On Cheeger-type inequalities for weighted graphs

Transcription

1 O Cheeger-type equaltes for weghted graphs Shmuel Fredlad Uversty of Illos at Chcago Departmet of Mathematcs 851 S. Morga St., Chcago, Illos USA Rehard Nabbe Fakultät für Mathematk Uverstät Belefeld Postfach Belefeld Germay Abstract We gve several bouds o the secod smallest egevalue of the weghted Laplaca matrx of a fte graph ad o the secod largest egevalue of ts weghted adjacecy matrx. We establsh relatos betwee the gve Cheeger-type bouds here ad the kow bouds the lterature. We show that oe of our bouds s the best Cheeger-type boud avalable. Keywords: spectral graph theory, Cheeger equalty, sopermetrc umbers 1 Itroducto Let G = (V,E(G)) be a fte, udrected, smple, coected graph wthout loops o vertces < >:= {1,...,}. We wll call such G a graph G. The Laplaca matrx L = [l,j ] of G s defed by l,j = emal: fredla@uc.edu emal: abbe@mathematk.u-belefeld.de f = j, 1 f(,j) E(G), 0 otherwse, 1

2 2 where s the degree of vertex. Wth = dag( 1,..., ), the matrx A = L s called the adjacecy matrx of G. Let 0 = λ 1 < λ 2... λ, µ 1 µ 2... µ, be the egevalues of L ad A respectvely. The spectra of both of these matrces ca be used to obta several formato about the graph, e.g. estmates for the dameter ot the graph (see the surveys by Merrs [Me] ad Mohar [Mo3]). I partcular estmates or bouds for λ 2 ad µ 1 are of great terest. Recetly, dscrete aalogs of Cheeger s [Che] fudametal equalty for λ 2 of the Laplaca o the compact Remaa mafold are establshed ([A], [D], [DK], [L], [Mo1], [Chu]). I these works, several dscrete versos of Cheeger costats are used. We brg three versos of Cheeger costats. Let the Cheeger costat h(g) of a fte graph G be: h(g) = m =U V E(U,Ū) m( U, Ū ) = m E(U,Ū) U V U 0 U 2, (1.1) where E(U,Ū) are the edges coectg vertces U wth vertces Ū, the complemet of U. The costat h(g) s called the sopermetrc umber of G by Mohar [Mo1] ad the modfed Cheeger costat by Chug [Chu] (p.34). Let (G) = m =U V E(U,Ū) m( E(U), E(Ū) ) = m U V E(U) 2 1 E(V ) E(U,Ū), (1.2) E(U) where E(U) =. (1.3) U Note that 1 2 E(V ) s the umber of edges G. The costat (G) s called the trasto sopermetrc umber of G by Mohar [Mo2] ad the Cheeger costat of G by Chug [Chu]. Obvously oe has where Fredlad used [Fr] a slghtly dfferet costat (G) h(g) (G), (1.4) = m ad = max. (1.5) ǫ(g) = m U V E(U,Ū) E(U) U 2. (1.6)

3 3 It s straghtforward to see that ǫ(g) (G). (1.7) Furthermore ǫ(g) h(g) ǫ(g). (1.8) For -regular graphs,.e. graphs for whch the degree of all vertces s, the three costats satsfy h(g) = (G) = ǫ(g). All these costats have a lot of varous terpretatos ad applcatos. They ca serve as measures of the coectvty of graphs ad are clearly related to the so-called bsecto wdth problem. (see Mohar [Mo1] for more applcatos). It seems that the computato of h(g),(g),ǫ(g) are ot polyomal. These costats ca be bouded by λ 2 or µ 1. We meto some of these (Cheeger type) equaltes. Alo proved [A] (see also [D]) the followg equalty λ h(g)2. (1.9) The related equalty usg (G) stead of h(g) s proved [DK]. From results by Mohar [Mo1] (see also [BZ]) we obta λ 2 2 h(g) 2, (1.10) f G K 1,K 2,K 3. (K s the complete graph o vertces.) Note that (1.10) mproves (1.9). Moreover Mohar [Mo1] showed µ 1 2 h(g) 2. (1.11) Oe ca exted the above results to postvely weghted graphs (possbly wth loops). Let G be a graph. A weghted graph G w s a complete, udrected graph wth loops, wth udrected edges < > < >, where each edge (,j) = (j,) s assged a oegatve weght w,j as follows: Frst w,j = 0 f j,(,j) / G. Secod, w,j > 0 f (,j) E(G). Thrd, w, 0 for < >. Let Ĝ be a smple, coected graph wth loops, obtaed from G by addg a loop to each vertex < >. Thus, E(Ĝ) = E(G) =1 (,). The Ĝ = G w for correspodg weghts w,j {0,1}. Defe the weghted degree = (,j) E(Ĝ) w,j.

