INTERNATIONAL CONFERENCE ON
APPLIED MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
ICAMTCS 2013
Editors
Dr. L. Mary Florida
Dr.V. Vilfred
ICAMTCS 2013
ST. XAVIER'S CATHOLIC COLLEGE OF ENGINEERING
NAGERCOIL, TAMILNADU, INDIA
International Conference on Applied Mathematics and Theoretical Computer Science – 2013
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© 2013 Bonfring This Conference is Dedicated to Srinivasa Ramanujan Srinivasa Ramanujan (22 December 1887 – 26 April 1920) National Mathematics Year Preface
The ICAMTCS Team warmly welcomes our distinguished delegates and guests to the
International Conference on Applied Mathematics and Theoretical Computer Science, ICAMTCS
2013, organized by the Department of Mathematics, St. Xavier’s Catholic College of Engineering,
Chunkankadai, Nagercoil-629 003, during January 24-25, 2013. This Conference is dedicated to the
great Mathematician Srinivasa Ramanujan.
This conference is organized to gather international scientists and researchers from around
the world to present their work and to gain insight into the significant challenges currently being
addressed.
This conference solicits technical research submissions related to diverse topics in Applied
Mathematics and Theoretical Computer Science. The technical research papers appear in the
conference Proceedings have been reviewed and selected by the members of the technical
committee on the basis of originality, significance and clarity. Special issues of the ISRO Journals
(International Organization of Scientific Research) ISRO Journal of Mathematics (ISRO-JM), ISRO
Journal of Computer Engineering (ISRO-JCE), ISRO Journal of Electronics and Communication
Engineering (ISRO-JECE), ISRO Journal of Electrical and Electronics Engineering (ISRO-JEEE),
International Journal of Electronics Communication and Computer Engineering (IJECCE) are
attached to the conference. Selected papers will appear in any of these journals after due refereeing
process as per the norms of the respective journals.
The conference will therefore be a unique event, where delegates will be able to appreciate
the latest results in their field of expertise and acquire additional knowledge in other fields. The
program has been structured to favour interactions among delegates coming from many varied
horizons, scientific, geographic, academia and industry.
We are grateful to all those who have contributed to the success of ICAMTCS 2013. We are
delighted to offer the proceeding of this Conference to the International Research Community.
Editors
Dr.L. Mary Florida
Dr.V. Vilfred
Ch
hairma
an’s Me
essage Most Rev. Peter R
Remigus, D.D of Kottar Bishop o
Itt gives me im
mmense pleaasure that Stt. Xavier’s Caatholic Colleege of Engineeering (SXCC
CE), a top‐notch
h institution of Kanyakum
mari District is organizingg on 24‐25 Jaanuary, 2013
3 an Internattional Conferen
nce on Appliied Mathem
matics and Theoretical T
C
Computer Sccience (ICAM
MTCS 2013)), Co‐
sponsoreed by DRDO, SERC and CSIR and Contr
S
ributed in co
ollaboration w
with IEEE – M
Madras section. In
n bringing th
he fields of Mathematiccs and Comp
puter Science within thee preview of this conferencce, the depaartment of Mathematics M
has demonstrated the im
mportance of o Mathematics in other maajor area of study as well as the need for inter‐dissciplinary collaboration aand research
h. The presence and particip
pation of eminent scholars and reseaarchers of intternational rrepute will en
nsure he Conference a memorab
ble experiencce for all. a high staandard of exccellence and will make th
I Congratulatee and wish all a those inv
volved in the organization of this con
nference for their d interest an
nd meticulo
ous work an
nd I am surre that it will w be an en
nriching learning sustained
experiencce for all the participantss. I pray for God
d’s blessings o
on this International Con
nference. With pray
yers & blessiing, + Bisho
op Peter Rem
migius Bisshop of Kottaar Prresiden
nt’s Me
essage Msgr.V. Mariadassan Vicar Gen
neral – Ko
ottar Dioceese I am delighted
d that St. Xav
vier’s Catholic College of f Engineeringg (SXCCE) is going to orgganize national Confference on “A
Applied Math
hematics and
d Theoretical Computer Science”. an Intern
This inter‐dissciplinary Co
onference, which w
brings together researchers s
an
nd experts in
