Spectral graph theory: Applications of Courant-Fischer

Transcription

1 Spectral graph theory: Applcatons of Courant-Fscher Steve Butler September 2006 Abstract In ths second talk we wll ntroduce the Raylegh quotent and the Courant- Fscher Theorem and gve some applcatons for the normalzed Laplacan. Our applcatons wll nclude structural characterzatons of the graph, nterlacng results for addton or removal of subgraphs, and nterlacng for weak coverngs. We also wll ntroduce the dea of weghted graphs. 1 Introducton In the frst talk we ntroduced the three common matrces that are used n spectral graph theory, namely the adjacency matrx, the combnatoral Laplacan and the normalzed Laplacan. In ths talk and the next we wll focus on the normalzed Laplacan. Ths s a reflecton of the bas of the author havng worked closely wth Fan Chung who popularzed the normalzed Laplacan. The adjacency matrx and the combnatoral Laplacan are also very nterestng to study n ther own rght and are useful for gvng structural nformaton about a graph. However, n many practcal applcatons t tends to be the spectrum of the normalzed Laplacan whch s most useful. In ths talk we wll ntroduce the Raylegh quotent whch s closely lnked to the Courant-Fscher Theorem and gve some results. We gve some structural results such as showng that f a graph s bpartte then the spectrum s symmetrc around 1) as well as some nterlacng and weak coverng results. But frst, we wll start by lstng some common spectra. Second of three talks gven at the Center for Combnatorcs, Nanka Unversty, Tanjn 1

2 Some common spectra When studyng spectral graph theory t s good to have a few basc examples to start wth and be famlar wth ther spectrums. Buldng a large encyclopeda of spectrums can help to recognze patterns. Here we lst several graphs and ther spectrums for ther normalzed Laplacans. Complete graphs: The complete graph on n vertces, denoted K n, s the graph whch contans all possble edges. The spectrum of the graph s 0 once) and n/n 1) n 1 tmes). Complete multpartte graph: The complete multpartte graph, denoted K m,n, has m + n vertces where the vertces are parttoned nto two groups A wth m vertces) and B wth n vertces) and as edges all possble edges connectng A and B. The spectrum of the graph s 0 once), 1 m+n 2 tmes) and 2 once). Cycles: The cycle on n vertces, denoted C n, s as ts name mples a cycle. The spectrum of the graph s 1 cos2πk/n) for k = 0, 1,..., n 1. Paths: The path on n vertces, denoted P n, s as ts name mples a path. The spectrum of the graph s 1 cos πk/n 1) ) for k = 0, 1,..., n 1. Petersen graph: Perhaps the most famous graph whch shows up n countless examples and counterexamples n graph theory. Ths s a graph on 10 vertces and s llustrated n Fgure 1. The spectrum of the graph s 0 once), 2/3 fve tmes), 5/3 four tmes). Hypercubes: The n-cube has 2 n vertces whch can be represented as all possble strngs of length n usng 0s and 1s wth an edge connectng two strngs f and only f they dffer n a sngle entry. The spectrum of the graph s 2k/n n k) tmes) for k = 0, 1,..., n. Fgure 1: From left to rght the graphs are K 7, C 7, K 3,4, Q 3, the Petersen graph, and on the bottom P 7. 2

