AN INTRODUCTION TO SPECTRAL GRAPH THEORY

Transcription

1 AN INTRODUCTION TO SPECTRAL GRAPH THEORY JIAQI JIANG Abstract. Spectral graph theory is the study of properties of the Laplacia matrix or adjacecy matrix associated with a graph. I this paper, we focus o the coectio betwee the eigevalues of the Laplacia matrix ad graph coectivity. Also, we use the adjacecy matrix of a graph to cout the umber of simple paths of legth up to 3. Cotets. Itroductio 2. Backgroud of Spectral Graph Theory 3. Basic Properties of The Laplacia Matrix 3 4. Eigevalues ad Eigevectors of the Laplacias of Some Fudametal Graphs 6 5. The Boudig of λ 9 6. Further Discussio o Simple Path Coutig Problem 4 7. Ackowledgmet 6 Refereces 6. Itroductio The eigevalues of the Laplacia matrix of a graph are closely related to the coectivity of the graph. Therefore, bouds for the smallest ozero eigevalue of the graph Laplacia give us ituitio o how well coected a graph is. Thus, we focus o fidig bouds for this eigevalue. Additioally, the adjacecy matrix of a graph provides iformatio about paths cotaied i the graph. Therefore, it is also oe of our goals to fid a way to cout the umber of paths of differet legth cotaied i the graph usig the adjacecy matrix. 2. Backgroud of Spectral Graph Theory We itroduce the basic cocepts of graph theory ad defie the adjacecy matrix ad the Laplacia matrix of a graph. Defiitio 2.. A graph is a ordered pair G=(V,E) of sets, where E {{x, y} x, y V, x y}. The elemets of V are called vertices (or odes) of the graph G ad the elemets of E are called edges. So i a graph G with vertex set {x, x 2,..., x }, {x i, x j } E if ad oly if there is a lie i G which coects the two poits x i ad x j. The graph we defied above is called a udirected graph. A directed graph is similar

2 2 JIAQI JIANG to a udirected graph except the edge set E V V. Uless specified, the graphs we deal with i this paper are all udirected ad fiite. Also, the vector spaces we are workig o are all real vector spaces. Now we will defie the cocept of adjacecy. Defiitio 2.2. I a graph G = (V, E), two poits x i, x j eighbors if {x i, x j } E. V are adjacet or If all the vertices of G are pairwise adjacet, the we say G is complete. A complete graph with vertices is deoted as K. For example, the graph of a triagle is K 3, the complete graph with three vertices. Defiitio 2.3. The degree d(v) of a vertex v is the umber of vertices i G that are adjacet to v. There are two matrices we ca get from a graph G. Oe is called adjacecy matrix, which we deote as A G. The other is called Laplacia matrix, which we deote as L G. Without loss of geerality, assume a graph G has the vertex set V = {, 2,..., }. Now we defie the adjacecy matrix ad the Laplacia matrix of G as follows: Defiitio 2.4. I the adjacecy matrix A G of the graph G, the etries a i,j are give by { if {i, j} E, a i,j = 0 otherwise. Defiitio 2.5. I the Laplacia matrix L G of the graph G, the etries l i,j are give by if {i, j} E, l i,j = d(i) if i = j, ad 0 otherwise. Oe use of the adjacecy matrix of a graph is to calculate the umber of walks of differet legth coectig two vertices i the graph. Before we defie the cocept of walk, we first itroduce the cocept of icidet. Defiitio 2.6. A vertex v V is icidet with a edge {v i, v j } E if v = v i or v = v j. Now we defie the cocept of the walk. Defiitio 2.7. A walk o a graph is a alteratig series of vertices ad edges, begiig ad edig with a vertex, i which each edge is icidet with the vertex immediately precedig it ad the vertex immediately followig it. A walk betwee two vertices u ad v is called a u v walk. The legth of a walk is the umber of edges it has. I a walk, we cout repeated edges as may times as they appear. Theorem 2.8. For a graph G with vertex set V = {, 2,..., m}, the etry a ij of the matrix A G obtaied by takig the th power of the adjacecy matrix A G equals the umber of i j walks of legth.