4 4 Let, be defed (1.5). The weghted Laplaca L w = [l,j ] s the gve by l,j = w,j for j l, = w,. Smlarly we defe A w := w L w, where w = dag( 1,..., ). It s worth to meto that weghted loops do ot fluece L w cotrast to A w. (Ths s the stadard coveto, e.g. [Chu], 1.4. A dfferet defto of the Laplaca of G w, whch takes accout weghted loops, s gve [Fr].) Wth E(U) = ad E(U,Ū) = w,j U U,j/ U oe ca defe the costats h(g w ), (G w ) ad ǫ(g w ) as (1.1), (1.2), ad (1.6) for the weghted graph G w. Fredlad establshed [Fr] that for ay dagoal matrx D = dag(d 1,...,d ) wth d > 0, λ 2 (D 1 L w ) m Recetly, Berma ad Zhag proved [BZ] that λ 2 (D 1 L w ) 2d ǫ(g w ) ( h(g w ) max d 2 ). (1.12) Aother costat for weghted graphs s troduced by Fedler [F]. g(g w ) = m =U V E(U,Ū) (1.13) E(U) E(Ū). Ths costat s called [F] the averaged mmal cut of G. Fedler proved that λ 2 (L w ) 2g(G w ). (1.14) It seems that g(g w ) s ot related to a dscrete verso of Cheeger s equalty. Ideed, assume that G s ay -regular graph o vertces. Choose the rght had sde of (1.13) U = {}. The 1 g(g) ( 1). Thus (1.14) fals to detect expaders for whch dscrete Cheeger costats were troduced orgally. (Expaders are a fte famly of -regular graphs whose secod egevalues of

5 5 the Laplaca s bouded below by a fxed a > 0, e.g. [A]). I the prevous paper [FN] the authors gave the followg bouds for λ 2 ad µ 1 for weghted graphs. Let τ m (L w ) = m λ 1(L w (U)), (1.15) U V, U =m where L w (U) s the prcpal submatrx of L w whose rows ad colums are U. Smlarly, It s show [FN] that ρ m (A w ) = max µ m(a w (U)). (1.16) U V, U =m λ 2 τ 2 (L w ) ad µ 1 ρ 2 (A w ). (1.17) Thus the secod smallest egevalue of the Laplaca ad the secod largest egevalue of the adjacecy matrx of a graph ca be bouded by cosderg egevalues of subgraphs of the half sze. The object of ths paper s twofold. Frst, we establsh ew Cheeger type bouds o λ 2 for weghted graphs, usg the costats (G w ) ad ǫ(g w ). These bouds geeralze ad mprove a boud by Chug ad a boud by Fredlad metoed above. We also gve a ew boud for µ 1 of a weghted graph. Ths boud mproves the boud gve by Mohar. Secod, we establsh a ew type of boud, whch s better tha all the above dscussed Cheeger-type bouds. Let σ(l w ) = m U V, U max(λ 1(L w (U)),λ 1 (L w (Ū))), (1.18) 2 The ν(a w ) = max U V, U m(µ U (A w (U)),µ Ū (Aw(Ū))), (1.19) 2 λ 2 σ(l w ) ad µ 1 ν(a w ). (1.20) 2 Improvemet ad comparso of kow equaltes We start ths secto wth some bouds o the secod largest egevalue of the Laplaca matrx of a weghted graph G w multpled by a postve dagoal matrx.

