Basic Concepts and Definitions of Graph Theory

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1 HAPTER Bsic ocepts d Defiitios of Grph Theory. INTRODUTION Grph theory is rch of thetics strted y Euler [5] s erly s 76. It took hudred yers efore the secod iportt cotriutio of Kirchhoff [9] hd ee de for the lysis of electricl etworks. yley [] d Sylvester [8] discovered severl properties of specil types of grphs kow s trees. Poicré [95] defied i priciple wht is kow owdys s the icidece trix of grph. It took other cetury efore the first ook ws pulished y Köig []. After the secod world wr, further ooks ppered o grph theory, Ore [8], Behzd d hrtrd [], Tutte [], Berge [], Hrry [7], Gould [6], d West [5], og y others. Grph Theory hs foud y pplictios i egieerig d sciece, such s cheicl, civil, electricl d echicl egieerig, rchitecture, geet d cotrol, couictio, opertiol reserch, sprse trix techology, coitoril optiistio, d coputer sciece. Therefore y ooks hve ee pulished o pplied grph theory such s those y Body d Murty [6], he [], Thulsir d Swy [5], Wilso d Beieke [5], Myed [7], hristofides [6], Godr d Mioux [6], Beieke d Wilso [], Deo [7], ooke et l. [], Kveh [-] d y others. I recet yers, due to the extesio of the cocepts d pplictios of the grph theory, y jourls such s Jourl of Grph Theory, Jourl of oitoril Theory A & B, Discrete d Applied Mthetics, SIAM Jourl of Discrete Mthetics, Europe Jourl of oitorics, d Grphs d oitorics re eig pulished to cover the dvces de i this field.

2 Structurl Mechics: Grph d Mtrix Methods I this chpter sic defiitios d cocepts of grph theory re preseted; however, for proofs d detils the reder y refer to textooks o this suject, Refs [6,7,5].. BASI DEFINITIONS There re y physicl systes whose perforce depeds ot oly o the chrcteristics of their copoets, ut lso o their reltive loctio. As exple, i structure, if the properties of eer re ltered, the overll ehviour of the structure will e chged. This idictes tht the perforce of structure depeds o the chrcteristics of its eers. O the other hd, if the loctio of eer is chged, the properties of the structure will gi e differet. Therefore the coectivity (topology) of the structure iflueces the perforce of the whole structure. Hece it is iportt to represet syste so tht its topology c e uderstood clerly. The grph odel of syste provides powerful es for this purpose... DEFINITION OF A GRAPH A grph S cosists of set N(S) of eleets clled odes (vertices or poits) d set M(S) of eleets clled eers (edges or rcs) together with reltio of icidece which ssocites ech eer with pir of odes, clled its eds. Two or ore eers joiig the se pir of is re kow s ultiple eer, d eer joiig ode to itself is clled loop. A grph with o loops d ultiple eers is clled siple grph. If N(S) d M(S) re coutle sets, the the correspodig grph S is fiite. Sice the gret jority of the results i this ook perti to fiite grphs with o loop d ultiple eers, oly siple fiite grphs re eeded, which re referred to s grphs. The ove defiitios correspod to strct grphs; however, grph y e visulized s set of poits coected y lie segets i Euclide spce; the poits re idetified with odes, d the lie segets without their ed poits re idetified with eers. Such cofigurtio is kow s topologicl grph. These defiitios re illustrted i Figure.. ultiple eers loop () A o-siple grph. () A siple grph. Fig.. No-siple d siple grphs.

3 HAPTER Bsic ocepts d Defiitios.. ADJAENY AND INIDENE Two odes of grph re clled djcet if these odes re the ed odes of eer. A eer is clled icidet with ode if it is ed ode of tht eer. Two eers re clled icidet if they hve coo ed ode. The degree (vlecy) of ode i of grph, deoted y deg ( i ), is the uer of eers icidet with tht ode. Sice ech eer hs two ed odes, the su of ode-degrees of grph is twice the uer of its eers (hdshkig le - kow s the first theore of grph theory) Fig.. A siple grph S. As exple, i Figure. two odes d 5 re djcet. Node is icidet with eer d 6, d deg ( ) =... ISOMORPHI GRAPHS Two grphs S d S re clled isoorphic if there exists oe-to-oe correspodece etwee their ode sets d djcecy is preserved. As exple, the three grphs show i Figure. re isoorphic. The word isoorphic is derived fro the Greek words se d for () () (c) Fig.. Three isoorphic grphs.

4 Structurl Mechics: Grph d Mtrix Methods.. GRAPH OPERATIONS A sugrph S i of S is grph for which N(S i ) N(S) d M(S i ) M(S), d ech eer of S i hs the se eds s i S. k The uio of sugrphs S, S,..., S k of S, deoted y S k = Si = S S... i= k k S k, is sugrph of S with N(S k ) = N(S i ) d M(S k ) = M(S i ). The i= i= itersectio of two sugrphs S i d S j is siilrly defied usig itersectios of ode-sets d eer-sets of the two sugrphs. The rig su of two sugrphs S i S j = S i S j S i S j is sugrph which cotis the odes d eers of S i d S j except those eleets coo to S i d S j. These defiitios re illustrted i Figure.. () S () S i (c) S j (d) S i S j (e) S i S j (f) S i S j Fig.. A grph, two of its sugrphs, uio, itersectio d rig su. There re y other useful opertios, such s rtesi product, direct product d strog rtesi product, successfully pplied to structurl egieerig [9]...5 WALKS, TRAILS AND PATHS A wlk w of S, is fiite sequece w = {,,,..., k, k } whose ters re ltertely odes i d eers i of S for i k, d i- d i re the two eds of i. A tril t i S, is wlk i which o eer of S ppers ore th oce. A pth P i S, is tril i which o ode ppers ore th oce. The legth of pth P i deoted y L(P i ) is tke s the uer of its eers. P i is clled the shortest pth etwee the two odes d k, if for y other pth P j