3 2 The Raylegh quotent and Courant-Fscher For a nonzero real) vector x and real) matrx M, the Raylegh quotent of x wth M s defned as: Rx) = xt Mx x T x. It s usually understood whch matrx we are workng wth and so we wrte Rx) nstead of Rx, M).) Comment. In general we should use x the conjugate transpose) nstead of x T for defnng the Raylegh quotent. In our case where we are dealng wth a real symmetrc matrx we can restrct ourselves only to real vectors and thus drop the conjugaton. Note that f x s an egenvector of M assocated wth egenvalue λ then Rx) = x T Mx/x T x = λx T x/x T x = λ. From ths t s easy to see that max x 0 Rx) max λ, whch s to say that the maxmum of the Raylegh quotent s at least as bg as the largest egenvalue of the matrx. In fact, much more can be sad f M s symmetrc. In ths case we have a full set of orthonormal egenvectors φ 1,..., φ n whch we can use to decompose x = a φ. We then have that ) a φ Rx) = xt Mx x T x = a λ φ ) ) ) = a φ a φ a 2 λ a 2 max λ. Combnng the two deas above t follows that for M real and symmetrc wth egenvalues λ 0 λ 1 λ n 1, whle a smlar dervaton wll gve max x 0 Rx) = λ n 1, 1) mn Rx) = λ 0. 2) x 0 Equatons 1) and 2) are smple cases of the Courant-Fscher Theorem whch generalzes the above statements to not only fndng the smallest and largest egenvalues of a graph but any sngle partcular egenvalue. In the statement of the theorem below the assumpton of beng real s not needed f we use conjugaton as noted above. Theorem 1 Courant-Fscher Theorem). Let M be a real) symmetrc matrx wth egenvalues λ 0 λ 1 λ n 1. Let X k denote a k dmensonal subspace of R n and x X k sgnfy that x y for all y X k. Then λ = mn X n 1 max x X n 1,x 0 Rx) ) = max X mn x X,x 0 Rx) ). 3) 3

4 From the proof of the Courant-Fscher Theorem t wll follow that when lookng for egenvalue λ, that X n 1 n the frst expresson s the span of the last n 1 egenvectors whle X n the second expresson s the span of the frst egenvectors. We wll not nclude the proof of the Courant-Fscher Theorem here. The nterested reader can fnd a proof n any major lnear algebra textbook. 3 Egenvalues of the normalzed Laplacan Recall from the last lecture that L = D 1/2 LD 1/2 = D 1/2 D A ) D 1/2. Then note that for a vector y = y ) that y T Ly = y T D A)y = y 2 d 2 j y y j = j y y j ) 2, where j ndcates that s adjacent to j. In general, y T My s sometmes referred to as the quadratc form of the matrx because t can always be decomposed nto a sum of squares. In our case snce all of the coeffcents are 1 t follows that y T Ly 0 from whch t follows that the egenvalues of L are nonnegatve. Relatng back to the normalzed Laplacan we have x T Lx x T x = D1/2 y)ld 1/2 y) D 1/2 y) T D 1/2 y) = yt Ly y T Dy = j y y j ) 2 y2 d where we made the substtuton x = D 1/2 y for some vector y. In partcular, f we assume that D s nvertble.e., no solated vertces) then D 1/2 maps a k-dmensonal subspace to some other k-dmensonal subspace. Usng ths we wll then get the followng: ) ) x T Lx y T Ly λ = mn max = mn max X n 1 x X n 1,x 0 x T x X n 1 D 1/2 y X n 1,D 1/2 y 0 y T y j = mn max y y j ) 2 ) Y n 1 y Y n 1,y 0 y2 d. 4) Smlarly, λ = max Y mn y Y,y 0 j y y j ) 2 ) y2 d. 5) To fnd the egenvector we fnd the y whch mnmzes or maxmzes the above expressons and take D 1/2 y. Lemma 2. The egenvalues of L are nonnegatve. 4