6 6 Theorem 2.1 Let G w be a weghted, udrected, coected graph ad V =< >. Let D = dag(d 1,...,d ) be a postve dagoal matrx. The λ 2 (D 1 L w ) m (1 1 (G w ) d 2 ). (2.1) Proof. Let v = [v ] be the egevector correspodg to λ 2 (D 1 L w ). Let U = { : v > 0} ad U = { : v < 0}. By terchagg U,v wth U, v f ecessary, we ca assume U j U j. Let Ū = { : v 0} U. The j. (2.2) U j Ū I the proof of Theorem 2.3 [FN] t s proved that λ 2 (D 1 L w ) max(λ 1 (D 1 L w (U)),λ 1 (D 1 L w (Ū))). (2.3) Here D 1 L w (U) stads for a prcpal submatrx of D 1 L w whose rows ad colums are U. Let λ 1 := λ 1 (D 1 L w (U)). Sce D 1 L w (U) s a M matrx (see [BP]), there exsts a postve egevector x correspodg to λ 1,.e. Reame the vertces of G such that U =< t > ad D 1 L w (U)x = λ 1 x. (2.4) x x +1 for < t 1. Set α = m 1 k 1 (,j) E(G), k<j w,j m( k k, >k k ). (2.5) Clearly, α (G w ). Let y R gve by y = { x f t 0 else. (2.6) Defe (,j) E(G) w,j(y y j ) 2 (,j) E(G),<j, t w,j(y y j ) 2 R = V y 2 = t=1 y 2. (2.7)

7 7 We ow show that R α2 2 R. (2.8) For techcal reasos we eed to modfy slghtly the weghts of loops. Let Use the Schwarz equalty to deduce ( ( = ( (,j) E(G) (,j) E(Ĝ) (,j) E(G) { w,j f j w,j = 1 2 w,j f = j. w,j y 2 y2 j )2 = ( w,j (y y j ) 2 ) ( w,j (y y j ) 2 ) ( For each edge (, j) E(G), < j, observe that The (,j) E(G) (,j) E(Ĝ) (,j) E(Ĝ) w,j y y j w,j y + y j ) 2 w,j (y + y j ) 2 ) (2.9) (,j) E(Ĝ) w,j (y + y j ) 2 ). j 1 y 2 y2 j = y2 y2 j = (yk 2 y2 k+1 ). k= w,j y 2 y2 j = = = j 1 (,j) E(G),<j k= 1 k=1 t k=1 (y 2 k y2 k+1 ) w,j (y 2 k y2 k+1 ) k,k<j (y 2 k y2 k+1 ) k,k<j where the last equalty follows from (2.6). As k t, (2.2) yelds that k l l=1 l=k+1 l. w,j (2.10) w,j, The defto of α yelds: k,k<j w,j k α, k t.

8 8 Usg the above equalty the last equalty of (2.10) we get t k t w,j y 2 yj 2 (yk 2 yk+1)α 2 = α j yj. 2 (2.11) (,j) E(G) k=1 =1 j=1 Observe ext (,j) E(Ĝ) t t w,j (y 2 + yj) 2 = w, y 2 + y 2 t w,j = y 2. (2.12) =1 =1 j,(,j) E(Ĝ) =1 From the defto of the modfed weghts w, (2.12) ad (2.7) we obta: (,j) E(Ĝ) w,j (y + y j ) 2 = 2 = ( (,j) E(Ĝ) t =1 y 2 )(2 R). w,j (y 2 + y 2 j) (,j) E(Ĝ) w,j (y y j ) 2 (2.13) Now use the defto of R (see (2.7)), the equalty (2.9), the equaltes (2.13) ad (2.11) to deduce (2.8) as follows: Hece Thus R = ( (,j) E(G) w,j(y y j ) 2 )( (,j) E(Ĝ) w,j(y + y j ) 2 ) ( t =1 y 2 )( (,j) E(Ĝ) w,j(y + y j ) 2 ) ( (,j) E(G) w,j y 2 y2 j )2 ( t =1 y 2 )2 (2 R) α2 2 R. Multplyg (2.4) wth x T D(U) we get R 2 2R + α 2 0. R 1 1 α (G w ) 2. x T L w (U)x = λ 1 x T D(U)x.

