

( xr − xr −1 -1) x r +

t mod x r and l ( y2 , yi ) = l ( y j , y i ) . 5. Labeling of M[ x3 × x3 ] is shown in Figure 3. Output: tes(M[ x r × x r ]) =  x r 2 + 2  .

edge irregularity strength of Square Grid Networks. This problem is under investigation for certain other architectures like Hexagonal and Circulant Networks. REFERENCES

 

 

Level 2

Level 1

n- 2

A n- 1

n 1

Figure 2

Figure 3: Edge irregular k-labeling of M[5× 5] square matrix when k = 14

3. CONCLUSION In this paper, we have proved that that Square Grid Networks are total edge irregular and we have also obtained the total

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Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 888-892 (2011)

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Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)