3 AN INTRODUCTION TO SPECTRAL GRAPH THEORY 3 Proof. We will prove the theorem by iductio. Whe =, the etry a ij is if {i, j} E. By defiitio, i {i, j} j is the a i j walk of legth ad this is the oly oe. So the statemet is true for =. Now, we assume the statemet is true for ad the prove the statemet is also true for +. Sice A + ij = A ij A ij, therefore, a + ij = m k= a ik a kj. Because a ki = 0 wheever {k, i} / E ad a ki = if {k, i} E, it follows that a ik a ki represets the umber of those i j walks that are i k walks of legth joied by the edge {k, j}. I particular, all walks from i to j of legth + are of this form for some vertex k. Thus a + ij = m k= a ik a kj ideed represets the total umber of i j walks of legth +. This proves the statemet for +. The by the priciple of iductio, we prove the statemet for all atural umbers. I a later sectio, we will discuss how to compute the umber of paths (a walk i which vertices are all distict from each other) of legth up to 3 betwee i ad j i a graph give the adjacecy matrix of the graph. 3. Basic Properties of The Laplacia Matrix Oe of the most iterestig properties of a graph is its coectedess. The Laplacia matrix provides us with a way to ivestigate this property. I this sectio, we study the properties of the Laplacia matrix of a graph. First, we give a ew way to defie the Laplacia matrix for a graph, which turs out to be much more useful tha the previous oe. Defiitio 3.. Suppose G = (V, E) is a graph with V = {, 2,..., }. For a edge {u, v} E, we defie a matrix L G{u,v} by if i = j ad i {u, v}, l G{u,v} (i, j) = if i = u ad j = v, or vice versa, 0 otherwise. It is easy to check that L G{u,v} has the ice property that x T L G{u,v} x = (x u x v ) 2 for all x R. Now give the edge set E of a graph G, we are ready to give our ew defiitio of the Laplacia matrix L G of the graph. Defiitio 3.2. For a graph G = (V, E), L G = L G{u,v}. {u,v} E It is readily see that this ew defiitio of the Laplacia matrix is ideed equivalet to the defiitio give i the previous sectio. However, we fid that may elemetary properties of the Laplacia matrix follow easily from the ew defiitio. We see that the eigevalues of the Laplacia matrix are all real by realizig that the Laplacia matrix of a graph is symmetric ad cosists of real etries. Thus, L G =L G where L G is the cojugate traspose of L G. Therefore, L G is self adjoit. By the followig theorem, all the eigevalues of L G are thus real. Theorem 3.3. The eigevalues of a self adjoit matrix are all real. Proof. Suppose λ is a eigevalue of the self adjoit matrix L ad v is a ozero eigevector of λ. The

4 4 JIAQI JIANG λ v 2 = λ v, v = λv, v = Lv, v = v, Lv = v, λv = λ v, v = λ v 2 Sice v 0, we have v 2 0. Therefore, λ = λ. This proves that λ is real. I fact, the eigevalues of the Laplacia matrix are ot oly real but also oegative. Recall the defiitio of positive-semidefiite. Defiitio 3.4. A matrix M is called positive-semidefiite if x T M x 0 for all x R. Sice x T L G{u,v} x = (x u x v ) 2 for all x R, we have x T L G x = x T L G{u,v} x = (x u x v ) 2 0. {u,v} E {u,v} E Therefore, the Laplacia matrix of a graph is positive-semidefiite. It follows that all the eigevalues of L G is o-egative. Theorem 3.5. For a graph G, every eigevalue of L G is o-egative. Proof. Suppose λ is a eigevalue ad x R is a ozero eigevector of λ. The x T L G x = x T (λ x) = λ( x T x) Sice x T L G x 0 ad x T x > 0, we have λ 0. As we have oted, the Laplacia matrix L G is self-adjoit ad cosists of real etries. Thus the Real Spectral Theorem states that L G has a orthoormal basis cosistig of eigevectors of L G. Therefore, for a graph G of vertices, we ca fid eigevalues (ot ecessarily distict) for L G. We deote them as λ, λ 2,..., λ. Sice they are all real ad o-egative, we assume that 0 λ λ 2 λ. Now we prove some fudametal facts about Laplacias. Recall that i the previous sectio we metioed that the eigevalues of the Laplacia tell us how coected a graph is. Now, we defie coectedess. First, we give the defiitio of a path. Defiitio 3.6. A path is a o-empty graph P = (V, E) of the form V = {x 0, x,..., x } E = {{x 0, x }, {x, x 2 },..., {x, x }}, where all the vertices x i are distict.

