LAPLACIAN MATRIX IN ALGEBRAIC GRAPH THEORY

Transcription

1 Aryabhatta Joural of Mathematcs & Iformatcs Vol 6, No, Ja-July, 04 ISSN : Joural Impact Factor (03) : 0489 LAPLACIAN MARIX IN ALGEBRAIC GRAPH HEORY ARameshkumar *, RPalakumar ** ad SDeepa *** * Head, Departmet of Mathematcs, Srmad Adava Arts ad Scece College, VKovl, rchy ** Asst Professor, Departmet of Mathematcs, Srmad Adava Arts ad Scece College, VKovl, rchy *** AsstProfessor, Sr Saratha Arts ad Scece College for wome, Perambalur *Emal Id: rameshmaths@ymalcom, palarkumar@yahooco, sdeepamathematcs@yahoocom ABSRAC: I ths paper we tur to the spectral decomposto of the Laplaca matrx We show how the elemets of the spectral matrx for the Laplaca ca be used to costruct symmetrc polyomals that are permutato varats he coeffcets of these polyomals ca be used as graph features whch ca be ecoded a vectoral maer We exted ths represetato to graphs whch there are uary attrbutes o the odes ad bary attrbutes o the edges by usg the spectral decomposto of a Hermta property matrx that ca be vewed as a complex aalogue of the Laplaca Key words: Laplaca Matrx, Spectral Matrx, Hermta Property Matrx, Permutato matrx, symmetrc polyomals MSC 0: 05CXX INRODUCION Graph structures have proved computatoally cumbersome for patter aalyss he reaso for ths s that before graphs ca be coverted to patter vectors, correspodeces must be establshed betwee the odes of structures whch are potetally of dfferet sze o embed the graphs a patter space, we explore whether the vectors of varats ca be embedded a low dmesoal space usg a umber of alteratve strateges cludg prcpal compoets aalyss (PCA), multdmesoal scalg (MDS) ad localty preservg projecto (LPP) Expermetally, we demostrate the embeddgs result well defed graph clusters he aalyss of relatoal patters, or graphs, has prove to be cosderably more elusve tha the aalyss of vectoral patters Relatoal patters arse aturally the represetato of data whch there s a structural arragemet, ad are ecoutered computer vso, geomcs ad etwork aalyss Oe of the challeges that arse these domas s that of kowledge dscovery from large graph datasets he tools that are requred ths edeavour are robust algorthms that ca be used to orgase, query ad avgate large sets of graphs I partcular, the graphs eed to be embedded a patter space so that smlar structures are close together ad dssmlar oes are far apart Moreover, f the graphs ca be embedded o a mafold a patter space, the the modes of shape varato ca be explored by traversg the mafold a systematc way he process of costructg low dmesoal spaces or mafolds s a route procedure wth patter-vectors A varety of well establshed techques such as prcpal compoets aalyss [], multdmesoal scalg [7] ad depedet compoets aalyss, together wth more recetly developed oes such as locally lear embeddg, so-map ad the Laplaca egemap [] exst for solvg the problem -65-