9 9 Hece max d λ 2 (D 1 L w )x T x 1 λ 2(D 1 L w )x T D(U)x 1 λ 1 x T D(U)x = 1 x T L w (U)x x T x T x xt L w (U)x x x T (U)x xt x = Rx T x (1 1 (G w ) 2 )x T x. Here, the frst equalty s straghtforward, the secod equalty uses (2.3), the thrd oe uses the precedg detty, the fourth uses the mmalty of ad the last two steps follow from prevous observatos. Thus λ 2 (D 1 L w ) m (1 1 (G w ) d 2 ). As a mmedate corollary we obta the result for the weghted Laplaca matrx of a graph. Corollary 2.2 Let G w be a weghted, udrected, coected graph. The λ 2 (L w ) (1 1 (G w ) 2 ). Now, as [Fr] for U V let E(W, ǫ(u) := m W) =W U E(W) (2.14) ad smlarly E(W, (U) := m W) =W U m( E(W), E( W ) ). (2.15) Lemma 2.3 Let G w be a weghted, udrected, coected graph ad V =< >. Let D = dag(d 1,...,d ) be a postve dagoal matrx. The for ay oempty subset U of V max(λ 1 (D 1 L w (U)),λ 1 (D 1 L w (Ū))) m (1 1 (U) U d 2 ). (2.16) λ 1 (D 1 L w (U)) m U (1 1 ǫ(u) d 2 ). (2.17)

10 10 Proof. It s clear that ether U or Ū satsfes (2.2) ad we obta (2.16) as the proof of Theorem 2.1. To prove (2.17) we ca ot assume geeral (2.2). Thus we defe (,j) E(G), k<j α = m w,j 1 k U k. k Use the argumets of the proof of Theorem 2.1 for α to obta (2.17). Theorem 2.4 Let G w be a weghted, udrected, coected graph ad V =< >. Let D = dag(d 1,...,d ) be a postve dagoal matrx. The λ 2 (D 1 L w ) σ(d 1 L w ). (2.18) Proof. Recall that D 1 L w has a real spectrum. Let D 1 L w v = λ 2 (D 1 L w )v, where v s a ozero vector. Sce G s a coected graph t follows that v has at least oe postve ad oe egatve coordate. Assume that U < > s the set where all the coordates of v are postve. The Ū s the set where all the coordates of v are opostve. The (2.3) holds. The m max characterzato of σ(d 1 L w ) yelds (2.18). The ext Theorem gves a comparso of our ew bouds wth the well-kow bouds metoed the prevous secto. Theorem 2.5 Let G w be a weghted, udrected, coected graph ad V =< >. Let D = dag(d 1,...,d ) be a postve dagoal matrx. The Furthermore λ 2 (D 1 L w ) σ(d 1 L w ) τ 2 (D 1 L w ) ( ) 1 1 ǫ(g w ) 2 m m d 2d ǫ(g w ) 2. λ 2 (D 1 L w ) σ(d 1 L w ) ( ) m 1 1 (G w ) 2 m m d d ( ) 1 1 ǫ(g w ) 2 2d ǫ(g w ) 2.