5 AN INTRODUCTION TO SPECTRAL GRAPH THEORY 5 Recall the defiitio of a walk, a path is i fact a walk with o repeatig vertices. Defiitio 3.7. A o-empty graph G is called coected if ay two of its vertices are cotaied i a path i G. Now we will see that the eigevalue 0 is closely related to this coectedess. Lemma 3.8. For ay graph G, λ = 0 for L G. If G = (V, E) is a coected graph where V = {, 2,..., }, the λ 2 > 0. Proof. Let x = (,,..., ) R. The the etry m i of the matrix M = L G x is m i = l ik. k= Recall the defiitio of L G give i the first sectio. It follows immediately that m i = 0 sice the row etries of L G should add up to zero. So L G x = 0. Therefore, 0 is a eigevalue of L G. Sice 0 λ λ 2 λ, it follows that λ = 0. Now we wat to show that λ 2 > 0 for a coected graph. Sice 0 is a eigevalue of L G, let z be a ozero eigevector of 0. The So z T L G z = z T L G z = z T 0 = 0. {u,v} E (z u z v ) 2 = 0. This implies that for ay {u, v} such that {u, v} E, z u = z v. Sice G is coected, this meas z i = z j for all i, j V. Therefore, z = α., where α is some real umber. So, U λ = Spa ((,,..., )), where U λ is the eigespace of λ. Therefore, the multiplicity of eigevalue 0 is. It follows that λ 2 0, so λ 2 > 0. I fact, the multiplicity of the eigevalue 0 of L G tells us the umber of coected compoets i the graph G. Defiitio 3.9. A coected compoet of a graph G = (V, E) is a subgraph G = (V, E ), (V V, E = {{x, y} E x, y V }), i which ay two vertices i, j V are coected while for ay i V ad k V \ V, i, k are ot coected. Corollary 3.0. Let G = (V, E) be a graph. The the multiplicity of 0 as a eigevalue of L G equals the umber of coected compoets of G. Proof. Suppose G = (V, E ), G 2 = (V 2, E 2 ),..., G k = (V k, E k ) are the coected compoets of G. Let w i be defied by { if j V i, (w i ) j = 0 otherwise.