2 ARameshkumar, RPalakumar ad SDeepa I addto to provdg geerc data aalyss tools, these methods have bee extesvely exploted computer vso to costruct shape-spaces for D or 3D objects, ad partcular faces, represeted usg coordate ad testy data Collectvely these methods are sometmes referred to as mafold learg theory However, there are few aalogous methods whch ca be used to costruct low dmesoal patter spaces or mafolds for sets of graphs NOAIONS PCA Prcpal Compoet Aalyss MDS Mult Dmesoal Scalg LPP Localty Preservg Projecto LDA Lear Dscrmat Aalyss P U Permutato Matrx L Laplaca Matrx H Hermta Matrx B K symmetrc Polyomals U s Orthogoal Matrces s Dagoal Matrces dagger operator 3 SPECRAL GRAPH REPRESENAIONS Cosder the udrected graph G = (V, E,W) wth ode-set V = {v, v,,v }, edgeset E = {e, e,, e m } V V ad weght fucto W : E (0, ] he adjacecy matrx A for the graph G s the symmetrc matrx wth elemets A ab, f ( va, vb ) 0, otherwse I other words, the matrx represets the edge structure of the graph Clearly f the graph s udrected, the matrx A s symmetrc If the graph edges are weghted, the adjacecy matrx s defed to be w( va, vb ), f ( va, vb ) Aab 0, otherwse he Laplaca of the graph s gve by L = D A where D s the dagoal ode degree matrx whose elemets D aa =b Aab are the umber of edges whch ext the dvdual odes he Laplaca s more sutable for spectral aalyss tha the adjacecy matrx sce t s postve semdefte I geeral the task of comparg two such graphs G ad G volves fdg a correspodece mappg betwee the odes of the two graphs, f : v v he recovery of the correspodece map f ca be posed as that of mmsg a error crtero -66-

3 Laplaca Matrx I Algebrac Graph heory he mmum value of the crtero ca be take as a measure of the smlarty of the two graphs he addtoal ode represets a ull match or dummy ode Extraeous or addtoal odes are matched to the dummy ode A umber of search ad optmsato methods have bee developed to solve ths problem [5, 9] We may also cosder the correspodece mappg problem as oe of fdg the permutato of odes the secod graph whch places them the same order as that of the frst graph hs permutato ca be used to map the Laplaca matrx of the secod graph oto that of the frst If the graphs are somorphc the ths permutato matrx satsfes the codto L =PL P Whe the graphs are ot exactly somorphc, the ths codto o loger holds However, the Frobeus dstace L = PL P betwee the matrces ca be used to gauge the degree of smlarty betwee the two graphs Spectral techques have bee used to solve ths problem For stace, workg wth adjacecy matrces the permutato matrx P U that mmses the Frobeus orm J(PU ) = PUAP U - A he method performs the sgular value decompostos A = UU ad A = U U, where the U s are orthogoal matrces ad the s are dagoal matrces [] Oce these factorzatos have bee performed, the requred permutato matrx s approxmated by U U I some applcatos, especally structural chemstry, egevalues have also bee used to compare the structural smlarty of dfferet graphs However, although the egevalue spectrum s a permutato varat, t represets oly a fracto of the formato resdg the ege system of the adjacecy matrx Sce the matrx L s postve semdefte, t has egevalues whch are all ether postve or zero he spectral matrx s foud by performg the egevector expaso for the Laplaca matrx L, e L e e where λ ad e are the egevectors ad egevalues of the symmetrc matrx L he spectral matrx the has the scaled egevectors as colums ad s gve by e e (), he matrx Φ s a complete represetato of the graph the sese that we ca use t to recostruct the orgal Laplaca matrx usg the relato L = ΦΦ he matrx Φ s a uque represetato of L ff all egevalues are dstct or zero hs follows drectly from the fact that there are dstct egevectors whe the egevalues are also all dstct Whe a egevalue s repeated, the there exsts a subspace spaed by the egevectors of the degeerate egevalues whch all vectors are also egevectors of L I ths stuato, f the repeated egevalue s o-zero there s cotuum of spectral matrces represetg the same graph However, ths s rare for moderately large graphs hose graphs for whch the egevalues are dstct or zero are referred to as smple e 4 NODE PERMUAIONS AND INVARIANS he topology of a graph s varat uder permutatos of the ode labels However, the adjacecy matrx, ad hece the Laplaca matrx, s modfed by the ode order sce the rows ad colums are dexed by the ode order If we relabel the odes, the Laplaca matrx udergoes a permutato of both rows ad colums Let the matrx P be the permutato matrx represetg the chage ode order he permuted matrx s L = PLP here -67-

