AN APPLICATION OF G d -METRIC SPACES AND METRIC DIMENSION OF GRAPHS

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1 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 AN APPLICATION OF -METRIC SPACES AND METRIC DIMENSION OF RAPHS Ms. Mausha R a Dr. Su Kuriakose A Research Scholar Uio Christia College Aluva Kerala Iia Dea Feeral Istitute of Sciece a Techolog Agamal Kerala Iia Abstract The iea of metric imesio i graph theor was itrouce b P J Slater i []. It has bee fou applicatios i optimiatio avigatio etwork theor image processig patter recogitio etc. Several other authors have stuie metric imesio of various staar graphs. I this paper we itrouce a real value fuctio calle geeralie metric R : where r v / W { v v v v... v v / v V } k eote a is use to stu metric imesio of graphs. It has bee prove that metric imesio of a coecte fiite simple graph remais costat if umbers of peat eges are ae to the o-basis vertices. Kewors Resolvig set Basis Metric imesio Ifiite raphs -metric.. Itrouctio raph theor has bee use to stu the various cocepts of avigatio i a arbitrar space. A work place ca be eote as oe i a graph a eges eote the coectios betwee places. The problem of miimum machie or Robots to be place at certai oes to trace each a ever oe eactl oce is worth ivestigatig. The problem ca be eplaie usig etworks where places are itercoecte i which a avigatig aget moves from oe oe to aother i the etwork. The places or oes of a etwork where we place the machies robots are calle lamarks. The miimum umber of machies require to locate each a ever oe of the etwork is terme as metric imesio a the set of all miimum possible umber of lamarks costitute metric basis. A iscrete metric like geeralie metric [4] is efie o the Cartesia prouct of a oempt set ito R is use to epa the cocept of metric imesio of the graph. The efiitio of a geeralie metric space is give i.6. I this tpe of spaces a o-egative real umber is assige to ever triplet of elemets. Several other stuies relevat to metric spaces are beig etee to -metric spaces. Differet geeraliatios of the usual otio of a metric space were propose b several mathematicias such as ahler [7 8] calle -metric spaces a Dhage [5 6] calle D-metric spaces have poite out that the results cite b ahler are iepeet rather tha geeraliatios of the correspoig results i metric spaces. Moreover it was show that Dhage s otio of D-metric space is flawe b errors a most of the results establishe b him a others are ivali. These facts are etermie b Mustafa a Sims [4] to itrouce a ew cocept i the area calle -metric space. DOI:0.5/graphoc.05.70

2 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 The cocept of metric imesio was itrouce b P J Slater i [] a stuie iepeetl b Harar a Melter i [3]. Applicatios of this avigatio of robots i etworks are iscusse i [4] a i chemistr while applicatios to problems of patter recogitio a image processig some of which ivolve the use of hierarchical structures are give i [5]. Besies Kuller et.al. provie a formula a a liear time algorithm for computig the metric imesio of a tree i []. O the other ha Chartra et.al. i [7] characterie the graph with metric imesio - a -. See also i [8] the tight bou o the metric imesio of uicclic graphs. Shamukha a Sooraaraaa [90] compute the parameters for wheels graphs costructe b oiig wheels with paths complete graphs etc. I 960 s a atural efiitio of the imesio of a graph state b Paul Eros a state some relate problems a usolve problems i []. Some other applicatio icluig coi weighig problems a combiatorial search a optimiatio []. The metric imesio of the Cartesia proucts of graph has bee stuie b Peters-Frase a Oellerma [3]. The metric imesio of various classes of graphs is compute i [ ]. I [4 5] the results of [3] are correcte a i [9 0] the results of [5] are refie.. Prelimiaries The basic efiitios a results require i subsequet sectio are give i this sectio... Defiitio A graph V E is a orere pair cosistig of a oempt set V V of elemets calle vertices a a set E E of uorere pair of vertices calle eges. Two vertices u v V are sai to be aacet if there is a ege uv E oiig them. The ege uv E is also sai to be iciet to vertices u a v egv is the umber of vertices i V aacet to it.. The egree of a verte v eote b A ege of a graph is sai to be a peat ege if it is iciet with ol oe verte of the graph. A uv -path is a sequece of istict vertices u vo v... v v so that vi is aacet to v i for all i i such a path is sai to be of legth. A uu -path of legth is a ccle eote b C. A graph is sai to be coecte if there is a path betwee ever two vertices. A complete graph is a simple graph a graph havig o loops a parallel eges i which each pair of istict vertices is oie b a ege... Defiitio A graph is ifiite if the verte set V is ifiite. A ifiite graph is locall fiite if ever verte has fiite egree. A ifiite graph is uiforml locall fiite if there eists a positive iteger M such that the egree of each verte is at most M. For eample the ifiite path P is both locall fiite a uiforml locall fiite b takig M..3. Defiitio If is a coecte graph the istace u v betwee two vertices u v V is the legth of the shortest path betwee them. Let W { w w... w k } be a orere set of vertices of a let v be