6 6 JIAQI JIANG The, it follows from the previous lemma that if x R is a o-zero eigevector of 0, the x i = x j for ay i, j V such that i, j are i the same coected compoet. So U λ = Spa({ w, w 2,..., w k }). It is clear that w, w 2,..., w k are liearly idepedet. Therefore, the multiplicity of 0 as a eigevalue of L G is the umber of coected compoets i G. 4. Eigevalues ad Eigevectors of the Laplacias of Some Fudametal Graphs Now we begi to examie the eigevalues ad the eigevectors of the Laplacia of some fudametal graphs. Defiitio 4.. A complete graph o vertices, K, is a graph G = (V, E) where V = {, 2,..., } ad E = {{i, j} i j, i, j V }. Propositio 4.2. The Laplacia of K has eigevalue 0 with multiplicity ad eigevalue with multiplicity. Proof. The first part of the propositio simply follows from Corollary 3.9. To prove the secod part of the propositio, cosider the Laplacia of K. It is a matrix defied by { if i j, a ij = if i = j. Therefore, L K I = M where M is the matrix with etries all equal -. Clearly, M is ot ivertible ad has rak. Thus is a eigevalue of L K. The by Rak-ullity Theorem, ull(m) =. It follows that the eigevalue has multiplicity. Defiitio 4.3. The path graph o vertices, P, is a graph G = (V, E) where V = {, 2,..., } ad E = {{i, i + } i < }. Defiitio 4.4. The cycle graph o vertices, C, is a graph G = (V, E) where V = {, 2,..., } ad E = {{i, i + } i < } {{, }}. Propositio 4.5. The Laplacia of C has eigevalues 2 2 cos( 2πk ) ad associated eigevectors of the form ( ) 2πki x i (k) = si ad, ( ) 2πki y i (k) = cos, where x i (k) deotes the ith compoet of the eigevector for the kth eigevalue ad k 2.

7 AN INTRODUCTION TO SPECTRAL GRAPH THEORY 7 Proof. Note the Laplacia matrix of C has the form Therefore, if λ is a eigevalue of L C ad x is a associated eigevector, the x should satisfy (4.6) (4.7) (4.8) 2x x 2 x = λx x x + 2x = λx, ad, x m + 2x m x m+ = λx m, for all < m <. Also, from the form of L C, we ca see that if P is a cyclic permutatio, the P x is also a eigevector for λ. This meas that x, P x,..., P x are all eigevectors of λ. However, the maximum dimesio of ay eigespace is sice 0 is a eigevalue of multiplicity. This implies that P j x Spa( { x, P x,..., P j x } ) for some j. This shall give us some idea about the possible form of x. We might try the particular form correspodig to the case j = such that (4.9) x m = A m, where A = x is some costat. Pluggig (4.9) ito (4.6), (4.7) ad (4.8), we get 2 A A = λ 2 A A = λ ad, 2 A A = λ for all < i <. Combiig these equatios, we actually get that A =. This meas A k = e 2πki for k are solutios. The associated eigevalues λ k = 2 2 cos( 2πk ) are ideed all real. Recall we are workig o the real vector space, but so far the x we get by the form x m = A m are complex as A k is complex. However, if we write x m (k) = A m usig Euler s Formula, we ca see that z m (k) = cos( 2πkm ) + i si(2πkm ). Sice L C z(k) = λ k z(k) ad both L C ad λ k are real, this meas that both the real part ad the imagiary part of z(k) are ivariat uder L C cosists of real etries. Therefore, x m (k) = cos( 2πkm both eigevectors for λ k = 2 2 cos( 2πk ). The, for k > 2, ( j 2 x(k) Spa({x(), x(2),..., x(j)}) y(k) Spa({y(), y(2),..., y(j)}) because L C ) are ) ad y m(k) = si( 2πkm ), ad, ( j ). 2

8 8 JIAQI JIANG So the eigevalues are 2 2 cos( 2πk ) with k 2. The list above is exhaustive sice we have formed idepedet eigevectors correspodigly to these eigevalues: Whe is odd, we have oe eigevector x(0) for λ 0 = 0 (y(0) = 0), ad two eigevectors x(k) ad y(k) for all 0 < k < 2. Whe is eve, we have oe eigevector x(0) for λ 0 = 0 (y(0) = 0) ad two eigevectors x(k) ad y(k) for all 0 < k < 2 ad oe eigevector x( 2 ) for λ = 4 2 (y( 2 ) = 0). Propositio 4.0. The Laplacia of P has the same eigevalues as C 2, ad the associated eigevectors ( πki x i (k) = cos πk ), 2 for 0 k <. Proof. I order to prove the propositio, we will treat P as a quotiet of C 2 by idetifyig vertex i of P with both vertices i ad 2 + i of C 2. The we fid a eigevector v of C 2 such that v i = v 2+ i for all vertices i of C 2. The, v v 2 x =. v is a eigevector of P. First, I am goig to show that x is a eigevector of P. Notice that L P has the form So if u is a eigevalue of P, the it must satisfy: u u 2 = λu u + u = λu, ad, u i + 2u i u i+ = λu i, for all < i <. Followig immediately from the previous proof, our x satisfies the last coditio. We oly eed to check whether it satisfies the first two coditios. As we otice, λu = u 2 + 2u u 2 = u + 2u u 2 = u u 2 λu = u + 2u u + = u + 2u u = u + u. So our x satisfies all the coditios. Now, we wat to make sure that there exists a eigevector v of C 2 satisfyig v i = v 2+ i, so we ca get our x from it. Fortuately, we oly eed to let v i (k) = cos ( πki πk 2 ),