5 Ths follows easly from notng that the expressons n 4) and 5) are always nonnegatve. More specfcally we have that 0 s an egenvalue, and t s easy to see that ths corresponds to the egenvector D 1/2 1 n equatons 4) and 5) ths would correspond to the choce of y = 1). Snce λ 0 s always 0 the two mportant egenvalues wll be λ n 1 and λ 1. In the thrd lecture we wll see that these control expanson and the closer these egenvalues are to 1 the more random-lke our graph wll be. From 4) we have that λ n 1 = max y 0 whle from 5) and the above comments λ 1 = mn y D1,y 0 j y y j ) 2 y2 d, j y y j ) 2 y2 d. If G s not connected t s smple to fnd a y so that λ 1 = 0. On the other hand f G s connected snce a vector y whch mnmzes the above expresson must have postve and negatve entres to satsfy y D1), there would be an j so that y > 0 and y j < 0, thus that term, as well as the complete sum, would be strctly postve. Ths establshes the followng. Lemma 3. Gven a graph G, λ 1 > 0 f and only f G s connected. More generally, λ = 0 f and only f there are at least + 1 connected components of G. Ths shows the power of spectral graph theory, egenvalues are analytc tools whle beng connected s a structural property, and as the above lemma shows they are closely connected. Smlarly we have the followng. Lemma 4. Gven a graph G, λ n 1 2 and λ n 1 = 2 f and only f G has a bpartte component. Ths makes use of the followng smple nequalty: a b) 2 2a 2 + 2b 2. j λ n 1 = max y y j ) 2 j y 0 y2 d max 2y2 + 2yj 2 ) y 0 y2 d = 2 Equalty wll hold f and only f y y j ) 2 = 2y 2 + 2y 2 j for all j, whch s equvalent to sayng that y = y j for all j. So we can then use the egenvector whch acheves the egenvalue 2 to fnd a bpartte component by gnorng any vertces whch have 0 entry n the egenvector and then puttng all vertces wth postve entry n one group and all vertces wth negatve entry n another group. It s easy to check that there then must be a nontrval bpartte component. 5

6 Comment. In the above argument we used the egenvector to help locate a bpartte component. Whle most of our dscusson wll focus on the egenvalues of the matrx, there s also mportant nformaton that can be derved by examnng the egenvalues of the graph. More generally, let G be a bpartte graph wth the vertces n two sets A and B. Then λ s an egenvalue of L f and only f 2 λ s an egenvalue. Ths follows by takng the egenvector for λ, say y, and modfyng t to create { y f A; y = y f B; then verfyng that y s an egenvector of L assocated wth egenvalue 2 λ. Lemma 5. If G s not a complete graph then λ 1 1. For ths suppose that k and l are not adjacent and consder the vector ŷ whch s d l at k and d k at l. Then we have that λ 1 = mn y D1,y 0 j y y j ) 2 y2 d j ŷ ŷ j ) 2 ŷ2 d = d kd 2 l + d ld 2 k d k d 2 l + d ld 2 k Note that earler we saw that the complete graph had λ 1 = n/n 1) > 1. Ths mples that G s a complete graph f and only f λ 1 > 1. 4 Interlacng nequaltes In ths secton and the next we wll compare egenvalues of two graphs. Intutvely we want to be able to say that f two graphs are very smlar then ther spectrums should be close. Before we begn ths we wll frst ntroduce the dea of weghted graphs. Weghted graphs So far we have worked wth smple unweghted graphs. For some applcatons t can be useful to allow weghts to be placed on the edges. For example, f we want to work wth the spectrum of multgraphs then we can model such graphs by assgnng the weght between vertex and vertex j to be the number of mult-edges. For another example, suppose we want to follow the flow of water through a system of ppes, snce not all ppes may be equal n sze we should account for the dsparty of flow, whch can be done by puttng weghts on the edges. In ths case one reasonable 6 = 1.