11 11 Proof. Cosder the frst set of equaltes. The frst equalty s gve the prevous Theorem. To prove the secod equalty assume that σ(d 1 L w ) = max(λ 1 (D 1 L w (W)),λ 1 (D 1 L w ( W))), W. (2.19) 2 Let U 1 be ay subset of V such that U 1 = 2 ad W U 1. The τ 2 (D 1 L w ) λ 1 (D 1 L w (U 1 )) λ 1 (D 1 L w (W)) σ(d 1 L w ). The thrd equalty follows mmedately from Lemma 2.3. Moreover, sce 1 1 ǫ(g w ) ǫ(g w) 2 = 1 2 ǫ(g w) 2, we obta the last equalty. Cosder ow the secod set of equaltes. The frst equalty s aga Theorem 2.4. To show the secod equalty let W be as (2.19). The choose U be ether W or W such that (2.2) holds. The argumets of the proof of Theorem 2.1 yeld λ 1 (D 1 L w (U)) m (1 1 (G w ) d 2 )). The thrd equalty follows from (1.7) ad the last equalty s proved above. Thus Theorem 2.5 shows that σ(d 1 L w ) s better tha the boud τ 2 ad better tha the Cheeger-type bouds usg the quatty ǫ(g w ) ad (G). As (2.2) does ot hold geeral, τ 2 s ot comparable wth the boud (2.1). Note that wth D = I we obta the equaltes for the weghted Laplaca matrx. Next we compare our ew bouds wth the boud (1.12) by Berma ad Zhag, whch uses h(g). For -regular graphs our Cheeger-type boud (2.1) ad the boud (1.12) are the same. However, we wll see the ext secto that for arbtrary graphs t s ot possble to compare these bouds. As [Chu] cosder the matrx 1 2L 1 2. We easly obta the followg comparso of our ew boud (2.1) ad (1.12). Corollary 2.6 Let G be a graph. The λ 2 ( 1 2L 1 2) 1 1 (G) 2 ) 1 1 h(g)2 2. Proof. We use D = Theorem 2.1 ad (1.12) ad use the fact that the egevalues of 1 2L 1 2 ad 1 L are the same. The secod equalty follows from (1.4).

12 12 Note that the frst equalty Corollarry 2.6 was proved by Chug [Chu]. Morover, followg the proof of Theorem 2.2 [BZ] we obta Theorem 2.7 Let G w be a weghted, udrected, coected graph ad V =< >. Let D = dag(d 1,...,d ) be a postve dagoal matrx. The λ 2 (D 1 L w ) σ(d 1 L w ) τ 2 (D 1 L w ) 1 max d ( 2 ) h(g w ) 2. (2.20) We ow estmate from above the secod largest egevalue of the weghted adjacecy matrx of G w. The freedom of usg a postve dagoal matrx (2.1) allows us to traslate the boud for λ 2 to a boud for µ 1 for oregular graphs. Theorem 2.8 Let G w be a weghted, udrected, coected graph. Let V =< >. The µ 1 (A w ) ν(a w ) ρ 2 (A w ) 1 ǫ(g w ) 2 (2.21) 2 h(g w ) 2. Furthermore µ 1 (A w ) ν(a w ) 1 (G w ) 2 1 ǫ(g w ) 2 (2.22) 2 h(g w ) 2. Proof. The the proof of the frst equalty of (2.21) s essetally the same as the proof of Theorem 2.4. The secod equalty of (2.21) follows by smlar argumets as the proof of Theorem 2.5. To prove the thrd equalty of (2.21) we use Lemma 2.3 wth D =. Thus, for every subset U wth U 2 Let I deote the detty matrx. Hece λ 1 ( 1 L w (U)) 1 1 ǫ(u) 2. µ U (( 1 A w I)(U)) ǫ(u) 2.

13 13 Therefore The µ U (( 1 2 Aw 1 2 )(U)) = µ U (( 1 A w )(U)) ( 1 ǫ(u) 2 )I ( 1 2 Aw 1 2 )(U)) 1 ǫ(u) 2. s a postve defte matrx. Hece s also postve defte. Therefore ( 1 ǫ(u) 2 ) (U) A w (U) µ U (A w (U)) max ( 1 ǫ(u) 2 ) U ( 1 ǫ(u) 2 ). As ǫ(g w ) ǫ(u) we obta ρ 2 (A) ( 1 ǫ(g w ) 2 ). The last equalty of (2.21) from the rhgt-had sde of (1.8). The proof of (2.22) s smlar to the proof of (2.21). Thus, smlarly to the weghted Laplaca matrx we see that for the weghted adjacecy matrx, ν(a w ) s the best of Cheeger-type bouds. Theorem 2.8 mproves the bouds by Mohar (1.11) ad geeralzes the boud [FN] for regular graphs. Note we have also establshed two ew Cheeger-type bouds, amely 1 (G w ) 2 ad 1 ǫ(g w ) 2, whch both mprove the well-kow boud by Mohar. 3 Examples I ths secto we wll compare the bouds stated the prevous secto. Example 1: Let G 1 be the graph gve as Fgure 1 ad let G 2 obtaed from G 1 by deletg vertex 5 ad edge (2,5). The related Laplaca matrces are L 1 = L 2 =