9 AN INTRODUCTION TO SPECTRAL GRAPH THEORY 9 the ( πk(2 + i) v 2+ i (k) = cos πk ) 2 ( ) πk( i ) = cos 2 ( ) 4πk πk(2i + ) = cos 2 2 ( ) 2πki + πk = cos 2πk ( πki = cos = v i (k), which satisfies our defiitio for v. Sice ( πki v i (k) = cos πk ) 2 ( ) ( πk 2πki = cos cos 2 2 πk 2 ) + si 2 ) ( ) πk si 2 ( ) 2πki, 2 we have v Spa ({x(k), y(k)}), where x(k) ad y(k) are the eigevectors for C 2 followig from propositio 4.5. The eigevalues associated are thus λ k = 2 2 cos( 2πk 2 ) where k. 5. The Boudig of λ As oted above, the smallest o-zero eigevalue of the Laplacia of a graph tells us how coected the graph is. Thus, it is importat to fid bouds for this value. First, we try to fid a lower boud for the largest eigevalue of the Laplacia of a graph. I order to do this, we itroduce the Courat-Fischer Theorem. Theorem 5. (Courat-Fischer). Let A be a symmetric matrix ad let k. Let λ λ 2 λ be the eigevalues of A ad v, v 2,..., v be the correspodig eigevectors. The, x T Ax λ = mi x 0 x T x x T Ax λ 2 = mi x 0 ad x v x T x x T Ax λ = max x 0 x T x. The proof of this theorem ca be foud i [5]. With Courat-Fischer Theorem, we ca get some easy lower bouds for the largest eigevalue λ of the Laplacia of a graph. Lemma 5.2. Let G = (V, E) be a graph with V = {, 2,..., } ad u V. If u has degree d, the λ (G) d.

10 0 JIAQI JIANG Proof. By Courat-Fischer Theorem, x T L G x λ (G) = max x 0 x T x. Let x = e u, where e, e 2,..., e is the stadard basis. Recall that x T L G x = (x u x v ) 2. Therefore, we have e u T L G e u e u T e u = {u,v} E = d = d {u,v} E (x u x v ) 2 x 2 u So λ (G) xt L G x x T x = d. I fact, we ca slightly improve the boud. Theorem 5.3. Let G = (V, E) be a graph with V = {, 2,..., } ad u V. If u has degree d, the λ (G) d +. Proof. It follows the same idea as i the proof for the previous lemma. However, this time we will cosider the vector x give by d if i = u, x i = if {i, u} E, 0 otherwise. The we have x T L G x x T x = = {u,v} E (x u x v ) 2 x 2 u d(d ( ))2 d( ) 2 + d 2 d(d + )2 = d(d + ) = d + So λ (G) xt L G x x T x = d +. Now we tur our focus oto the boudig of λ 2. By Courat-Fischer Theorem, we ca roughly get a upper boud for λ 2 of the path graph P. ( ) Propositio 5.4. Let P be the path graph, the λ 2 (P ) = O 2.