3 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 a verte of. The represetatio r v / W of v respect to W is the k-tuple v w v w... v w. If istict vertices of have istict represetatios co-oriates k with respect tow the W is calle a resolvig set or locatio set for. A resolvig set of miimum carialit is calle a basis for a this carialit is calle the metric imesio or β. locatio umber of a is eote b im or For each lamark the cooriate of a oe v i havig the elemets equal to the carialit of the set W a i th elemet of cooriate of v equal to the legth of the shortest path from the th i lamark to the verte v i. For eample cosier the graph of figure. The set W v } is ot a resolvig set of { v Figure. Figure. Sice r v / W r v /. Similarl we ca show that a set cosistig of two istict 3 4 W vertices will ot give istict cooriates for the vertices i. O the other ha W v v } { v3 form a resolvig set for i figure sice the represetatio for the vertices i with respect to W are r v / W 0 r v / W 0 r v3 / W 0 r v4 / W a it is the miimum resolvig set implig that im Remark A graph ca have more tha oe resolvig set. For eample cosier the graph i figure 3. Here we obtaie two resolvig sets amel {ab} a {ac}. Figure 3. A graph with two resolvig sets 3

4 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March Defiitio Let be a oempt set. A - Metric or geeralie metric is a fuctio from ito R havig the followig properties: for if 0 with for all 0 with for all the three variables...smmetr i all & Rectagle iequalit a a a a.6. Illustratio Let be a metric space. Defie R : b is a -metric satisfig the above five coitios. Coversel if is a -metric space it is eas to verif that is a metric space where For a 0 b ii b 0 b i c b iv [ ] Sice Similarl b v Now we recall a few results alrea publishe i [3].7. Theorem [7] The metric imesio of graph is if a ol if is a path. Figure 4. black colore vertices shows the metric basis for P.8. Theorem [7] If K is the complete graph with > the K β.

5 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March Theorem [] If C is a ccle of legth > the β. C.9. Theorem [0] If is a ifiite graph with fiite metric imesio the it is uiforml locall fiite. The ifiite graph P is uiforml locall fiite with metric imesio equal to two. Figure 4. The coverse of the above theorem is ot true. That is a uiforml locall fiite graph ee ot have fiite metric imesio. For eample the ifiite comp is uiforml locall fiite but its metric imesio is ifiite. 3. Mai Results 3.. Theorem Figure 5. The metric imesio of the graph obtaie b aig peat eges to each of the vertices i the complete graph K > is same as that of K. Proof: We have β. Let W { v v }\ { } K... v v i for some i i be a basis for K. Sice ever vertices K are aacet to each other the cooriate of - vertices i i W th has - compoets at which compoet takes the value 0 a the other compoets are s with respect to W. Now the verte v i W is aacet to the vertices i W its cooriate vector also has - compoets a that will be. Suppose m m... m are the peat eges ae correspoigl to the vertices v v... v such that m v u. Let the graph obtaie i this wa is eote b K K m. We kow that the cooriate of th v is 0 place. So for some v u a v 5

