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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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