n two mportance off Mathematiccs and Compu
uter Science. areas of sstudy, foregrounds the im
Itt is my immeense joy and
d pride to congratulate th
he participan
nts of the co
onference an
nd the organizin
ng committeee for makingg it possible to bring Matthematics fro
om differentt fields under one roof. I wiish them eveery success aand pray thatt God will blless their effo
orts with succcess, so thaat this conferencce may be a w
wonderful leearning experrience for alll the particip
pants. With pray
yers and besst wishes, Msgr.V
V. Mariadasaan Viccar General Corrrespondent's Messa
age Rev.Fr.A
A. Jesu Ma
arian The National year of Maathematics ended e
on 22.12.2012 th
hat marked the 125th birth was a pioneerr for many m
mathematiciaans to anniversaary of Matheematician Sriinivasa Ramaanujan. He w
distinguissh themselvees by perform
ming at very
y high levels. It is pertinen
nt that at thiis juncture H
H & Sc and R & D
D of our colleege is jointly
y organizing an “Internattional Conferrence on App
plied Mathem
matics and Theo
oretical Comp
puter Sciencee”. Everyone willl agree that tthe competen
nt mathematticians are in
nadequate in
n the country
y. It is ng that many
y do not purssue this disccipline at thee advanced levels, resultiing in the fieeld of saddenin
mathemaatics. I’m sure, the conference will w impresss and convince the participants c
that mathematics is a p
a
basis for all sciences and technology, appreciate its nicetiies and the pivotal r
ole it plays in various branchess of studies an
nd remove th
he illusion an
nd terrifying fear of the subject. R
Researches, inventions, i
innovations and appliccations of scciences and
d technologies in different ways are need of the hou
ur to overcom
me current crisis. Our colllege, that beecomes a plattform mpart and excchange the laatest develop
pments by exxperts from in and n
for the learned and leearning to im
abroad, w
will boost eveeryone to hav
ve more tastte of the subject and inten
nsification off their researrch. I appreciate the t organizeers and participants and
d wish them maximum harvest h
from
m this unique ev
vent. With prayers & blessi
y
ing, Fr.A. Jesu Marian, C
Corresponden
nt Prrincipa
al's Me
essage Dr.S. Josseph Sekh
har Today, the facets of basiccs sciences and a engineerring are chaanging rapidly, and scien
nce & Technolo
ogy are becom
ming interdissciplinary ass well as multidisciplinary
y. The majorr developmen
nts in engineeriing and Tecchnology aree closely reelated to thee continuingg revolutions in the fieeld of informatiion and com
mmunication technologies. The devellopments of mathematiccal concepts have profound
d impact on tthe developm
ment of break
kthrough tecchnologies in
n engineeringg fields, espeecially in compu
uter sciencee and engin
neering. Kno
owing the importance of fundameental researcch in mathemaatics, St. Xav
vier’s Catholic College of Engineering is comm
memorating the 125th birth anniversaary of the gen
nius of math
hematics, Srin
nivasa Ramanujan by orgganizing ICAM
MTCS – 2013
3. The meticulou
usly planned
d activities su
uch as invited
d talks and p
paper presen
ntations durin
ng the conferrence will defin
nitely bring significant development
d
ts in the ressearches relaated to theo
oretical comp
puter science. or their maid
den attempt for holding this internattional I congratulatee the organizzing team fo
o all the conttributors, rev
viewers and
d experts for their conferencce. I expresss my sincere gratitude to
unconditional supporrt in organizzing this megga event. Also I extend m
my profound thanks to D
DRDO, SERC, CSIIR and IEEE Madras Sectiion, for theirr financial and technical ssponsorship. With pray
yers & best wiishes, Dr..S. Joseph Sekh
har, Principal Bursarr's Messsage Rev.Dr.A
A.S. Micha
aelraj Itt gives me unbounded joy to know that our college is organizing the t
internattional conferencce on Applied Mathematics and Theo
oretical comp
puter sciencee on 24th and 25th of Jan
nuary 2013. Eucclid once said
d that the law
ws of nature are but the m
mathematicaal thoughts o
of God. Theree is no doubt thaat God in His Goodness haas shared wiith us the maathematical tthoughts. Wee use Mathem
matics in our homes, in our w
work place aand in life in general. Besides we know
w that Matheematics is playing mportant role in buildingg up our mo