5 5 HAPTER Bsic ocepts d Defiitios 5 etwee these odes L(P i ) L(P j ). The distce etwee two odes of grph is defied s the uer of its eers of shortest pth etwee these odes. As exple, i Figure.5, w = (, 5, 5, 6,, 7, 6,, 5, 6,,, ) is wlk etwee d, i which eer 6 d odes d 5 re repeted twice () A wlk w i S. () A tril t i S. (c) A pth P i S. Fig..5 A wlk, tril d pth i S. t = (, 5, 5, 6,, 7, 6,, 5,, ) is tril etwee d i which ode 5 is repeted twice. P = (, 5, 5, 6,, 7, 6 ) is pth of legth i which o ode d o eer is repeted...6 ONNETEDNESS Two odes i d j re sid to e coected i S if there exists pth etwee these odes. A grph S is clled coected if ll pirs of its odes re coected. A copoet of grph S is xil coected sugrph, i.e. it is ot sugrph of y other coected sugrph of S. These defiitios re illustrted i Figure.6. () A coected grph. () A discoected grph. Fig..6 A coected grph d discoected grph with copoets.

6 6 6 Structurl Mechics: Grph d Mtrix Methods..7 YLES AND UTSETS A cycle is pth (,,,..., p, p ) for which = p d p ; i.e. cycle is closed pth. Siilrly, closed tril (higed cycle) d closed wlk c e defied, Figure.7. () A cycle of S. () A higed cycle of S. Fig..7 Two cycles of S. A cutset i grph S is set of eers whose reovl fro the grph icreses the uer of coected copoets of S, Figure.8(). If cutset results i two copoets S d S, the it is kow s prie cutset, Figure.8(). A lik is eer with its eds i two copoets produced y cutset. Liks re show i old lies. I prie cutset, if oe of the copoets S or S cosists of sigle ode, the the prie cutset is clled cocycle, Figure.8(c). () A cutset of S. () A prie cutset of S. (c) A cocycle of S. Fig..8 Differet cutsets of S...8 TREES, SPANNING TREES AND SHORTEST ROUTE TREES A tree T of S is coected sugrph of S, which cotis o cycle. A set of trees of S fors forest. If tree cotis ll the odes of S, it is clled spig tree of S. For siplicity it will e referred to s tree, fro ow o. A shortest route tree (SRT) rooted t specified ode of S, is tree for which the distce etwee every ode j of T d is iiu. A SRT of grph c e geerted y the followig siple lgorith: Lel the selected root s "" d the djcet odes s "". Record the eers icidet to "" s tree eers. Repet the process of lellig with "" the uuered eds of ll the eers icidet with odes lelled s "", gi

7 7 HAPTER Bsic ocepts d Defiitios 7 recordig the tree eers. This process terites whe ech ode of S is lelled d ll the tree eers re recorded. The lel of the lst ode idictes the legth of the SRT d the xiu uer of odes with the se lel is defied s the width of the SRT. The ove defiitios re illustrted i Figure.9. The legth d width of the SRT i Figure 9(d) re d, respectively. It is esy to prove tht for tree T, M(T) = N(T), (-) where M(T) d N(T) re the uers of eers d odes of T, respectively. The copleet of T i S is clled cotree, deoted y T. The eers of T re kow s rches d those of T re clled chords. For coected grph S, the uer of chords is give y: Sice, N(T) = N(S), hece, M(T) = M(S) M(T). (-) M(T) = M(S) N(S) +, (-) where M(S) d N(S) re the uers of eers d odes of S, respectively. Notice tht for set d its crdility the se ottio is used d the differece should e ovious fro the cotext. () A grph S. () A tree of S. (c) A spig tree of S. (d) A SRT rooted fro. (e) A cotree show i dshed lies. Fig..9 A tree d cotree of S.

8 8 8 Structurl Mechics: Grph d Mtrix Methods. DIFFERENT TYPES OF GRAPHS I order to siplify the study of properties of grphs, differet types of grphs hve ee defied. Soe iportt oes re s follows: A ull grph is grph which cotis o eers. Thus N k is grph cotiig k isolted odes. A cycle grph is grph cosistig of sigle cycle. Therefore k is polygo with k eers. A pth grph is grph cosistig of sigle pth. Hece P k is pth with k odes d (k) eers. A coplete grph is grph i which every two distict odes re coected y exctly oe eer, Figure.. K K K K K 5 Fig.. Soe coplete grphs. A coplete grph with N odes is deoted y K N. It is esy to prove tht coplete grph with N odes hs N(N )/ eers. A grph is clled iprtite, if the correspodig ode set c e split ito two sets N d N i such wy tht ech eer of S jois ode of N to ode of N. A coplete iprtite grph is iprtite grph i which ech ode N is joied to ech ode of N y exctly oe eer. If the uer of odes i N d N re deoted y r d s, respectively, the coplete iprtite grph is deoted y K r,s. Exples of iprtite d coplete iprtite grphs re show i Figure.. () A iprtite grph. () A coplete iprtite grph K,. Fig.. Two iprtite grphs.

9 9 HAPTER Bsic ocepts d Defiitios 9. VETOR SPAES ASSOIATED WITH GRAPHS I this sectio, it is show tht vector spce c e ssocited with grph, d the properties of two iportt suspces of this vector spce, ely cycle d cutset spces, is studied. For this purpose, siple defiitios fro sets, groups, fields d vector spces re riefly preseted. The teril preseted i this sectio is sed o the work of Thulsir d Swy [5]... GROUPS AND FIELDS osider fiite set S = {,,c,.}, d defie iry opertio + o S. This opertio ssigs to every pir ( d ) S uique eleet deoted y +. The set S is sid to e closed uder + if the eleet (+) S, wheever ( d ) S. The opertio + is sid to e ssocitive if +(+c) = (+)+c for ll, d c i S. The opertio is clled couttive if + = + for ll d i S. Defiitio : A set S with iry opertio +, clled dditio, is group if the followig postultes hold:. S is closed uder +.. The opertio + is ssocitive.. There exists uique eleet e S such tht +e = e+ = for ll S.. For ech eleet S there exits uique eleet such tht + = + = e. The eleet is kow s the iverse of, d vice vers. Oviously the idetity eleet e is its ow iverse. A group is clled eli if the opertio + is couttive. Exples: The set S = {,,,,+,+, } cosistig of ll iteger uers uder usul dditio for + fors group. Here, is the idetity eleet d is the iverse of S. This group is lso eli. Aother exple is the set Z p = (,,,,p} of itegers with odulo p dditio opertio. If = p+q for q p, the i odulus rithetic = q(odulo p). I this group, is the idetity eleet d the iteger p is the iverse of, except which is its ow iverse. As exple, the dditio tle of Z is show i Tle..