11 AN INTRODUCTION TO SPECTRAL GRAPH THEORY Proof. Cosider the vector u such that u i = ( + ) 2i. The u = i ( + ) 2i = 0, where = (,,..., ). As we have show before, spas the eigespace for the eigevalue λ = 0. The by Courat-Fischer Theorem, we get Recall that So x T L P x λ 2 (P ) = mi x x T x ut L P u u T u x T L G x = u T L P u u T u = < {u,v} E i< i< (x u x v ) 2. (u i u i+ ) 2 (u i ) 2 i 2 2 ( + 2i) 2 The deomiator (+ 2i) 2 is of order 3, therefore we ca give a upper-boud of λ 2 by ( ) λ 2 = O 2 Now, we try to get a lower boud for λ 2 of P. Defiitio 5.5. For two symmetric by matrices A ad B, we write A B if A B is positive semidefiite. So if A B, the x T A x x T B x for all x. Lemma 5.6. If G ad H are both graphs with vertices such that c L G L H, the c λ 2 (G) λ 2 (H). Proof. By Courat-Fischer Theorem, c λ 2 (G) = c x T L G x mi x 0 ad x v x T x = mi x 0 ad x v x T (c L G ) x x T x

12 2 JIAQI JIANG Assume whe x = u, xt (c L G ) x x T reaches miimum for x 0 ad x v. Sice x c L G L H, by defiitio, c λ 2 (G) = ut (c L G ) u u T u ut (L H ) u u T u x T L H x mi x 0 ad x v x T x λ 2 (H). By usig this lemma, we will be able to fid a lower boud for λ 2 (P ) by comparig it with λ 2 (K ). Recall our previous defiitio of L G{u,v} i defiitio 3.. For coveiece, I will deote L G{u,v} simply as L {u,v} from ow o. Lemma 5.7. For a graph G, let c, c 2,..., c > 0. The where c ( ) c i L {i,i+} L {,}, i= c = i=. c i Proof. First, we should see the statemet is equivalet to c i L {i,i+} ( i= )L {,}. c i i= We shall prove this equivalet statemet by iductio. Whe = 2, it is trivial. Assume the statemet is true for, the 2 c i L {i,i+} = c i L {i,i+} + c L {,} i= Now, we wat to show that ( 2 i= i= which is equivalet to showig that M = ( 2 )L {, } + c L {,}. c i i= )L {, } + c L {,} c i ( 2 i= ( )L {, } + c L {,} c i )L {,}, c i i= ( i= )L {,} c i

13 AN INTRODUCTION TO SPECTRAL GRAPH THEORY 3 is positive semi-defiite. As we kow, x T M x = ( 2 )(x x ) 2 + c (x x ) 2 ( )(x x ) c i c i i= i= 2. Let a = 2 c i i= ad a 2 = c ad let y = x x ad y 2 = x x, the x T M x = a y 2 + a 2 y 2 2 = a + a 2 + a (y + y 2 ) 2 2 ) (a y 2 + a 2 y2) 2 (y + y 2 ) 2]. a [( a + a 2 By Cauchy-Schwartz Iequality, we get a 2 2 a 2 y 2 a 2 a 2 2 y a 2 2 a 2 2 y 2 a 2 2 a 2 2 y 2 ( a + a ) 2 (a y 2 + a 2 y2) 2 (y + y 2 ) 2. Therefore, x T M x 0. So M 0. Thus, we have proved the statemet for. By priciple of iductio, we have proved the statemet for all. Propositio 5.8. For a path graph P, λ Proof. We will prove this propositio by comparig the path graph P to the complete graph K usig the lemma we have just proved. Suppose K = (V, E) where V = {, 2,..., }. For each edge {u, v} E with u < v, we apply Lemma 5.0, with c, c 2,..., c =. The v (v u) L K{i,i+} L K{u,v}. i=u The summig over all pairs of u, v with u < v, we get u<v v (v u) L K{i,i+} i=u u<v L K{u,v} = L K.