4 ARameshkumar, RPalakumar ad SDeepa s a famly of Laplaca matrces whch are ca be trasformed to oe aother usg a permutato matrx he spectral matrx s also modfed by permutatos, but the permutato oly reorders the rows of the matrx Φo show ths, let L be the Laplaca matrx of a graph G ad let L = PLP be the Laplaca matrx obtaed by relabellg the odes usg the permutato P Further, let e be a ormalzed egevector of L wth assocated egevalue λ, ad let e = Pe Wth these gredets, we have that Le PLP Pe = PLe = e Hece, e s a egevector of L wth assocated egevalue λ As a result, we ca wrte the spectral matrxφ of the permuted Laplaca matrx L =Φ Φ as Φ = PΦ Drect comparso of the spectral matrces for dfferet graphs s hece ot possble because of the ukow permutato he egevalues of the adjacecy matrx have bee used as a compact spectral represetato for comparg graphs because they are ot chaged by the applcato of a permutato matrx he egevalues ca be recovered from the spectral matrx usg the detty j he expresso j s fact a symmetrc polyomal the compoets of egevector e A symmetrc polyomal s varat uder permutato of the varable dces [3] I ths case, the polyomal s varat uder permutato of the row dex I fact the egevalue s just oe example of a famly of symmetrc polyomals whch ca be defed o the compoets of the spectral matrx However, there s a specal set of these polyomals, referred to as the elemetary symmetrc polyomals (S) that form a bass set for symmetrc polyomals I other words, ay symmetrc polyomal ca tself be expressed as a polyomal fucto of the elemetary symmetrc polyomals belogg to the set S We therefore tur our atteto to the set of elemetary symmetrc polyomals [4] For a set of varables {v, v v } they ca be defed as j S ( v, v, v ) v S ( v, v, v ) v v v j ( r v, v, v ) v v S v r j r -68-

5 he power symmetrc polyomal fuctos S ( v, v, v ) has roots v, v,, v Multplyg out Equato (3) gves -69- v P ( v, v, v ) v P ( v, v, v ) v P ( v r P ( v, v, v ), v, v) v v r Laplaca Matrx I Algebrac Graph heory also form a bass set over the set of symmetrc polyomals As a cosequece, ay fucto whch s varat to permutato of the varable dces ad that ca be expaded as a aylor seres, ca be expressed terms of oe of these sets of polyomals he two sets of polyomals are related to oe aother by the Newto-Grard formula: r r ( ) k Sr Pk Srk r ( ) () where we have used the shorthad S r for S r(v,, v ) ad Pr for P r(v,, v ) As a cosequece, the elemetary symmetrc polyomals ca be effcetly computed usg the power symmetrc polyomals I ths paper, we ted to use the polyomals to costruct varats from the elemets of the spectral matrx he polyomals ca provde spectral features whch are varat uder ode permutatos of the odes a graph ad utlse the full spectral matrx hese features are costructed as follows; each colum of the spectral matrx Φ forms the put to the set of spectral polyomals For example the colum {Φ,, Φ,,, Φ, } produces the polyomals S (Φ,,Φ,,,Φ, ), S (Φ,, Φ,,, Φ, ),, S (Φ,, Φ,,, Φ, ) he values of each of these polyomals are varat to the ode order of the Laplaca We ca costruct a set of spectral features usg the colums of the spectral matrx combato wth the symmetrc polyomals Each set of features for each spectral mode cotas all the formato about that mode up to a permutato of compoets hs meas that t s possble to recostruct the orgal compoets of the mode gve the values of the features oly hs s acheved usg the relatoshp betwee the roots of a polyomal x ad the elemetary symmetrc polyomals he polyomal k ( x ) 0 (3) v