odern civilization by perrfecting all sciences. s
Scieences a very im
progressees only with
h the aid of Mathematics
M
s. So it is ap
ptly remarked by the fam
mous philoso
opher Kant “A SScience is exaact only in so
o far as it employs
p
Mathematics”. m
I wish to co
ongratulate the t
Departm
ment of Matthematics fo
or taking leeadership ro
ole in ng such an important in
nternational conference on applied Mathematics
M
s and Theoretical organizin
computerr Science. Maay this conference give an
n ample opportunity to eexchange thee valuable ideeas of Mathemaaticians, Reseearchers from
m Science an
nd Engineering. Congratss to the edito
orial board o
of this splendid Souvenir. yers & best w
wishes, With pray
Dr.A.S. Michaellraj, Bursar Co­Sp
ponsored by The Scie
ence and Eng
gineering Ressearch Counccil (SERC) waas established
d in 1974 and
d is an apex body thrrough which th
he Departmen
nt of Science aand Technologgy (DST), Gov
vt. of India pro
omotes R&D program
mmes in newly
y emerging and
d challenging areas of scien
nce and engineeering. SERC is composed of emineent scientistss, technologists drawn fro
om various universities/na
u
ational laboraatories and Industry
y. This Councill is assisted by
y Programme Advisory Com
mmittees (PAC
Cs) in variouss disciplines of Sciencce & Engineeriing. Th
he Council of o Scientific & Industria
al Research (CSIR) ‐ thee premier in
ndustrial R&D
D orgganization in India was co
onstituted in 1942 by a reesolution of th
he then Centrral Legislativee Assembly. It is an autonom
mous body registered under the Registtration of Societies Act of 18
860.CSIR aimss to provide industrial co
ompetitivenesss, social welfare, strong S&T base forr strrategic sectorss and advanceement of fundaamental know
wledge. Defence
e Research & & Development Organisattion (DRDO) works underr Department of Defence Research
h and Develop
pment of Ministry of Defen
nce. DRDO dedicatedly worrking towardss enhancing self‐reliaance in Defence Systems and a
undertakees design & development d
leading to prroduction of world cllass weapon systems and equipment in i accordancee with the exxpressed neeeds and the qualitativ
ve requiremen
nts laid down by the three sservices. DRDO iss working in various v
areass of military technology t
w
which include aeronautics, armaments, com
mbat vehiclees, electronicss, instrumenttation engineeering system
ms, missiles, materials, naval systemss, advanced com
mputing, simu
ulation and liffe sciences. DR
RDO while striving to meet the Cutting ed
dge weapons technology reequirements pro
ovides ample spinoff benefiits to the socieety at large thereby contributing to the nation buildingg. ICA
AMTCS‐2013 ttechnically co
o‐sponsored by IEEE Madraas Section. IEE
EE Madras Section is one of the t most activ
ve 10 Section
ns in India, co
oming under Asia A
–Pacific Region, R
the Region R
10 of IEE
EE. It covers southern s
partt of India, con
nsisting of thee Capital Terrritory of Tamiil Nadu and Po
ondicherry. IEEE Madras Seection was fo
ormally starteed, as a Subseection of Banggalore Section
n, in 1973 an
nd was later eleevated into a ffull Section, in
n 1978.IEEE M
Madras Section
n has 183 stud
dent Branchess, which is the highest in thee world. (As of 15th Feb. 20
012), our Mem
mbership streength is 11,73
34. (Life Fello
ow‐2, Fellow‐‐2, Life seniorr ‐9, Life mem
mber ‐5, Sr. Meember ‐101, Member M
‐2321, Affiliate – 42, Associate – 149, Gradu
uate student members – 1295, 1
Student members ‐
78
808). The Secttion has, 13 Society Chapteers, besides GOLD & WIE Affinity A
groups and OE and
d TMC councills. The IEEE Maadras Section organizes speecial technicall lecturers, tuttorials, faculty
y developmen
nt programs, p
professional development and training, conferences, exh
hibition etc., aand also givess financial assistance to stu
udent branchees to conduct cconferences and FDPs and project p
fundin
ng for the stu
udent membeers. It publish
hes a compreh
hensive montthly bulletin highlighting h
various professiional activities carried out under IEEE M
Madras section
n. In the year 2010, it receiived the presttigious R‐10 Disstinguished Laarge Section A
Award for its o
overall activitiies. Dr. Lowell W. Beineke, Schrey Chair of Mathematical Sciences, Professor of Mathematics,
Indiana University ‐ Purdue University
Fort Wayne, Indiana, USA.