10 Structurl Mechics: Grph d Mtrix Methods Tle. Tle of Z. + Defiitio : A set F with two opertios of dditio (+) d ultiplictio ( o ) is field if the followig postultes hold:. F is eli group uder +, with the idetity eleet deoted s e.. The set F {e} is eli group uder o, the ultiplictio opertio.. The ultiplictio opertio is distriutive with respect to dditio, i.e. o (+c) = ( o ) + ( o c) for ll, d c i F. Exple: osider Z p ={,,,,p} gi with dditio (odulo p) d ultiplictio (odulo p) s the two opertio. It c e show tht the set Z p {}={,, p} is eli group if p is prie. Therefore Z p is field if p is prie. The set Z of itegers odulo, deoted GF(), is iportt field i our study with: + =, + = + =, d +=, o =, o = o =, d o =... VETOR SPAES osider set S with iry opertio. Let F e field with + d o eig the dditio d ultiplictio opertios, respectively. A ultiplictio opertio, deoted y, is lso defied etwee the eleets of F d those of S. This opertio ssigs to ech ordered pir (α,s) uique eleet deoted y α s, where α is i F d s is i S. The set S is vector spce over F if the followig postultes hold:. S is eli group uder.

11 HAPTER Bsic ocepts d Defiitios. For y eleets α d β i F, d y eleets s d s i S the followigs hold: α (s s ) = ( α s) ( α s) d ( α + β) s = ( α s) ( β s). For y eleet α d β i F d y eleet s i S: ( α β) s = α ( β s). For y eleet s i S, s = s, where is the ultiplictive idetity i F. osider vector spce S, over the field F. The eleets of S re clled vectors d those of F re kow s sclrs. If eleet s of S is expressile s, s = (α s ) (α s ). (α j s j ), (-) where s i s re vectors d α i s re sclrs, the s is sid to e lier coitio of s,s,,s j. Vectors s,s,,s j re sid to e lierly idepedet if o vectors i this set is expressile s lier coitio of the reiig vectors i the set. Vectors s,s,,s for sis i the vector spce S if they re lierly idepedet d every vector i S is expressile s lier coitio of these vectors. The vectors s,s,,s re kow s sis vectors. The diesio of the vector spce S, deoted y di(s), is the uer of vectors i sis of S. If S is suset of the vector spce S over F, the S is suspce of S if S is lso vector spce o F. The direct su S S of two suspces S d S of S is the set of ll vectors of the for s s, where s S d s S. It c e proved tht S S is lso suspce, d its diesio is give y di(s S ) = di(s ) + di(s ) di(s S ). (-5) Note tht S S is lso suspce wheever S d S re suspces... VETOR SPAE OF A GRAPH osider grph S = (N,M) d let W S deote the collectio of ll susets of M, icludig the epty set. Uder rig su opertio etwee sets, i the followig it is show tht W S is eli group. Defiig suitle ultiplictio etwee eleets of the field Z d those of W S, it c e show tht W S is vector spce over Z.

12 Structurl Mechics: Grph d Mtrix Methods It c e show tht W S is closed uder. The opertor is ssocitive d couttive. Further for y eleet M i i W S, M i = M i d M i M i =. Therefore for the opertio, is the idetity eleet, d ech M i is its ow iverse. Hece W S is eli group uder. Let, ultiplictio opertio etwee the eleets of Z d those of W S e defied s follows: M i = M i d M i =. With this defiitio of oe c verify tht the eleets of W S stisfy the followig other requireets of vector spce:. (α+β) M i = (α M i ) (β M i ).. α (M i M j ) = (α M i ) (α M j ).. (α.β) M i = α (β M i ).. M i = M i. (Notice tht is the ultiplictive idetity i Z.) Therefore W S is vector spce over Z. The diesio of this spce is equl to the uer of eers of the grph S. Sice ech eer-iduced sugrph of S correspods to uique suset of M, d y defiitio the rig su of y two eer-iduced sugrphs correspods to the rig su of their correspodig eer sets, it is ovious tht the set of ll eers-iduced sugrphs of S is lso vector spce over Z if the ultiplictio is defied s follows: M i = M i d M i =, the ull grph hvig o odes d o eers. This vector spce will lso e referred to y the syol W S... YLE SUBSPAE AND UTSET SUBSPAE OF A GRAPH Now we study two iportt suspces of W S, ely cycle spce d cutset spce of grph. Theore : The set of ll siple cycles d uio of eer-disjoit cycles of grph, W, is suspce W S of S.