6 ARameshkumar, RPalakumar ad SDeepa x -S x - - S x (-) S 0 (4) where we have aga used the shorthad S r for S r(v,, v ) By substtutg the feature values to Equato (4) ad fdg the roots, we ca recover the values of the orgal compoets he root order s udetermed, so as expected the values are recovered up to a permutato 5 UNARY AND BINARY ARIBUES Attrbuted graphs are a mportat class of represetatos ad eed to be accommodated graph matchg We have suggested the use of a augmeted matrx to capture addtoal measuremet formato Whle t s possble to make use of a such a approach cojucto wth symmetrc polyomals, aother approach from the spectral theory of Hermta matrces suggests tself A Hermta matrx H s a square matrx wth complex elemets that remas uchaged uder the jot operato of trasposto ad complex cojugato of the elemets, e H = H Hermta matrces ca be vewed as the complex umber couterpart of the symmetrc matrx for real umbers Each off-dagoal elemet s a complex umber whch has two compoets, ad ca therefore represet a -compoet measuremet vector o a graph edge he o-dagoal elemets are ecessarly real quattes, so the ode measuremets are restrcted to be scalar quattes here are some costrats o how the measuremets may be represeted order to produce a postve semdefte Hermta matrx Let {x, x,,x } be a set of measuremets for the ode-set V Further, let {y,, y,3,, y, } be the set of measuremets assocated wth the edges of the graph, addto to the graph weghts [6] Each edge the has a par of observatos (W a,b, y a,b) assocated wth t here are a umber of ways whch the complex umber H a,b could represet ths formato, for example wth the real part as W ad the magary part as y However, we would lke our complex property matrx to reflect the Laplaca As a result the off-dagoal elemets of H are chose to be H a,b = W a,be yab (5) I other words, the coecto weghts are ecoded by the magtude of the complex umber H a,b ad the addtoal measuremet by ts phase By usg ths ecodg, the magtude of the umbers s the same as the orgal Laplaca matrx Clearly ths ecodg s most sutable whe the measuremets are agles If the measuremets are easly bouded, they ca be mapped oto a agular terval ad phase wrappg ca be avoded If the measuremets are ot easly bouded the ths ecodg s ot sutable he measuremets must satsfy the codtos π y a,b < π ad y a,b = y b,a to produce a Hermta matrx o esure a postve defte matrx, we requre H aa > H hs codto s satsfed f b a ab Whe defed ths way the property matrx s a complex aalogue of the weghted Laplaca for the graph For a Hermta matrx there s a orthogoal complete bass set of egevectors ad egevalues obeyg the egevalue equato He = λe I the Hermta case, the egevalues λ are real ad the compoets of the egevectors e are complex here s a potetal ambguty the egevectors, that ay multple of a -70- H x W, ad x a 0 (6) aa a ba a b

7 Laplaca Matrx I Algebrac Graph heory egevector s also a soluto, e Hαe = λαe I the real case, we choose α such that e s of ut legth I the complex case, α tself may be complex, ad eeds to determe by two costrats We set the vector legth to e = ad addto we mpose the codto arg e 0, whch specfes both real ad magary parts hs represetato ca be exteded further by usg the four-compoet complex umbers kow as quateros As wth real ad complex umbers, there s a approprate ege decomposto whch allows the spectral matrx to be foud I ths case, a edge weght ad three addtoal bary measuremets may be ecoded o a edge It s ot possble to ecode more tha oe uary measuremet usg ths approach However, for the expermets ths paper, we have cocetrated o the complex represetato Whe the egevectors are costructed ths way the spectral matrx s foud by performg the egevector expaso H e e where λ ad e are the egevectors ad egevalues of the Hermta matrx H [8] We costruct the complex spectral matrx for the graph G usg the egevectors as colums, e e e e (7), We ca aga recostruct the orgal Hermta property matrx usg the relato H = ΦΦ Sce the compoets of the egevectors are complex umbers ad therefore each symmetrc polyomal s complex Hece, the symmetrc polyomals must be evaluated wth complex arthmetc ad also evaluate to complex umbers Each S r therefore has both real ad magary compoets he real ad magary compoets of the symmetrc polyomals are terleaved ad stacked to form a feature-vector B for the graph hs featurevector s real 6 GRAPH EMBEDDING MEHODS We explore three dfferet methods for embeddg the graph feature vectors a patter space wo of these are classcal methods Prcpal compoets aalyss (PCA) fds the projecto that accouts for the varace or scatter of the data Multdmesoal scalg (MDS), o the other had, preserves the relatve dstaces betwee objects he remag method s a ewly reported oe that offers a compromse betwee preservg varace ad the relatoal arragemet of the data, ad s referred to as localty preservg projecto [0] I ths paper we are cocered wth the set of graphs G, G,, G k,, G N he k th graph s deoted by G k = (V k, E k) ad the assocated vector of symmetrc polyomals s deoted by B k 6 Prcpal compoet aalyss We commece by costructg the matrx X = [B B B B B k B B N B ], wth the graph feature vectors as colums Here B s the mea feature vector for the dataset Next, we compute the covarace matrx for the elemets of the feature vectors by takg the matrx product C = XX We extract the prcpal compoets drectos by performg the egedecomposto C N l u u o the covarace matrx C, where the l are the egevalues ad the u are the egevectors We use the frst s leadg egevectors to represet the graphs extracted -7-