Line Graphs and Beyond ­ A Survey Abstract The line graph operation, in which the edges of one graph are taken as the vertices of a new graph, with adjacency preserved, is arguably the most interesting of graph transformations. In this survey, we will begin by looking at various characterizations of line graphs, focusing on results related to our set of nine forbidden subgraphs. In contrast to many concepts in graph theory, the directed counterpart is quite different to the undirected concept, and we look at some of the differences between the two, including characterizations of line digraphs. This will be followed by a discussion of some of the most important properties of line graphs, including planarity and Hamilton city. Line graphs have been generalized in many interesting ways, and we conclude with some of our results on two of these: (a) the Krausz dimension, for which line graphs are those with value 1 or 2, and (b) super line graphs, in which the vertices correspond to sets with more than one edge. Dr.S. Arumugam, Director, n‐CARDMATH Senior Professor (Research), Kalasalingam University, Tamil Nadu, India, Conjoint Professor, School of Electrical Engineering and Computer Science,
The University of NewcastleNSW 2308, Australia.
Visual Cryptography Scheme ­ A Survey Abstract Let P be a set of n elements, called participants. Consider a secret image which consists of black and white pixels. A (k, n)‐ Visual Cryptography Scheme is a method of constructing n shares of the secret image, one for each participant, in such a way that the secret image can be reconstructed from k or more shares but not from less than k shares. The process of reconstruction is purely visual in the sense that if each share is xeroxed on a transparency, the recovering of the image is just by stacking the respective transparency sheets. In this talk we give a brief survey of Visual Cryptography Scheme and its applications. Dr. Jay Bagga, Professor of Computer Science,
Department of Computer Science,
Ball State University,
Muncie, Indiana, 47306‐0450, USA.
Graphical Models in Bioinformatics Abstract Graphical models are used extensively in investigations of biological networks and associated processes. Several problems related to DNA sequences have been modeled by graphs and those are solved by well‐known techniques such as dynamic programming. For networks based on biological processes, an investigation of smaller local structures, or graphlets, leads to a better understanding of the entire network. Network motifs are patterns of interconnections that occur with signi_cantly larger frequencies than those in randomized networks. Such network motifs have been found in biological networks, sequential logic circuit networks, and the networks of the world wide web. Biological networks have been investigated using network models such as the Erdos‐R_enyi model, the geometric random network model, and exponential random graph model. In particular, the Erdos‐R_enyi and the geometric random network models were used in the study of graphlets in Saccharomyces cerevisiae protein‐protein interaction (PPI) networks, and exponential random graph models have been employed to study biological databases such as RegulonDB. In this talk we present some extensions. We present a brief survey of graph algorithms and methodologies that are used in the investigations of certain biological networks. We discuss recent work in structural graph theory that has been motivated by applications in DNA sequencing, text compression and scheduling problems. Dr. Shariefuddin Pirzada, Professor of Applied Mathematics,
Department of Mathematics,
University of Kashmir,
Srinagar, India.