6 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 v V K are aacet to v v u for all those vertices v v. Hece sice ever verte th the cooriate of u will be place. That is the cooriate of u is u is u is respectivel. Thus the vertices i the graph K obtaie b aig peat eges to each of the vertices i K has istict cooriates with respect to W. Therefore W itself is the basis for K a hece β K. 3.. Illustratio Cosier K 5 Figure 6. Here five peat eges m v u 5 are ae at each of the vertices v v3 v4 a v5 v respectivel a show that K K β β 5 m. Figure 6. The followig corollar is about ifiite graph with costat metric imesio Corollar The above theorem hols for a ifiite graph obtaie b aig peat eges m v u successivel at each u. Thus there eist ifiite graphs with fiite metric imesio. The evelopmet of uiforml locall fiite ULF[9] graphs is base o the aacec operator A actig o the space of boue sequeces efie o the vertices. It has several applicatios i spectral theor. The followig theorem gives a simple result o uiforml locall fiite graph Theorem The ifiite graph K K m metioe i theorem 3. is uiforml locall fiite graph with fiite metric imesio. Proof: B theorem 3. K β K m β where m v u. Sice ever verte is aacet to each other i K v for v V K a the egree of the vertices u which is oe of the e verte i each of the ege ae to K is. Now fi a positive iteger M where >. The v M for all v K. Thus K is uiforml locall fiite. 6

7 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March Theorem Let be coecte graph with β k W { v v... v k } be the basis a r v / W { v v v v... v vk / v V }. Defie geeralie metric or -metric : R b r r r mi { } r r r k R r r r r r r Where is the -metric efie from R b r r k i i. i r r r If m the the metric imesio of the super graph ~ obtaie b aoiig at most m ~ peat eges to the vertices W β β. Proof: Let W { v v v3... v k } r v / W { v v v v... v vk / v V } v is same as that of with respect tow. That is be the basis for. The the cooriate space. Sice β k the cooriate of each verte i cotais k compoets a the are istict. Let m. Now we a m peat eges are ae to suitable vertices v W. Suppose the first peat ege e is ae at v W a e v ve. The cooriate of v is v v v v... v v a it is istict from the cooriate of other vertices i. k Figure 7. Thus the cooriate of v e will be v... v v v v vk with respect to W a is ifferet from all other cooriates of the vertices i sice v v v v... v v is k istict from vi v vi v... vi vk i... i. Hece β e k. If the seco peat ege is ae at v sa e e v e v e the b the same argumet as i the case of v e the cooriate of v e will be v... e v ve v ve v k a it is istict from all other cooriates v v v v... v vk for... e. The obviousl the cooriate of the ew verte is istict from all other vertices sice each compoet i the cooriate of ve is icrease b oe. Thus β e e k. Suppose the seco peat ege e is ae to i i e cooriates of the vertices i the super graph e e... em. Thus 7 v i a e v i v e. Here also the cooriate of v e will be v... i v vi v vi vk. Hece β e e k. Therefore the result is true for m. Assume that β e e... em k where v v e l e l for v W l... m. If e m is ae at a v e the each of the k compoets i l the cooriate of the verte is icrease b oe a hece it is istict from other cooriates. vem If e m is ae to a verte v i ot i W a ot the e verte of a of e l l... m the the cooriate of v will be v v v v... v v a istict from all other em k

8 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 β e e... em em k. Hece the result is true for m. Thus the theorem is true for a r r r itegral value of R. 3.6 Eample Cosier a 5- verte Kite sa H Figure 8. There { } r Mi{ } where i i a Thus the miimum umber of peat eges that ae to the Kite is. If these eges are ae to those vertices which are ot i W amel v 3 a v4 with β H e e. i 3.7. Eample Cosier C4 C 4 Figure 8. β with respect to W { }Figure 9. The r v / W { 0 0 } v v B the efiitio of : Figure 9. R we have r Mi{ } where i i So oe peat ege is ae to C 4. Suppose the peat ege is ae at i v W a v v e. e The the cooriate of ve is with respect to W but that is similar to the cooriate of v 3 Figure 0. Therefore β C e with respect to W. 4 8