8 ARameshkumar, RPalakumar ad SDeepa from the mages he coordate system of the egespace s spaed by the s orthogoal vectors U = (u, u,, u s ) he dvdual graphs represeted by the log vectors B k, k=,,, N ca be projected oto ths egespace usg the formula y k = U (B k B ) Hece each graph G k s represeted by a s-compoet vector y k the egespace Lear dscrmat aalyss (LDA) s a exteso of PCA to the multclass problem We commece by costructg separate data matrces X,X, X Nc for each of the N c classes hese may be used to compute the ~ Nc correspodg class covarace matrces C X X he average class covarace matrx C C N s foud hs matrx s used as a spherg trasform We commece by computg the egedecomposto N ~ C luu UU where U s the matrx wth the egevectors of C ~ as colums ad Λ= dag(l, l,, l ) s the dagoal egevalue matrx he sphered represetato of the data s X = Λ / U X Stadard PCA s the appled to the resultg data matrx X he purpose of ths techque s to fd a lear projecto whch descrbes the class dffereces rather tha the overall varace of the data 6 Multdmesoal scalg Multdmesoal scalg (MDS) s a procedure whch allows data specfed terms of a matrx of parwse dstaces to be embedded a Eucldea space Here we ted to use the method to embed the graphs extracted from dfferet vewpots a lowdmesoal space o commece we requre parwse dstaces betwee graphs We do ths by computg the orms betwee the spectral patter vectors for the graphs For the graphs dexed ad, the dstace s d = (B B ) (B B ) he parwse dstaces d, are used as the elemets of a, N N dssmlarty matrx R, whose elemets are defed as follows d, f R, (8) 0 f I ths paper, we use the classcal multdmesoal scalg method [7] to embed the graphs a Eucldea space usg the matrx of parwse dssmlartes R he frst step of MDS s to calculate a matrx whose elemet wth row r ad colum c s gve by ˆ N ˆ ˆ rc drc dr d c d where dˆ r c d rc s the average dssmlarty N value over the r th row, d ˆ c s the dssmlarty average value over the c th N N colum ad dˆ r c d r, c s the N average dssmlarty value over all rows ad colums of the dssmlarty matrx R 63 Localty preservg projecto he LPP s a lear dmesoalty reducto method whch attempts to project hgh dmesoal data to a low dmesoal mafold, whle preservg the eghbourhood structure of the data set he method s relatvely sestve to outlers ad ose hs s a mportat feature our graph clusterg task sce the outlers are usually troduced by mperfect segmetato processes he learty of the method makes t computatoally effcet -7- c