Combinatorics of Digraph Sequences Abstract Ranking of objects is a typical practical problem. One of the popular ranking methods is the pairwise comparison of the objects. Many authors describe di_erent applications: e.g. Landau biological, Hakimi chemical , Kim et al. and Newman et al. network modeling, Bozoki, Fulop, Keri, Poesz, Ronyai et al. economical , Liljeros et al. human relation modeling , while Ivanyi et al. and Pirzada et al. sport applications. A tournament is a complete oriented graph and the score of a vertex in a tournament is its outdegree. Landau characterized the scores of tournaments, which is known as Landau's theorem for which there exist more than a dozen proofs in the literature. One interpretation of a tournament is as a competition where n participants play each other once in a match that cannot end in a tie and score one point for each win. Player v is represented in the tournament by vertex v and an arc from vertex u to vertex v means that u defeats v. Avery characterized scores in oriented graphs and like the above type of interpretation, an oriented graph is a competition where now ties are included. The author extended these concepts to various types of digraphs. One of such extensions is the concept of mark in digraphs, the mark being associated to the vertex of a digraph. The sequences of marks in non‐decreasing order or non‐
increasing order is called the mark sequence of the digraph. Here again among other representations of such digraphs, one is that in which n participants play each other more than once including the ties. We provide various characterizations of mark sequences in di_erent types of digraphs. These also provide algorithms for the constructions of such digraphs. Further we interpret an oriented graph as the result of a football tournament with teams represented by vertices in which the teams play each other once, with an arc from team u to team v if and only if u defeats v. A team receives three points for each win and one point for each draw (tie). With this f‐scoring system, team v receives a total of fv points. We call the sequence F = [f1; f2; _ _ _ ; fn] as the football sequence, if fi is the f‐score of some vertex vi. Thus a sequence F = [f1; f2; _ _ _ ; fn] of non‐negative integers in non‐
decreasing order is a football sequence if it realizes some oriented graph. References [1]
P. Avery, Score sequences of oriented graphs, J. Graph Theory, 15, 3 (1991) 251‐257. [2]
S. Bozoki, J. Fulop, A. Poesz, On pairwise comparison matrices that can be made consistent by the modi_cation of a few elements, Cent. Eur. J. Oper. Res., 19 (2011) 157‐175. [3]
S. Bozoki, J. Fulop, L. R_onyai, On optimal completion of incomplete pairwise comparison matrices, Math. Comput. Modelling, 52 (2010) 318‐333. [4]
S. L. Hakimi, On the realizability of a set of integers as degrees of the vertices of a simple graph, SIAM Appl. Math., 10 (1962) 496‐506. [5]
A. Iv_anyi, Deciding football sequences, Acta Univ. Sapientiae,Inform.,4, 1 (2012) 130‐183. [6]
H. Kim, Z. Toroczkai1, P. L. Erdos, I. Miklos and L. A. S_zekely, Degreebased graph construction, J. Phys. A: Math. Theor. 42(2009), 392001 [7]
H. G. Landau, On dominance relations and the structure of animal societies. III. The condition for a score sequence, Bull. Math.Biophys., 15, (1953) 143‐ 148. [8]
F. Liljeros, C. R. Edling, L. Amaral, H. E. Stanley, Y. Aberg, The web of human sexual contacts, Nature 411 (2001) 907‐908. [9]
M. E. J. Newman, A. L. Barabasi, D. J. Watts, The structure and Dynamics of networks (Princeton Studies in Complexity, Princeton UP) (2006) pp 624. [10]
S. Pirzada, An Introduction to Graph Theory, Orient BlackSwan, Hyderabad, 2012. [11]
S. Pirzada, A. Iv_anyi, M. A. Khan, Score sets and kings, in (ed. A. Iv_anyi) Algorithms of Informatics, Vol. 3, AnTonCom, Budapest, 2011, pp. 1451‐ 1490. [12]
S. Pirzada, M. A. Khan, Football sequences, 2012 (manuscript under preparation). [13]
S. Pirzada, Merajuddin and Y. Jainhua, On the scores of oriented bipartite graphs, J. Math. Study, 33(4) (2000), 354‐359. [14]