9 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 Figure 0. Similarl if e is ae to v W the cooriate of v e will be a that is similar to the cooriate of v 4 Figure. Thus e must be ae to a of v 3 or v 4. It will give a istict represetatio for the cooriates of the vertices i C 4 e Figure. That is e must be ae to the vertices ot i W. Figure. Note: Sice W is ot uique C4 e a v / W { } Figure. β with respect to aother resolvig set W { v } r Figure 3. e v Figure Coclusio This paper gives a measure that ca be use i avigatio space where the umber of robots require to avigate a work place kept costat. Etesio of avigatio space will lea us to 9

10 Iteratioal Joural o Applicatios of raph Theor i Wireless A hoc Networks a Sesor NetworksRAPH-HOC Vol.7 No. March 05 ifiite graphs a its properties. With the help of geeral cocepts a results. Refereces -metric a its properties we establishe [] J. Caceres C. Herao M. Mora I. M. Pelao M. L. Puertas C. Seara a D. R. Woo O the Metric Dimesio of Some Families of raphs Electroic Notes i Discrete Mathematics Vol. 005 pp [] P. J. Slater. Leaves of trees. I: Proc. 6th Southeaster Cof. o Combiatorics raph Theor a Computig 4 pp [3] F. Harar a R. A. Melter O the Metric Dimesio of a raph Ars Combiatorica Vol. 976 pp [4] I. Javai M. T. Rahim a K. Ali Families of Regular raphs with costat Metric Dimesio Utilitas Mathematica Vol pp [5] S. Khuller B. Raghavachari a A. Rosefel Localiatio i raphs Techical Report CS-Tr- 336 Uiversit of Marla at College Park 994. [6] S. Khuller B. Raghavachari a A. Rosefel. Lamarks i raphs. Discrete Appl. Math 703 pp [7]. Chartra L. Eroh M. A. Johso a O. R. Oellerma. Resolvabilit i graphs a Metric Dimesio of raph. Discrete Appl. Math pp [8] C. Poisso a P. Zhag. The metric imesio of uicclic graphs. J. Combi. Math. Combi. Compute. 40 pp [9] B Sooraaraaa B. Shamuka A Note o metric imesio Far. East Joural of Applie Mathematics [0] B Sooraaraaa B. Shamuka Metric imesio of a wheel Far. East Joural of Applie Mathematics [] P. Eros O sets of istaces of poits i Eucliea space Publ. Math. Ist. Hug. Aca. Sci [] A. Sebo E. Taier O metric geerators of raphs Mathematics of Operatio Research [3] J. Peters-Frase a O. R. Oellerma. The metric imesio of Cartesia proucts of raphs. Util. Math. 69 pp [4] Z. Mustafa H. Obieat a F. Awaweh Some fie poit theorem for mappig o complete - metric spaces Fie Poit Theor a Applicatios Volume 008 Article ID oi:0.55/008/ [5] B.C.Dhage eeralise metric spaces a mappigs with fie poit Bulleti of the Calcutta Mathematical Societ vol.84 o. 4 pp [6] B.C.Dhage eeralise metric spaces a topological structure- I Aalele Stiitifice ale Uiversitˇatii Al.I.Cua i Iasi. Serie Nouˇa. Matematicˇa vol.46 o. pp [7] S. ahler -metrische R aume u ihre topologische Struktur Mathe- matische Nachrichte vol.6 pp [8] S. ahler Zur geometric -metrische r aume Revue Roumaie e Math ematiques Pures et Appliqu ees vol.40 pp [9] R.Diestel E. Directios i Ifiite raphs a Combiatorics. Topics i Discrete Mathematics 3. Elsevier-North Holla 99. [0] D. Koig. Theor of Fiite a Ifiite raphs. Birkhauser Bosto990. [] C.St.J.A. Nash-Williams. Ifiite raphs. A surve. Joural of Combiatorial Theor 3: [] C. Thomasse. Ifiite raphs. Further Selecte Topics i raph Theor Acaemic Press Loo 983. [3] Mausha. R Dr. Su Kuriakose A O metric imesio of some special graphs a its isomorphism Uiversit rat Commissio of Iia sposore Natioal coferece o Fu Logic a its Applicatios to Computer Sciece04. 0