13 HAPTER Bsic ocepts d Defiitios osider d s two cycles of W. To prove the theore oe should show tht is lso cycle elogig to W. Let e ode of. This ode is preset t lest i of the two cycles d. Let M i (i=,,), deotes the eers icidet to i i. Let i shows the uer of eers of i, thus M i is the uer of eers icidet to i i. Note tht M d M re oth eve d oe of the y e zero. Furtherore M is o-zero. Sice =, we hve: M = M M. Therefore: M = M M M M. Sice M d M re oth eve, thus M is lso eve. This is true for ll the odes of, d it follows tht it is cycle i W d the theore is prove. Theore : The set of ll cutsets d the uio of eer-disjoit cutsets W i grph S, is suspce of the vector spce W S of S. It c e show tht the rig su of y two cuts i grph is cut i S. Siilrly the uio of y two eer-disjoit cuts i grph S is lso cut i S. Sice W is closed uder the rig su opertio thus the proof is copleted...5 FUNDAMENTAL YLE BASES A specil cycle sis kow s fudetl cycle sis c esily e costructed correspodig to tree T of S. I coected S, chord of T together with T cotis cycle kow s fudetl cycle of S. Moreover, the fudetl cycles otied y ddig the chords to T, oe t tie, re idepedet, ecuse ech cycle hs eer which is ot i the others. Also, every cycle i depeds o the set of fudetl cycles otied y the ove process, for i is the syetric differece of the cycles deteried y the chords of T which lie i i. Thus the cycle rk (cyclotic uer, first Betti uer, ullity) of grph S which is the uer of cycles i sis of the cycle spce of S, is give y, d if S cotis (S) copoets, the: (S) = M(S) N(S) +, (-6) (S) = M(S) N(S) + (S). (-7)

14 Structurl Mechics: Grph d Mtrix Methods A forl proof is provided i Sectio..7. As exple, the selected tree T d four fudetl cycles of S re illustrted i Figure.. S T Fig.. A grph S d fudetl cycle sis of S...6 FUNDAMENTAL UTSET BASES A sis c e costructed for the cutset spce of grph S. osider the tree T d its cotree T. The sugrph of S cosistig of T d y eer of T (rch) cotis exctly oe cutset kow s fudetl cutset. The set of cutsets otied y ddig rches of T to T, oe t tie, fors sis for the cutset spce of S, kow s fudetl cutset sis of S. The cutset rk (rk of S) is the uer of cutsets i sis for the cutset spce of S, which is give y d for grph with (S) copoets: A forl proof is provided i Sectio..7. ρ(s) = N(S), (-8) ρ(s) = N(S) (S). (-9) A grph S d fudetl cutset sis of S re show i Figure.. A rch of the tree sudivides the odes of the tree ito two susets. The eers of cutest should hve oe ed i ech susets.

15 5 HAPTER Bsic ocepts d Defiitios 5 () A grph S. () A tree T of S. (c) otree T of T. 5 6 Fig.. A grph S d fudetl cutset sis of S...7 DIMENSION OF YLE AND UTSET SUBSPAES osider T s spig tree of coected grph S with N odes d M eers. The rches of T re deoted y,,, N- d the chords y c,c,,c MN+. Let i d i refer to the fudetl cycle d the fudetl cutset with respect to c i d i, respectively. Sice ech fudetl cycle cotis exctly oe chord, d this chord is preset i o other fudetl cycle, therefore the fudetl cycles,,, MN+ re idepedet. Usig siilr resoig for cutsets, it ecoes ovious tht ll the fudetl cutsets,,, N re lso idepedet. Now we eed to prove tht every sugrph i cycle (cutset) suspce of S c e expressed s rig su of i ( i ). For this purpose cosider y sugrph i the cycle spce of S. Let coti the chords c i,c i,,c ir. Let deote the rig su of the fudetl cycle i, i,, ir. Oviously the chords c i,c i,,c ir re preset i, d cotis o other chord of the T. Sice lso cotis these chords d o others, cotis o chords.

16 6 6 Structurl Mechics: Grph d Mtrix Methods Now it is clied tht is epty. If this is ot true, the y the precedig rguets, cotis oly rches d hs o cycle. O the other hd, eig rig su of cycles, is cycle of the uio of eer-disjoit cycles. Therefore the ssuptio tht is ot epty leds to cotrdictio. Hece is epty. This iplies tht = = i i ir, i.e. every sugrph i the cycle spce of S c e expressed s rig su of the fudetl cycles. I siilr er it c e proved tht every sugrph i the cutset suspce of S c e expressed s rig su of the fudetl cutsets. The followig fct c ow e cocluded:. The fudetl cycles with respect to spig tree of S costitute sis for the cycle suspce of S, d therefore the diesio of the cycle suspce of S is equl to MN+.. The fudetl cutsets with respect to spig tree of S costitute sis for the cutset suspce of S, d therefore the diesio of the cutset suspce of S is equl to N. For grphs which re ot coected, spig forest will replce the spig tree, d the diesios for cycle suspce d cutset suspce will the e ullity of S = (S) = MN+ (S) d rk of S = ρ(s) = N (S), respectively. Here, (S) is the uer of copoets of S...8 ORTHOGONALITY PROPERTY Two vectors re clled orthogol if their sclr product is zero. It c e show tht vector is cycle set (cutset) vector, if d oly if it is orthogol to every vector of cutset (cycle set) sis. Sice the cycle set d cutset spces of grph S cotiig M(S) eers re oth suspces of the M(S)-diesiol spce of ll vectors which represet susets of the eers, therefore the cycle set d cutset spces re orthogol copoets of ech other..5 MATRIES ASSOIATED WITH A GRAPH Mtrices ply doit role i the theory of grphs d i prticulr i its pplictios to structurl lysis. Soe of these trices coveietly descrie the coectivity properties of grph d others provide useful ifortio out the ptters of the structurl trices, d soe revel dditiol ifortio out trsfortios such s those of equiliriu d coptiility equtios.