S. Pirzada, U. Samee, Mark sequences in digraphs, Seminare Lotharingien de Combinatoire, 55 (2006), Art.B55c. [15]
S. Pirzada, Mark sequences in multidigraphs, Discrete Mathematics and Applications, 17, 1 (2007), 71‐
76. [16]
K. B. Reid, Tournaments: Scores, kings, generalizations and special topics, Congr. Numer., 115 (1996) 171‐211. Dr.S. Ramachandran Professor of Mathematics,
Noorul Islam University, India
Degree Associated Reconstruction Abstract Let G be a graph/digraph and v be a vertex of G. G – v (as an unlabeled graph/digraph) is called a card of G. G – v with which degree of v ( (od(v), id(v)) in the case of digraphs) is also given is called a degree associated card or dacard of G. The multiset of cards (dacards) of G is called the deck (dadeck) of G. Ulam’s graph reconstruction conjecture (URC) (1942) claims that graphs having the same deck are isomorphic. The digraph reconstruction conjecture (DRC) (1964) claims that digraphs having the same deck are isomorphic. In 1976, six infinite families of counterexample pairs to DRC were found by Stockmeyer and consequently, truth of URC became doubtful. (1) In 1979, dacards were used while devising a test to decide whether a deck has actually come from a graph/digraph. (2) It was also proposed (Degree Associated Reconstruction Conjecture (DARC)) that “all graphs/digraphs having the same dadeck are isomorphic” and it is still open. DARC is stronger than URC but weaker than the disproved DRC. All the digraphs forming part of the known counterexample pairs to DRC were proved to obey DARC. (3) Degree associated reconstruction number was introduced in 2000 to study URC/DARC from a different angle and it has been calculated for some classes. (4) In 2012, a proposed characterization of similar vertices in graphs has been proved to be equivalent to URC and the corresponding proposal for digraphs is false. However, the corresponding proposal for digraphs using degree association is open and has been proved to be equivalent to DARC. Thus some form of degree association not only overcomes the counterexamples to the DRC, but also that it might be essential in order to register more progress in the area. We discuss some of these in this talk. Dr.V. Vilfred, Professor of Mathematics,
St.Jude’s College,Thoothoor,
Tamil Nadu, India
New Abelian Groups from Isomorphism of Circulant Graphs Abstract Abstract Adam’s isomorphism of circulant graphs is taken as Type­1 isomorphism. Vilfred defined type­2 isomorphism, a new type of isomorphism different from already known Adam’s isomorphism on circulant graphs and made extensive study on this. This study helps us to find more graphs as circulant graphs even though they may not be in the form of Cn(R), circulant graph of order n with R as the set of jump sizes, R ⊆ {1,2,…,⎣n/2⎦}. VB program POLY215.EXE demonstrates the transformation used to define type­2 isomorphism and how it acts on circulant graph C8n(R) for R = {2,2r­1, 4n­(2r­1)}, 2r,n∈2N. We obtain abelian groups (Adn(S),o) and (Adn(Cn(R)), o) related to Adam’s isomorphism and (Vn,r(Cn(R)), o) related to type­2 isomorphism. We call the group (Vn,r(Cn(R)), o) as type­2 group on Vn,r(Cn(R) under the operation ’o’. It is proved that for 2 ≤ n, 3 ≤ k, 1 ≤ 2r­1 ≤ 2n­1, and n ≠ 2r­1, (V8n,2(C8n(R)), o) = (V8n,2(C8n(S)), o) is a group of type­2 w.r.t. r = 2 where n,p1,p2,…,pk­2∈N, gcd(p1,p2, ... , pk­2) = 1, R = {2r­1, 4n­2r+1, 2p1, 2p2 .. , 2pk­2} and S = {2n­2r+1, 2n+2r­1, 2p1, 2p2 ,…,2pk­2}. An important result is that for R = {2, 2r­1, 2s­1}, 1 ≤ t ≤ [n/2], 1 ≤ 2r­1 < 2s­1 ≤ [n/2], n,r,s,t∈N if (Vn,2(Cn(R)), o) is a group of type­2 w.r.t. 2, then n ≡ 0(mod 8), 2r­1+2s­1 = n/2, t = n/8 or 3n/8 and 2r­1 ≠ n/8, 1 ≤ 2r­1 ≤ n/4, 16 ≤ n, n∈N. Keywords Adam’s Isomorphism, Adam’s Conjecture, Type‐1 Isomorphism, Type‐2 Isomorphism, Reflexive Modular Reduction, Symmetric Equidistance Condition, Abelian Groups (Adn(S), o), Adn(Cn(R), o), (Vn,r(S), o), (Vn,r(Cn(R)), o), Type‐2 Group. Introduction Investigation of symmetries of structure yields powerful results in Mathematics. Circulant graphs form a class of highly symmetric mathematical (graphical) structures. In 1846 Catalan (cf.[8]) introduced circulant matrices, and their properties have been investigated by many authors since. An excellent account can be found in the book by Davis [8]. In 1994, Vilfred completely settled a conjecture due to Sachs [19] on the existence of self‐complementary circulant graphs (See pages 45‐