9 Laplaca Matrx I Algebrac Graph heory he graph feature vectors are used as the colums of a data matrx X = ( B B B N ) he relatoal structure of the data s represeted by a proxmty weght matrx W wth elemets W, = exp[ kd, ], where k s a costat If Q s the dagoal degree matrx wth the row weghts N Q k k j W k, j as elemets, the the relatoal structure of the data s represeted usg the Laplaca matrx J = Q W he dea behd LPP s to aalyse the structure of the weghted covarace matrx XWX he optmal projecto of the data s foud by solvg the geeralsed egevector problem XJX u = lxqx u (9) We project the data oto the space spaed by the egevectors correspodg to the s smallest egevalues Let U = (u, u,, u s ) be the matrx wth the correspodg egevectors as colums, the projecto of the kth feature vector s y k = U B k 7 SHOCK GRAPHS 7 ree Attrbutes Before processg, the shapes are frstly ormalsed area ad alged alog the axs of largest momet After performg these trasforms, the dstaces ad agles assocated wth the dfferet shapes ca be drectly compared, ad ca therefore be used as attrbutes o abstract the shape skeleto usg a tree, we place a ode at each jucto pot the skeleto he edges of the tree represet the exstece of a coectg skeletal brach betwee pars of juctos he odes of the tree are charactersed usg the radus r(a) of the smallest btaget crcle from the jucto to the boudary Hece, for the ode a, x a = r(a) he edges are charactersed by two measuremets For the edge (a, b) the frst of these, y a,b s the agle betwee the odes a ad b, e y a,b = θ(a, b) Sce most skeleto braches are relatvely straght, ths s a approxmato to the agle of the correspodg skeletal brach Furthermore, sce π θ(a, b) < π ad θ(a, b) = θ(b, a), the measuremet s already sutable for use the Hermta property matrx I order to compute edge weghts, we ote that the mportace of a skeletal brach may be determed by the rate of chage of boudary legth l wth skeleto legth s, whch we deote by dl/ds hs quatty s related to the rate of chage of the btaget crcle radus alog the skeleto, e dr/ds, by the formula dr ds he edge weght W a,b s gve by the average value of dl/ds alog the relevat skeletal brach dl ds -73-

10 ARameshkumar, RPalakumar ad SDeepa Fgure A: Examples from the shape database wth ther assocated skeletos We extract the skeleto from each bary shape ad attrbute the resultg tree the maer outled Fgure A shows some examples of the types of shape preset the database alog wth ther skeletos We commece by showg some results for the three shapes show Fgure A he objects studed are a had, some puppes ad some cars he dog ad car shapes cosst of a umber of dfferet objects ad dfferet vews of each object he had category cotas dfferet had cofguratos We apply the three embeddg strateges outle to the vectors of permutato varats extracted from the Hermta varat of the Laplaca 7 Expermets wth shock trees Our expermets are performed usg a database of 4 bary shapes Each bary shape s extracted from a D vew of a 3D object here are 3 classes the database, ad for each object there are a umber of vews acqured from dfferet vewg drectos ad a umber of dfferet examples of the class Fgure B: MDS (left), PCA (mddle) ad LPP (rght) appled to the shock graphs We commece the left had pael of Fgure B by showg the result of applyg the MDS procedure to the three shape categores he had shapes form a compact cluster the MDS space here are other local clusters cosstg of three or four members of the remag two classes hs reflects the fact that whle the had shapes -74-