17 7 HAPTER Bsic ocepts d Defiitios 7 I this sectio vrious trices re studied which reflect the properties of the correspodig grphs. For siplicity, ll the grphs re ssued to e coected, sice the geerliztio to o-coected grphs is trivil d cosists of cosiderig the direct su of the trices for their copoets..5. MATRIX REPRESENTATION OF A GRAPH A grph c e represeted i vrious fors. Soe of these represettios re of theoreticl iportce, others re useful fro the progrig poit of view whe pplied to relistic proles. I this sectio six differet represettios of grph re descried. Node Adjcecy Mtrix: Let S e grph with N odes. The djcecy trix A is N N trix i which the etry i row i d colu j is if ode i is djcet to j, d is otherwise. This trix is syetric d the row sus of A re the degrees of the odes of S. The djcecy trix of the grph S, show i Figure., is 5 5 trix s: A = 5 5 (-) Fig.. A grph S. It c e oted tht A is syetric trix of trce zero. For two isoorphic grphs S d S', the djcecy trix A of S c e trsfored to A' of S' y siulteous peruttios of the rows of A. The (i,j)th etry of A shows the uer of wlks of legth with i d j s ed odes. Siilrly, the etry i the

18 8 8 Structurl Mechics: Grph d Mtrix Methods (i,j) positio of A k is equl to the uer of wlks of legth k with i d j s ed odes. The polyoil, φ ( λ) = det( λi A), (-) is clled the chrcteristic polyoil of S. The collectio of N(S) eigevlues of A is kow s the spectru of S. Sice A is syetric, the spectru of S cosists of N(S) rel uers. The su of eigevlues of A is equl to zero. The eigevectors of A re orthogol. Node-Meer Icidece Mtrix: Let S e grph with M eers d N odes. The ode-eer icidece trix B is N M trix i which the etry i row i d colu j is if ode i is icidet with eer j, d is otherwise. As exple, the ode-eer icidece trix of the grph i Figure. is 5 7 trix of the for: B = (-) Oviously, the ptter of icidece trix depeds o the prticulr wy tht its odes d eers re lelled. Oe icidece trix c e otied fro other y siply iterchgig rows (correspodig to relellig the odes) d colus (correspodig to relellig the eers). The icidece trix B d the djcecy trix A of grph S re relted y, t B B = A + V, (-) where V is digol trix of order N(S), kow s the degree trix, whose typicl o-zero etry v ii is the vlecy of the ode i of S for i=,...,n(s). The rows of B re depedet d oe row c ritrrily e deleted to esure the idepedece of the rest of the rows. The ode correspodig to the deleted row is clled dtu (referece) ode. The trix otied fter deletig depedet row is clled icidece trix of S d it is deoted y B. Although A d B re of gret theoreticl vlue, however, the storge requireets for these trices re high d proportiol to N N d M (N),

19 9 HAPTER Bsic ocepts d Defiitios 9 respectively. I fct lrge uer of uecessry zeros is stored i these trices. I prctice oe c use differet pproches to reduce the storge required, soe of which re descried i the followig. Meer List: This type of represettio is coo pproch i structurl echics. A eer list cosists of two rows (or colus) d M colus (or rows). Ech colu (or row) cotis the lels of the two ed odes of ech eer, i which eers re rrged sequetilly. For exple, the eer list of S i Figure. is: i ML =. j 5 5 (-) It should e oted tht eer list c lso represet oriettios o eers. The storge required for this represettio is M. Soe egieers prefer to dd third row cotiig the eer's lels, for esy ddressig. I this cse the storge is icresed to M. A differet wy of preprig eer list is to use vector cotiig the ed odes of eers sequetilly; e.g. for the previous exple this vector ecoes: (, ;,5 ;, ;, ;, ;,5 ;, ). (-5) This is copct descriptio of grph; however, it is iprcticl ecuse of the extr serch required for its use i vrious lgoriths. Adjcecy List: This list cosists of N rows d D colus, where D is the xiu degree of the odes of S. The ith row cotis the lels of the odes djcet to ode i of S. For the grph S show i Figure., the djcecy list is: The storge eeded for djcecy list is N D. AL = 5 (-6) 5 5 N D opct Adjcecy List: I this list the rows of AL re cotiully rrged i row vector R, d dditiol vector of poiters P is cosidered. For exple, the copct djcecy list of Figure. c e writte s:

20 Structurl Mechics: Grph d Mtrix Methods R = (,,,,,6,,,5,,,5,, ), P = (,,6,,,5). (-7) P is vector (p, p, p, ) which helps to list the odes djcet to ech ode. For ode i oe should strt redig R t etry p i d fiish t p i+. A dditiol restrictio c e put o R, y orderig the odes djcet to ech ode i i scedig order of their degrees. This orderig c e of soe dvtge, exple of which is odl orderig for dwidth optiistio. The storge required for this list is M + N YLE BASES MATRIES The cycle-eer icidece trix of grph S, hs row for ech cycle or higed cycle d colu for ech eer. A etry c ij of is if cycle i cotis eer j d it is otherwise. I cotrst to the ode djcecy d ode-eer icidece trix, the cycle-eer icidece trix does ot deterie grph up to isoorphis; i.e. two totlly differet grphs y hve the se cycle-eer icidece trix. (S) For grph S there exists cycles or higed cycles. Thus is (S) ( ) M trix. However, oe does ot eed ll the cycles of S, d the eleets of cycle sis re sufficiet. For cycle sis, cycle-eer icidece trix ecoes (S) M trix, deoted y, kow s the cycle sis icidece trix of S. As exple, trix for the grph show i Figure., for the followig cycle sis, is give y: = (,, ) = (, 5, 6 ) = (, 5, 7 ) = (-8)

21 HAPTER Bsic ocepts d Defiitios The cycle djcecy trix D = t W is (S) (S) trix, ech etry d ij of which is if i d j hve t lest oe eer i coo, d it is otherwise. For the ove exple, t D = W = =, (-9) where W is digol trix, i which typicl o-zero etry w ii is the legth of the cycle i. The trce of t is equl to the totl legth of the cycles i the sis. A iportt theore c ow e proved which is sed o the orthogolity property etioed i Sectio..8. Theore: Let S hve icidece trix B d cycle sis icidece trix. The: B t = (od ). (-) Proof: osider the ith row of d the jth colu of B t, which is the jth row of B. The rth etry i these two rows re oth o-zero if d oly if r is i cycle i d is icidet with j. If r is i i, the j is lso i i, ut if j is i the cycle, the there re two eers of i icidet with j so tht the (i,j)th etry of B t is + = (od ), d this copletes the proof. Mtrix for fudetl cycle sis with specil lels for its tree eers d chords, fids prticulr ptter. Let S hve tree T whose eers re M(T) = (,,..., p ) d cotree for which M(T) = ( p+, p+, p+m(s) ). The there is uique fudetl cycle i i S M(T) + i, p+ i M(S) d this set of cycles fors sis for the cycle spce of S. As exple, for the grph S of Figure. (pge ) whose eers re lelled s show i Figure.5, the fudetl cycle sis cosists of: = (,, 7 ) = (,, 8 ) = (,, 9, 5, ), = (, 5,, 6, ). The correspodig for the selected tree T is deoted y d hs the followig for:

22 Structurl Mechics: Grph d Mtrix Methods M(T) M(T) ] [ T I = = (-) Fig..5 A grph d its tree eers..5. UTSET BASES MATRIES The cutset-eer icidece trix for grph S, hs row for ech cutset of S d colu for ech eer. A etry ij c of is if cutset i cotis eer j d it is otherwise. This trix like does ot deterie grph copletely. Idepedet rows of for cutset sis, deoted y, for trix kow s cutset sis icidece trix, which is ρ(s) M trix, ρ(s) eig the rk of grph S. As exple, for the cutset of Figure. with eers lelled s i Figure.5, is give elow: = (-)

23 HAPTER Bsic ocepts d Defiitios t The cutset djcecy trix D = is ρ(s) ρ(s) trix defied logously to cycle djcecy trix D. For fudetl cutset sis with pproprite lellig of the eers i T d T, the prticulr ptter of ecoes: [ ] = = I c (-) Fro the orthogolity coditio, t =, hece: I T t c [ I] =. (-) t Hece T + c = (od ), resultig i : t T = c. (-5) Therefore, for grph hvig, oe c costruct d vice vers. There exists very siple sis for the cutset spce of grph, which cosists of N cocycles of S. As exple, for the grph of Figure., cosiderig 5 s dtu ode, we hve, = 5 6 7, (-6) which is the se s the icidece trix B of S. The siplicity of the displceet ethod of structurl lysis is due to the existece of such siple sis.

24 Structurl Mechics: Grph d Mtrix Methods.6 DIRETED GRAPHS AND THEIR MATRIES A orieted or directed grph is grph i which ech eer is ssiged oriettio. A eer is orieted fro its iitil ode to its fil ode, s show i Figure.6(). The iitil ode is sid to e positively icidet o the eer d the fil ode egtively icidet, s show i the Figure: j i i () () (c) Fig..6 A orieted eer, directed grph d directed tree. The choice of oriettio of eers of grph is ritrry; however, oce it is chose, it ust e retied. ycles d cutsets c lso e orieted s show i Figure.6(). As exple, 7 is positively orieted i cycle i, d is egtively orieted i cutset i. All the trices B, B, d c e defied s efore, with the differece of hvig +, d s etries, ccordig to whether the eer is positively, egtively d zero icidet with cutset or cycle. As exple, for grph S i Figure.6() the trix B with s dtu ode is fored: = B (-7)

25 HAPTER Bsic ocepts d Defiitios 5 5 osider tree s show i Figure.6(c). The correspodig cycle sis icidece trix c e writte s: c T = (-8) Oviously, B t = B t = (od ), (-9) with siilr proof s tht of the o-orieted cse. A cutset-eer icidece trix is siilrly otied s: = c T (-) It c esily e proved tht: c t T =. (-).7 GRAPHS ASSOIATED WITH MATRIES Mtrices ssocited with grphs re discussed i the previous sectios. Soeties it is useful to cosider the reverse of this process d thik of the grph ssocited with ritrry trix H. Such grph hs ode ssocited with ech row of the trix d if h ij is o-zero, the there is coectig eer fro ode i to ode j. I the cse of syetric trix, there is lwys coectio fro i to j wheever there is oe fro j to I ; therefore oe c siply use udirected

26 6 Structurl Mechics: Grph d Mtrix Methods 6 eers. Two siple exples re illustrted i Figure.7 d Figure.8. The directed grph ssocited with o-syetric trix is usully clled digrph d the word grph is used for the udirected grph ssocited with syetric trix: = H Fig..7 A o-syetric trix H d its ssocited digrph. = H Fig..8 A syetric trix H d its ssocited grph. For H to e viewed s the djcecy trix, due to the presece of digol etries, oe loop should e dded to ech ode. However, sice the structurl odels hve lwys o-zero digol etries d coti o loops, this dditio is disregrded. With rectgulr trix E iprtite grph S = (A,B) c e ssocited. For ech row of E ode of A d with ech colu of E ode of B is ssocited. Two odes of A d B re coected with eer of S if e ij is o-zero. A exple of this is show i Figure.9. = E B A

27 7 HAPTER Bsic ocepts d Defiitios 7 Fig..9 A rectgulr trix E d its ssocited iprtite grph. A weighted iprtite grph c e defied for y rectgulr trix H. The odes r,r,,r d odes c,c,,c of the grph correspod to rows d colus of H, respectively. If h ij, the r i is joied to c j y eer whose weight is h ij..8 PLANAR GRAPHS - EULER S POLYHEDRA FORMULA Grph theory d properties of plr grphs, were first discovered y Euler i 76. After 9 yers Kurtowski foud criterio for grph to e plr. Whitey developed soe iportt properties of eeddig grphs i the ple. McLe expressed the plrity i ters of the grph s cycle sis. I this sectio soe of these criteri re studied, d Euler s polyhedr forul is prove..8. PLANAR GRAPHS A grph S is clled plr if it c e drw (eedded) i the ple i such wy tht o two eers cross ech other. As exple, coplete grph K show i Figure. is plr sice it c e drw i the ple s show: () K. () Plr drwigs of K. Fig.. K d two of its drwigs. O the other hd K 5, Figure., is ot plr, sice every drwig of K 5 cotis t lest oe crossig. () K 5. () Two drwigs of K 5 with oe crossig.