46, 55‐56 of [20].). Let n be a positive integer and R a subset of {1,2,…, [n/2]}. The circulant graph Cn(R) has vertices v0,v1,v2,...,vn‐1 with vi adjacent to vi+r for each r∈R, subscript addition taken modulo n, 0 ≤ i ≤ n–1. If a graph G is circulant, then its adjacency matrix A(G) is circulant. It follows that if the first row of the adjacency matrix of a circulant graph is [a1,a2,…,an], then a1 = 0 and ai = an‐i+2 , 2 ≤ i ≤ n. When discussing circulant graph Cn(R) for a set R = {r1,r2,…,rk}, we will often assume, with‐out further comment, that the vertices are the corners of a regular n‐gon, labeled clockwise. The circulant graphs C7(1,3), C8(2,4) and C10(1,4,5) are shown in Figures 1, 2 and 3. Through‐out this paper, for a set R = {r1,r2,…,rk}, Cn(R) denotes Cn(r1,r2,…,rk) where 1 ≤ r1 < r2 < … < rk ≤ [n/2] and for x∈N and R ⊂ N, xR ={xr : r∈R}, we consider only connected circulant graphs of finite order and all cycles have length at least 3, unless otherwise specified. However when n/2∈R, edge vi vi+n/2 is taken as a single edge for considering the degree of the vertex vi or vi+n/2 and as a double edge while counting the number of edges or cycles in Cn(R), 0 ≤ i ≤ n–1 [2‐12,14‐25]. We generally write Cn for Cn(1). It was originally noted by Adam [1] that if x and n are relatively prime, then Cn(R) and Cn(xR) for R = {r1,r2,...,rk} are isomorphic circulant graphs. Two circulant graphs Cn(R) and Cn(S) for R = {r1,r2,...,rk} and S = {s1,s2,...,sk} are Adam isomorphic if there exists a positive integer x relatively prime to n with S = {xr1,xr2,…,xrk}n* where < ri >n*, the reflexive modular reduction of a sequence < ri > is the sequence obtained by reducing each ri modulo n to yield ri’ and then replacing all resulting terms ri’ which are larger than n/2 by n‐ri’ [1]. Adam conjectured that any two isomorphic circulant graphs are Adam isomorphic [1]. Elspas and Turner [10] noted that the two circulant graphs C16(1,2,7) and C16(2,3,5) are isomorphic but not of Adam. Muzychuk [15,16] proved that Adam’s conjecture is valid if n is either square‐free or twice square‐free. Thus there exists at least one more type of isomorphism, other than Adam isomorphism among circulant graphs. Main Results Hereafter Adam isomorphism on circulant graphs is called as type­1 isomorphism. Vilfred [20] defined a new type of circulant graph isomorphism, different from Adam isomorphism, and studied them under the heading ‘generalized circulant graph isomorphism’. This new type of isomorphism is called Type­2 isomorphism. In this paper, we prove the following results. For 3 ≤ k, circulant graphs C8n(R) and C8n(S) are type‐2 isomorphic for the sets R = {2r‐1, 4n–2r+1, 2p1,2p2,…,2pk‐2} and S = {2n‐
2r+1, 2n+2r‐1,2p1, 2p2,..., 2pk‐2} where gcd(p1,p2,...,pk‐2) = 1, 2 ≤ n, 1 ≤ 2r‐1 ≤ 2n–1, n ≠ 2r‐1 and r,n,p1,p2,...,pk‐2∈N. If Cn(R) and θn,2,t(Cn(R)) for R = {2,2r‐1,2s‐1} are isomorphic circulant of type‐2 for some t, 1 ≤ t ≤ [n/2], 1 ≤ 2r‐1 < 2s‐1 ≤ [n/2] and r,s∈N, then n ≡ 0(mod 8), 2r‐1+2s‐1 = n/2, t = n/8 or 3n/8, 2r‐1 ≠ n/8, 1 ≤ 2r‐1 ≤ n/4 and 16 ≤ n, n∈N where θn,ri,t(vr) = vj+(q+jt)mi using subscript modulo n, r = j+qmi, 0 ≤ r ≤ n‐1, 0 ≤ j ≤ mi‐1 and 0 ≤ qmi ≤ n‐1. We define abelian groups (Adn(S), o) and (Adn(Cn(R)), o) related to Adam’s isomorphism and (Vn,r(Cn(R)), o) related to type‐2 isomorphism. The elements of the groups (Adn(Cn(R)), o) and (Vn,r(Cn(R)), o) may represent different forms of circulant graph Cn(R) including type‐1 and type‐2 isomorphic graphs to Cn(R). It is proved that for 2 ≤ n, 3 ≤ k, 1 ≤ 2r‐1 ≤ 2n‐1, n ≠ 2r‐1, r,n,p1,p2,…,pk‐2∈N where gcd(p1,p2 ,…,pk‐2) = 1, R = {2r‐1,4n‐2r+1, 2p1,2p2 ,...,2pk‐2} and S = {2n‐2r+1, 2n+2r‐1, 2p1, 