11 Laplaca Matrx I Algebrac Graph heory have very smlar shock graphs, the remag two categores have rather varable shock graphs because of the dfferet objects he mddle pael of Fgure B shows the result of usg PCA Here the dstrbutos of shapes are much less compact Whle a dstct cluster of had shapes stll occurs, they are geerally more dspersed over the feature space here are some dstct clusters of the car shape, but the dstrbutos overlap more the PCA projecto whe compared to the MDS space he rght-had pael of Fgure B shows the result of the LPP procedure o the dataset he results are smlar to the PCA method Oe of the motvatos for the work preseted here was the potetal ambgutes that are ecoutered whe usg the spectral features of trees o demostrate the effect of usg attrbuted trees rather tha smply weghtg the edges, we have compared the LDA projectos usg both types of data Fgure C : A comparso of attrbuted trees wth weghted trees Left : trees wth edge weghts based o boudary legths Rght : Attrbuted trees wth addtoal edge agle formato I Fgure C the rght-had plot shows the result obtaed usg the symmetrc polyomals from the egevectors of the Laplaca matrx L=D W, assocated wth the edge weght matrx he left-had plot shows the result of usg the Hermta property matrx he Hermta property matrx for the attrbuted trees produces a better class separato tha the Laplaca matrx for the weghted trees he separato ca be measured by the Fsher dscrmat betwee the classes, whch s the squared dstace betwee class cetres dvded by the varace alog the le jog the cetres able I Separatos Classes Hermta property matrx Weghted matrx Car/dog 077 Car/had Dog/had

12 ARameshkumar, RPalakumar ad SDeepa 8 CONCLUSIONS We have show how graphs ca be coverted to patter vectors by utlzg the spectral decomposto of the Laplaca matrx ad bass sets of symmetrc polyomals hese feature vectors are complete, uque ad cotuous However, ad most mportat of all, they are permutato varats We vestgate how to embed the vectors a patter space, sutable for clusterg the graphs Here we explore a umber of alteratves cludg PCA, MDS ad LPP I a expermetal study we show that the feature vectors derved from the symmetrc polyomals of the Laplaca spectral decomposto yeld good clusters whe MDS or LPP are used here are clearly a umber of ways whch the work preseted ths paper ca be developed For stace, sce the represetato based o the symmetrc polyomals s complete, they may form the meas by whch a geeratve model of varatos graph structure ca be developed hs model could be leared the space spaed by the permutato varats, ad the mea graph ad ts modes of varato recostructed by vertg the system of equatos assocated wth the symmetrc polyomals REFERENCES ) AD Bagdaov ad M Worrg, Frst Order Gaussa Graphs for Effcet Structure Classfcato, Patter Recogto, 36, pp 3-34, 003 ) MBelk ad P Nyog, Laplaca Egemaps for Dmesoalty Reducto ad Data Represetato, Neural Computato, 5(6), pp , 003 3) N Bggs, Algebrac Graph heory, CUP 4) P Bott ad R Morrs, Almost all trees share a complete set of maatal polyomals, Joural of Graph heory, 7, pp , 993 5) W J Chrstmas, J Kttler ad M Petrou, Structural Matchg ComputerVso usg Probablstc Relaxato, IEEE rasactos o Patter Aalyss ad Mache Itellgece, 7, pp , 995 6) FRK Chug, Spectral Graph heory, Amerca Mathmatcal Socety Ed,CBMS seres 9, 997 7) Cox ad M Cox, Multdmesoal Scalg, Chapma-Hall, 994 8) D Cvetkovc, P Rowlso ad S Smc, Egespaces of graphs, Cambrdge Uversty Press, 997 9) S Gold ad A Ragaraja, A Graduated Assgmet Algorthm for Graph Matchg, IEEE rasactos o Patter Aalyss ad Mache Itellgece, 8, pp , 996 0) XHe ad P Nyog, Localty preservg projectos, Advaces Neural IformatoProcessg Systems 6, MI Press, 003 ) D Heckerma, D Geger ad DM Chckerg, Learg Bayesa Networks: hecombato of kowledge ad statstcal data, Mache Learg, 0, pp 97-43, 995 ) G Nrmala ad M Vjya Strog Fuzzy Graphs o Composto, esor Ad Normal Products Aryabhatta J Maths & Ifo Vol 4 () 0, pp ) K hlakam & A Sumath Weer Idex of Cha Graphs Aryabhatta J Maths & Ifo Vol 5 () 03, pp