28 8 8 Structurl Mechics: Grph d Mtrix Methods Fig.. K 5 d two of its drwigs. Siilrly K,, Figure., is ot plr, s illustrted: () K,. () Two drwigs of K, with oe crossig. Fig.. K, d its drwigs. A plr grph S drw i the ple divides the ple ito regios ll of which re ouded d oly oe is uouded. If S is drw o sphere, ll the regios will e ouded; however, the uer of regios will ot chge. The cycle oudig regio is clled regiol cycle. Oviously the su of the legths of regiol cycles is twice the uer of eers of the grph. There is outstdig forul tht reltes the uer of regios, eers d odes of plr grph, i the for, R(S) M(S) + N(S) =, where R(S), M(S) d N(S) re the uers of regios, eers d odes of plr grph S, respectively. This forul shows tht for differet drwigs of S i the ple, R(S) reis costt. Origilly the ove reltioship ws give for polyhedr, i which R(S), M(S) d N(S) correspod to fces, edges d corers of polyhedro, respectively. However, the theore c esily e expressed i grph-theoreticl ters s follows. Theore (Euler [5]): Let S e coected plr grph. The: R(S) M(S) + N(S) =. (-) Proof: For proof, S is refored i two stges. I the first stge, spig tree T of S is cosidered i the ple for which R(T) M(T) + N(T) =. This is true sice R(T) = d M(T) = N(T). I secod stge chords re dded oe t tie. Additio of chord icreses the uer of eers d regios ech y

29 9 HAPTER Bsic ocepts d Defiitios 9 uity, levig the left hd side of Eq. (-) uchged durig the etire process, d the result follows..8. THEOREMS FOR PLANARITY I order to check the plrity of grph, differet pproches re ville which re sed o the followig theores. These theores re oly stted d the reder y refer to textooks o grph theory for proofs. Theore (Kurtowski [8]): A grph S is plr if d oly if it hs o sugrph cotrctile to K 5 or K,. otrctig eer k = ( i, j ) is opertio i which the eer is reoved d i is idetified with j so tht the resultig ode is icidet to ll eers (other th k ) tht were origilly icidet with i or j. If grph S c e otied fro S y successio of eer cotrctios, the S is cotrctile to S. The process of the cotrctio of eer ( i, j ) of grph is show i Figure.(), d the cotrctio of the Peterse grph to K 5 is illustrted i Figure.(). k i j i j i, j () otrctio of eer k. () otrctio of the Peterse grph to K 5. Fig.. The cotrctio of eer i grph.

30 Structurl Mechics: Grph d Mtrix Methods Theore (McLe [67]): A coected grph is plr if d oly if every lock of S with t lest three odes hs cycle sis,,,..., (S) d oe dditiol cycle, such tht every eer is cotied i exctly two of these (S)+ cycles. A lock is xil o-seprle grph, d o-seprle grph is grph tht hs o cut-poits. A cut-poit is ode whose reovl icreses the uer of copoets d ridge is such eer. I Figure., grph d its locks re illustrted: ridge cut-poit Fig.. A grph d its locks. Defiitios: A grph S is dul grph of grph S if there is - correspodece etwee the eers of S d those of S, such tht set of eers i S is cycle vector of S if d oly if the correspodig set of eers i S is cutset vector of S. Theore (Whitey [5]) - A grph is plr if d oly if it hs coitoril dul. () A plr grph S. () The dul grph of S. Fig..5 A plr grph d its dul.

31 HAPTER Bsic ocepts d Defiitios For coected plr grph S, the dul grph S is costructed s follows: To ech regio r i of S there is correspodig ode r i of S d to ech eer j of S there is correspodig eer j i S, such tht if the eer j occurs o the oudry of two regios r d r, the the eer correspodig odes r d r i S, Figure.5. j jois the.9 MAXIMAL MATHING IN BIPARTITE GRAPHS.9. DEFINITIONS As defied efore, grph is iprtite if its set of odes c e prtitioed ito two sets A d B, such tht every eer of the grph hs oe ed ode i A d other i B. Such grph is deoted y S = (A,B). A set of eers of S is clled tchig if o two eers hve coo ode. The size of y lrgest tchig i S is clled the tchig uer of S, deoted y ψ(s). A suset N'(S) N(S), is the ode cover of S, if ech eer of S hs t lest oe ed ode i N'(S). The crdility of y sllest ode cover, deoted y τ(s), is kow s the ode coverig uer of S..9. THEOREMS ON MATHING I this sectio, three theores re stted, d the proofs y e foud i the ook y Lovsz d Pluer [6]: Theore (Köig [,]): For iprtite grph S, the tchig uer ψ(s) is equl to the ode coverig uer τ(s). Theore (Hll [69]): Let S = (A,B) e iprtite grph. The S hs coplete tchig of A ito B if d oly if Γ(X) X for ll X A. Γ(X) is the ige of X, i.e. those eleets of B which re coected to the eleets of X i S. Figure.6() shows iprtite grph for which tchig exists d Figure.6() illustrtes cse where coplete tchig does ot exist, ecuse X = (, ) re tched to, i.e. Γ(X) X :

32 Structurl Mechics: Grph d Mtrix Methods A B A B () () Fig..6 Mtchig i iprtite grphs. A perfect tchig is tchig which covers ll odes of S. Theore (Froeius [5]): A iprtite grph S = (A,B) hs perfect tchig if d oly if A = B d for ech X A, Γ(X) X. () Perfect tchig exists. () Perfect tchig does ot exist. Fig..7 Perfect tchig i iprtite grphs. This is lso kow s the rrige theore. Figures.7() d () show cses whe perfect tchig exists d does ot exist, respectively. Therefore Froeius s theore chrcterizes those iprtite grphs which hve perfect tchig. Hll s theore chrcterizes those iprtite grphs tht hve tchig of A ito B. Köig s theore gives forul for the tchig uer of iprtite grph..9. MAXIMUM MATHING