2p2,...,2pk‐2}, (V8n,2(C8n(R)), o) = (V8n,2(C8n(S)), o) is a group of type‐2 w.r.t. 2. Another important result is that if (Vn,2(Cn(R)), o) is a group of type‐2 w.r.t. 2 for R = {2, 2r‐1, 2s‐1}, 1 ≤ t ≤ [n/2], 1 ≤ 2r‐1 < 2s‐1 ≤ [n/2], n,r,s,t∈N, then n ≡ 0 (mod 8), 2r‐1+2s‐1 = n/2, t = n/8 or 3n/8 and 2r‐1 ≠ n/8, 1 ≤ 2r‐1 ≤ n/4, 16 ≤ n∈N. Thus this study helps us to show more graphs as circulant graphs even though they may not be in the form of Cn(R). We give a VB program file, POLY215.EXE to give a clear understanding of the transformation used to define type‐2 isomorphism how it acts on circulant graph C8n(R) for R = {2, 2r‐1, 4n‐(2r‐1)}, 2r,n∈2N. Effort to understand the isomorphism that exists between C16(1,2,7) and C16(2,3,5) and to develop its general theory are motivation to do this work. For all basic ideas in graph theory, we follow [4,11,13]. Acknowledgement We express our sincere thanks to Professors, Dr.M.I.Jinnah, University of Kerala, Trivandrum, India, Dr.V.Mohan, Thiagarajar College of Engineering, Madurai, India, Dr.Lowell W Beineke, Indiana‐
Purdu University, U.S.A. and Dr.Brian Alspach, University of Regina, Canada for their valuable suggestions and guidance and to Lerroy Wilson Foundation, Nagercoil, India (Please visit Lerroy Wilson Foundation page on Facebook) for its support to this research work. References [1]
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Department of Applied Mathematics,
Motilal Nehru National Institute of Technology,
Allahabad‐ (U.P) India.
Computational Methods for Solving Boundary Value Problems Abstract During the past half‐century, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science and engineering. Computational Analysis naturally finds applications in all fields of engineering and the physical sciences. Numerical analysis is the area of mathematics and computer science that creates, analyzes and implements algorithm for solving numerically the problems of complicated mathematics. Such type of problems originate generally from real‐world applications of algebra, geometry and calculus, and they involve variables which vary continuously; these problems occur throughout the natural sciences, social sciences, engineering, medicine, and business. In this talk, numerical methods with computer applications will be discussed. Dr .Nicholas Sabu, Associate Professor of Mathematics, Indian Institute of Space Science and Technology,
Trivandrum, Kerala, India.
Two Dimensional Approximations of Piezoelectric Membrane Shells Using Gamma Convergence Abstract Lower dimensional models are preferred to three dimensional model when the thickness of the elastic plates or shells or rods is very small. There are two main reasons for this. One is their amenability for numeral methods and the other is that their simpler structure produce richer variety of results. In this context, depending on the geometry of the domain and the applied body and surface forces, various lower dimensional models has been proposed as approximation to the actual three dimensional models; for eg; bending, flexural, membrane, Koiter, Nagdhi models etc. But before computing numerical solution of any of these lower dimensional models we should first know whether this model is close enough to the three dimensional model we have in hand. Thus one is lead to the question of mathematically deriving the two dimensional model as the limit of the three dimensional model when the thickness parameter goes to zero. In this connection a lot of work has been done on the lower dimensional approximation of thin linear elastic plates, shells and rods by P.G.Ciarlet and co‐workers using the method of asymptotic analysis. But this method is not applicable to nonlinear problems. Another method to derive lower dimensional model is using gamma convergence theory which can be applied to nonlinear problems as well and in this talk I will discuss on "Deriving two dimensional model of thin elastic shallow shells using gamma convergence." Applied Mathematics and Theoretical Computer Science-2013
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