7 weght to put on the edges s the dameter of the ppe, snce a larger ppe should allow for more water flow. In general, a weghted graph s a graph wth a nonnegatve weght functon w where w, j) = wj, ).e., stll undrected at ths tme), and there s an edge jonng and j f and only f w, j) > 0. We also allow loops, these correspond to when w, ) > 0. The degree of vertex wll now be the sum of the weghts of the ncdent edges,.e., d = w, j). j We then proceed as before by lettng A,j = w, j), D the dagonal degree matrx usng our new defnton of degree) and L = D 1/2 D A)D 1/2. Ths s smlar to what we have done before and many of the results easly generalze ndeed everythng we have done so far corresponds to the case when the weghts are 0 and 1). In partcular, we can now wrte 4) and 5) as λ = mn Y n 1 = max Y max y Y n 1,y 0 mn y Y,y 0 j y y j ) 2 ) w, j) y2 d j y y j ) 2 w, j) y2 d Subtractng out a weghted subgraph 6) ). 7) Gven a smple graph G suppose we remove one edge. Ths wll change the graph and so also change the spectrum. What can we say about the relaton between the two dfferent spectrums? We wll show that there s a relaton and that the egenvalues nterlace. That s we wll show that an egenvalue of the new graph les n an nterval between two specfed egenvalues of the orgnal graph. In fact we wll show somethng more general, nstead of removng only one edge we wll remove some of the weghts off a set of edges. Frst we need to ntroduce some termnology needed for the statement of our result. Gven a weghted graph G a subgraph H of G s a weghted graph where w H, j) w G, j) for all and j, whle the graph G H has weght functon w G H, j) = w G, j) w H, j). Theorem 6. Let G be a weghted graph and H a connected subgraph of G wth V H) = t. If λ 0 λ 1 λ n 1 and θ 0 θ 1 θ n 1 7

8 are the egenvalues of LG) and LG H) respectvely, then for k = 0, 1,..., n 1 we have { λk+t 1 H s loopless and bpartte, λ k t+1 θ k otherwse, λ k+t where λ t+1 = = λ 1 = 0 and λ n = = λ n+t = 2. One nterestng thng to note s that the result s ndependent of the amount of weght subtracted. Ths s one consequence of normalzaton. Instead of subtractng out a graph we could also add a graph. The followng result mmedately follows from Theorem 6 workng wth the graphs G + H and G + H) H = G. Corollary 7. Let G be a weghted graph and H a connected graph on a subset of the vertces of G wth V H) = t. If λ 0 λ 1 λ n 1 and θ 0 θ 1 θ n 1 are the egenvalues of LG) and LG + H) respectvely, then for k = 0, 1,..., n 1 we have { λk t+1 H s loopless and bpartte, λ k+t 1 θ k otherwse, λ k t where λ t = = λ 1 = 0 and λ n = = λ n+t 1 = 2. The proof of Theorem 6 s establshed n three parts. We wll do one part here and leave the other two to the nterested reader. In partcular, we wll show that θ k λ k t+1. Suppose that { 1, j 1 }, { 2, j 2 },..., { t 1, j t 1 } are edges of a spannng subgraph of H, and let Z = {e 1 e j1, e 2 e j2,..., e t 1 e jt 1 } be a set of vectors where e j s the vector whch s 1 n the jth poston and 0 otherwse. Then usng 6) we have j θ k = mn max y y j ) 2 ) w G H, j) Y n k 1 y Y n k 1,y 0 y2 d G H, j = mn max y y j ) 2 w G, j) j y y j ) 2 ) w H, j) Y n k 1 y Y n k 1,y 0 y2 d G, y2 d H, j mn max y y j ) 2 ) w G, j) Y n k 1 y Y n k 1,y Z,y 0 y2 d G, y2 d H, j mn max y y j ) 2 ) w G, j) Y n k 1 y Y n k 1,y Z,y 0 y2 d G, j mn max y y j ) 2 ) w G, j) Y n k+t 2 y Y n k+t 2,y 0 y2 d = λ k t+1. G, 8