33 HAPTER Bsic ocepts d Defiitios Let M e y tchig i iprtite grph S = (A,B). A pth P is clled ltertig pth with respect to M, or M-ltertig pth if its eers (edges) re ltertely chose fro the tchig M d outside M. A ode is exposed (utched, ot covered) with respect to tchig M if o eer of M is icidet with tht ode. A ltertig tree reltive to the tchig, is tree which stisfies the followig two coditios: first, the tree cotis exctly oe exposed ode fro A, which is clled its root, secod, ll pths etwee the root d y other ode i the tree re ltertig pths. As exple, i Figure.8() the pth is ltertig pth with respect to the tchig show i old lies;,, d re exposed odes. () A ritrry tchig. () A ugeted tchig. Fig..8 Opertio for xiu tchig. A M-ltertig pth joiig two exposed odes is clled M-ugetig pth. For every such pth the correspodig tchig c e de lrger y discrdig the eers of P M d ddig those of P M, where P is M-ltertig pth, Figure.8(). Thus, if S cotis y M-ltertig pth P joiig two exposed odes, the M c ot e xiu tchig, for oe c redily oti lrger tchig M y discrdig the eers of P M d ddig those of P M. Theore (Berge [-]): Let M e tchig i grph S. The M is xiu tchig if d oly if there exists o ugetig pth i S reltive to M. The ove result provides ethod for fidig xiu tchig i S. The coputtiol procedure for costructio of xiu tchig egis with cosiderig y fesile tchig, possily the epty tchig. Ech exposed ode of A is de the root of ltertig tree, d odes d eers re dded to the trees y es of lellig techique. Evetully, the followig two cses ust occur: either exposed ode i B is dded to oe of the trees, or else

34 Structurl Mechics: Grph d Mtrix Methods it is ot possile to dd ore odes d eers to y of the trees. I the forer cse, the tchig is ugeted d the fortio of trees is repeted with respect to the ew tchig. I the ltter cse, the trees re sid to e Hugri d the process is terited. As exple, cosider the tchig show i Figure.9(), i which old lies represet eers i the tchig. Altertig trees re costructed, with the exposed odes d 5 of A s roots, s show i Figure.9(). A ugetig pth is foud, s idicted i the Figure. Nturlly, severl differet sets of ltertig trees could hve ee costructed. For exple, the tree rooted t ode could hve cotied the eer (, ) () A iprtite grph S. () Altertig trees. Fig..9 A iprtite grph d its ltertig tree. Augeted tchig is show i Figure.. Whe the ltertig tree of Figure. is used for the ugeted tchig, it ecoes Hugri Fig.. Augeted tchig. 5 5 Fig.. Altertig tree for ugeted tchig.

35 5 HAPTER Bsic ocepts d Defiitios 5 BIPARTITE MATHING ALGORITHM Let X e y tchig, possily the epty tchig of iprtite grph S = (A,B). No odes re lelled. Step (lellig):. Give the lel to ech exposed ode i A.. If there re o usced lels, go to Step. Otherwise, fid ode i with usced lel. If i A, go to Step.; if i B, go to Step... Sc the lel o ode i (i A) s follows: for ech eer (i,j) X icidet to ode i, give ode j the lel "i", uless ode j is lredy lelled. Retur to Step... Sc the lel o ode i (i B) s follows: if ode i is exposed, go to Step. Otherwise, idetify the uique eer (i,j) X icidet to ode i d give ode j the lel "i". Retur to Step.. Step (Augetig): A ugetig pth hs ee foud, teritig t ode i (idetified i Step.). The odes precedig ode i i the pth, re idetified y cktrckig. Tht is, if the lel o ode i is "j", the secod-to-lst ode i the pth is j. If the lel o ode j is "k", the third-to-lst ode is k, d so o. The iitil ode i the pth hs the lel " ". Auget X y ddig to X ll eers i the ugetig pth tht re ot i X d reovig those which re i X. Reove ll lels fro odes. Retur to Step.. Step (Hugri Lellig): The lellig is Hugri, o ugetig pth exists, d the tchig X is of xiu crdility. For further study, the reder y refer to the origil pper of Hopcroft d Krp [8] or Lwler [57]. A lgorith usig differet pproch y e foud i Ref. [6].

36 6 6 Structurl Mechics: Grph d Mtrix Methods EXERISES. I the followig grph, which eers re icidet with ode (idetify with their ed odes)? Which odes re djcet to ode? Wht is the degree of ode? 5 6. Are the followig grphs isoorphic? Drw tree, spig tree d SRT rooted t O for the followig grph. Use O s the root of secod SRT d copre its legth d width with those of the first oe. O O'. Wht types of grph re the followig?

37 7 HAPTER Bsic ocepts d Defiitios 7.5 List ll the cycles of the followig grph:.6 Prove tht for plr grph eedded o sphere with ll trigulr fces, M(S) = N(S) 6..7 Fid fudetl cycle sis of the followig grph usig ritrry spig tree: O O'.8 I the ove exple use two SRTs rooted t O d O d copre the legth of the correspodig fudetl cycle ses..9 Write the djcecy d eer-ode icidece trices of the grphs i Exercise.. Use ritrry ode d eer uerig. Wht c you sy out the resultig trices?. Write,, d orthogolity property: trices for the followig grph d exie the

38 8 8 Structurl Mechics: Grph d Mtrix Methods. Idetify the plr grphs i the followig figure: () () (c) (d). Prove tht K 5 d K, re ot plr.. Euler s forul s i Eq. (-) fils for discoected grphs. If plr grph S hs (S) copoets, how c the forul e djusted?. Fid xil tchig for the followig iprtite grphs. Which oe is coplete d which oe is perfect tchig? () ().5 Why is there o coplete tchig for the followig iprtite grphs?

39 9 HAPTER Bsic ocepts d Defiitios 9 () ()