14 The costats for ths graphs are as follows We obta the followg bouds Fgure 1 G 1 G 2 h(g) (G) ǫ(g) g(g) /3 L 1 L 2 λ σ(l) τ 2 (L) g(L) (1 ) 1 (G) (1 ) 1 ǫ(g) ǫ(g) ( 2 ) h(g) h(g) We ow show that for certa graphs G, t s possble to mprove the boud λ 2 (1 1 (G) 2 ), (3.1)

15 15 by troducg weghted loops ad usg Corollary 2.2. Cosder the graph G 2. We showed that (3.1) s λ 2 (G 2 ) 0.4. Let G = (G 2 ) w be a obtaed from G 2 by addg loops o vertces 3,4. (The degree of each vertex G s 3.) Clearly λ 2 (G ) = λ 2 (G 2 ) = 2. A straghtforward calculato shows that (G ) = 0.5. Hece λ 2 (G 2 ) = λ 2 (G ) (G )(1 1 (G ) 2 ) > > 0.4. Example 2: Cosder a cycle wth = 2m vertces. We get the costats h(g) = 2 m (G) = ǫ(g) = 1 m g(g) = 2 m 2. The egevalues of the related Laplaca matrx are λ j = 2 2cos 2πj j = 0,..., 1. For large we get λ 2 π2 m 2 ad Sce ths graph s 2-regular we obta σ(l) = τ 2 (L) π2 m 2. ( ) ( ) ( 1 1 (G) 2 = 1 1 ǫ(g) 2 = 2 ) h(g) 2 = m 2. Moreover, 2g(L) = 4 m 2 2 ǫ(g)2 = 1 2 h(g)2 = 1 m 2. Refereces [A] N. Alo, Egevalues ad expaders, Combatorca 6:83 96 (1986). [BP] [BZ] A. Berma, R.J. Plemmos, Noegatve Matrces the Mathematcal Sceces, Academc Press, New York, A. Berma, X.-D.Zhag, Lower bouds for the egevalues of Laplaca matrces, preprt (1999).

16 16 [D] [DK] [Che] [Chu] [F] [Fr] J. Dodzuk, Dfferece equatos, Isopermetrc equalty ad trascece of certa radom walks, Tras. AMS 284: (1984). J. Dodzuk, W.S. Kedal, Combatoral Laplaca ad sopermetrc equalty, From local tmes to global geometry, cotrol ad physcs Ed. K.D. Elworthy, Ptma Research Notes Mathematc Seres 150,1986, pp J. Cheeger, A lower boud for the smallest egevalue of the Laplaca, Problems Aalyss (R.C. Gug, Ed.), Prceto U.P., 1970, pp F.R.K. Chug, Spectral Graph Theory, CBMS Regoal Coferece Seres Mathematcs, 92, AMS, M. Fedler, A estmate for the ostochastc egevalues of doubly stochastc matrces, Lear Algebra Appl., 214: (1995). S. Fredlad, Lower bouds for the frst egevalue of certa M-matrces assocated wth graphs, Lear Algebra Appl., 172:71-84 (1992). [FN] S. Fredlad, R. Nabbe, O the secod real egevalue of oegatve ad Z- matrces, Lear Algebra Appl., 255: (1997). [L] A. Lubotzky, Dscrete Groups, Expadg Graphs ad Ivarat Measures, Progress Mathematcs Vol. 125, Brkäuser, Berl, [Me] R. Merrs, Laplaca matrces of graphs: a survey Lear Algebra Appl. 199: (1994). [Mo1] B. Mohar, Isopermetrc Numbers of Graphs, J. Combatoral Theory, Seres B, 47: (1989). [Mo2] B. Mohar, Isopermetrc Iequaltes, Growth, ad the Spectrum of Graphs, Lear Algebra Appl. 103: (1998). [Mo3] B. Mohar, Laplaca egevalues of graphs a survey, Dscrete Math. 109: (1992).