9 In gong from the second to the thrd lne we added the condton that y also be perpendcular to Z so that we are maxmzng over a smaller set. Wth the condton that y Z then y = y j for all, j n H, n partcular the second term n the numerator drops out. Gong from the thrd to the fourth lne we make the denomnator larger. Whle n gong from the fourth to the ffth lne we consdered a broader optmzaton that would nclude the fourth lne as a case. The other two statements n Theorem 6 use 7) and are proved smlarly. As an example of what can be proved usng nterlacng, t s an easy exercse usng Theorem 6 to show that f G s a smple graph on n vertces and more than n/2 of the vertces are connected to every other vertex then n/n 1) s an egenvalue of G. Ths works well because n/n 1) has hgh multplcty n the complete graph and then we punch out a subgraph. In general to use nterlacng to show a graph contans a partcular egenvalue we must start wth a graph whch has an egenvalue wth hgh multplcty.) 5 Weak coverngs and egenvalues In addton to removng subgraphs as we dd n the prevous secton we can also condense graphs, that s we dentfy vertces together and then reassgn weghts approprately. We call ths process a coverng, the condensed graph wll be referred to as the covered graph and the orgnal graph the coverng graph. Comment. We wll dscuss weak coverngs whch make the fewest assumptons on the structure of the coverng. By makng more assumptons we get a rcher structure and more can be sad about how egenvalues relate. For example, one type of coverng s known as a strong regular coverng and for such a coverng both graphs have the same egenvalues. These types of coverngs have been extensvely studed n algebrac graph theory wth great success n beng able to determne the egenvalues of a large graph by computng the egenvalues of a small graph. One drawback however s that such coverngs requre workng wth graphs wth nce structures, such as Cayley graphs, and the results are not wdely applcable to real-world graphs. We say that G s a weak coverng of H f and only f there s some onto mappng π : V G) V H) such that for all u, v V H), w H u, v) = w G x, y). x π 1 u) y π 1 v) From ths defnton t follows that d H v) = x π 1 v) d Gx). 9

10 Alternatvely, we group the vertces of G n some manner then collapse the ndvdual groups of vertces nto a sngle vertex. To fnd the edge weghts of the resultng edges we add the weghts of any resultng parallel edges that are formed. An example s shown n Fgure 2 note that the loop formed wll be counted twce when tallyng the edge weght) Fgure 2: The graph on the left s the coverng graph wth all edge weghts 1 whle the graph on the rght s the covered graph wth edge weghts as ndcated. Here we are weak n that we are puttng few restrctons on the mappng π. However, even wth these base assumptons we are stll able to relate the egenvalues of G and H the coverng graph and the covered graph). Theorem 8. Let G be a weak cover of H wth V G) = n and V H) = m, and further let λ 0 λ 1 λ n 1 and θ 0 θ 1 θ m 1 be the egenvalues of LG) and LH). Then for k = 0, 1,..., m 1 we have the followng λ k θ k λ k+n m). To establsh ths result, for = 1,..., m let V = π 1 v ),.e., these are the groupngs of the vertces of G, and let Z = {e 1 e 2, e 1 e 3,..., e 1 e j } where V = {v 1, v 2,..., v j } V G). Further we wll let Z = Z. It s easy to check that the dmenson of the span of Z s n m. θ k = max Y k R m = max Y k R n mn y Y k,y 0 mn y Y k,y Z,y 0 ) y y j ) 2 w H, j) y 2 d H ) ) y y j ) 2 w G, j) y 2 d G ) 8) What we have done n the second step s used the defnng property of weak covers to lft vectors from H up to G so that we stll satsfy the same Raylegh quotent. Our only condton n lftng s that y = y π) whch s to say that for a functon on H we lft the value at a vertex up to all of the vertces n G whch cover t. In partcular 10

11 f πv ) = πv j ), then we need y π) = y πj). Ths last condton s easly acheved by requrng that the lfted vector be perpendcular to Z, and hence the form lsted. Lookng at 8) we can take two approaches, the frst s to drop the requrement that we reman perpendcular to Z, thus we are mnmzng over a larger set and so we have θ k max Y k mn y Y k,y 0 y y j ) 2 w G, j) y 2 d G ) ) = λ k. The other approach s to maxmze over some larger set that wll also consder the stuaton gven n 8),.e., ) y y j ) 2 w G, j) θ k max mn = λ Y k+n m y Y k+n m,y 0 y 2 k+n m. d G ) Combnng the two nequaltes above establshes Theorem 8. 11