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International Journal of Computer Applications (09758887)

Volume 49No.3, July 2012

20

Total Edge Irregularity Strength of Butterfly Networks

IndraRajasinghDepartment of Mathematics

Loyola CollegeChennai

Bharati RajanDepartment of Mathematics au

Loyola CollegeChennai

S.Teresa ArockiamaryDepartment of Mathematics

Stella Maris CollegeChennai

ABSTRACTGiven a graph G (V, E) a labeling : VE{1, 2 k} is

called an edge irregular total k-labeling if for every pair of

distinct edges uv and xy, (u) + (uv) + (v) (x) + (xy) +

(y). The minimum k for which Ghas an edge irregular total

k-labeling is called the total edge irregularity strength of G. In

this paper we examine the butterfly network which is a well

known interconnection network, and obtain its total edge

irregularity strength.

KeywordsIrregular Total Labeling, Interconnection Networks, Butterfly

Networks , Labeling, Irregularity strength.

1. INTRODUCTIONLabeled graphs are becoming an increasingly useful family ofMathematical Models for a wide range of applications.Whilethe qualitative labelings of graph elements have inspired

research in diverse fields of human enquiry such as conflict

resolution in social psychology, electrical circuit theory andenergy crisis, these labelings have led to quite intricate fieldsof application such as coding theory problems, including the

design of good radar location codes, synch-set codes; missileguidance codes and convolution codes with optimalautocorrelation properties. Labeled graphs have also been

applied in determining ambiguities in X-Ray crystallographic

analysis, to design communication network addressingsystems, in determining optimal circuit layouts, radio-Astronomy, etc.

Interconnection network is a scheme that connects the units

of a multiprocessing system. It plays a central role in

determining the overall performance of a multicomputer

system. An interconnection network is modeled as a graph in

which vertices represent the processing elements and edges

represent the communication channel between them.

Interconnection networks play an important role for thearchitecture of parallel computers and PC-Clusters or

Networks of Workstations. Much of the early work on

interconnection networks was motivated by the needs of the

communications industry, particularly in the context of

telephone switching. With the growth of the computer

industry, applications for interconnection networks within

computing machines began to become apparent. Amongst the

first of these was the sorting of sequences of numbers, but as

interest in parallel processing grew, a large number of

networks were proposed for processor to memory and

processor to processor interconnection.

For a graph G (V, E), Baca et al.[1] define a labeling : V

E{1, 2 k} to be an edge irregulark-labelingof the

graph G if (u) + (uv) + (v) (x) + (xy) + (y) for every

pair of distinct edges uvandxy. The minimum kfor which thegraph G has an edge irregular total k-labeling is called the

total edge irregularity strength of the graph G, and is denotedby tes(G). More complete results on irregular total labelingscan be seen in the survey paper by Gallian [4] In this paper

we determine the total edge irregularity strength ofbutterflynetworks. As our main result we prove that the bound on tesforbutterfly networksis sharp.Theorem 1[1]Let Gbe a graph with medges. Then tes(G)

2

3

m

.

Theorem 2[1] Let Gbe a graph with maximum degree .

Then tes(G) 1

2

.

Theorem 3[2] A graph G(V, E) of order n, size m, and

maximum degree

n

m

8

100

3

satisfies tes(G) =2

3

m

Theorem 4[2]Every graph G(V,E) of order n, minimumdegree > 0, and maximum degree such that

24

10 3

n

satisfies tes(G) =

3

2m.

Theorem 5[2]For every integer 1, there is some n() suchthat every graph G(V, E) without isolated vertices with ordern n(), size m, and maximum degree at most satisfies

tes(G) =

3

2m .

Conjecture (Ivanco and Jendrol[5])For every graph Gwith

size mand maximum degree that is different from K5, thetotal edge irregularity strength equals

max

+23

,

+12

.

ForK5, the maximum of the lower bounds is 4 while tes(K5) =5.

Conjecture has been verified for trees by Ivanco andJendrol[5] and for complete graphs and complete bipartitegraphs by Jendrolet al. in [6]. As our main result, we provethe Conjecture for large graphs whose maximum degree is not

too large in terms of the order and the size. We now considerthe butterfly networks BF(r), and prove that for these

networks the bound on tesgiven in Theorem1 is sharp.

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