14 4 JIAQI JIANG Notice that u<v Therefore, ( ) ( 3 3 L P = 6 6 (v u) = i( i) ) i= = i = i= ( ) 2 = i 2 i= L K{i,i+} i= So ( 3 6 ( )(2 ) 6 u<v u<v L K ) L P L K. By Lemma 5.9, we have ( ) 3 λ 2 (P ) λ 2 (K ). 6 Propositio 4.2 implies that λ 2 (K ) =, therefore ( ) 3 λ 2 (P ), 6 ad so, λ 2 (P ) 6 2. (v u) i= v (v u) i=u L K{i,i+} L K{i,i+} We ca readily see that this lower boud has the same order as our previous rough upper boud of λ 2 (P ). 6. Further Discussio o Simple Path Coutig Problem I the previous sectio, we have show that for a give graph G = (V, E) where V = {, 2,..., }, the etry a k i,j of the expoetial of the adjacecy matrix Ak G tells us the umber of i j walks of legth k. However, from our defiitio, we ca see these i j walks cosist of repeatig edges. It would be iterestig to cout the umber of i j walks with o repeated vertices, which we will call simple paths. Defiitio 6.. Let G = (V, E) be a graph ad suppose i, j V. A simple path betwee i j i G is a subgraph which is a path coectig i ad j. The legth of the simple path is just the legth of the subgraph path.

15 AN INTRODUCTION TO SPECTRAL GRAPH THEORY 5 We ca easily see that i a graph G all the i j walks of legth 2 where i j are simple paths betwee i j of legth 2. Thus the etries a 2 i,j of A2 G where i j give us the correct umber. Sice ay closed walk (i i walk for ay i V ) is ot couted as a simple path, the umber of simple path betwee i ad i is always 0. Therefore, the matrix coutig the umber of simple paths of legth 2 is S (2) G = A2 G D G where D G is the degree matrix defied by { d(i) if i = j, d i,j = 0 otherwise, where d(i) deotes the degree of vertex i. Now we cosider simple paths of legth 3. A simple path of legth 3 betwee i ad j cosists of a simple path of legth 2 betwee i ad a poit u where {u, j} E. Thus, we would expect that s (3) i,j = s (2) i,p a p,j p= where s (2) i,p is the umber of simple path of legth 2 betwee i, p ad a p,j is the etry of adjacecy matrix A G. However, we should otice that if j is adjacet to i, the the formula above will iclude the umber of i j walks which cotai a closed walk at j. That is they are walks of the form i j p j. Therefore, we have to subtract the umber of such walks. We ca easily see this umber is give by a i,j (d(j) ). So i fact s (3) i,j = s (2) p= i,p a p,j a i,j (d(j) ). where i j ad, Therefore, we could see that s (3) i,i = 0. S (3) G = S(2) G A G A G (D G I) Diag(S (2) G A G) where Diag deotes the matrix give by { 0 if i j, Diag(M) i,j = m i,j if i = j. We ca easily see this simply meas that S (3) G is obtaied by chagig the diagoals of S (2) G A G A G (D G I) to all 0s. So far, I have oly computed S (k) G for up to 3 as the case for k > 3 is too complicated to compute. The problem may require differet methods or tools to tackle i the future. However, we ca reach a rough upper boud for the umber of simple paths of legth k. Cosider the graph G = (V, E) where V = {, 2,..., }. It is obvious that the umber m G of simple paths of legth k (k ) cotaied i G is

16 6 JIAQI JIANG less tha the umber m K of simple paths of legth k cotaied i the complete graph K. We ca easily compute m K, m K = ( ) ( k + ) 2 = P,k 2, where P,k deotes k permutatios of. So, m G m K = P,k 2. There still remais the possibility to get eve fier bouds for certai families of graphs. 7. Ackowledgmet It is a pleasure to thak my metor, Robi Walters, for itroducig me to the topic of spectral graph theory ad for his help throughout the paper writig process. I would ot possibly have fiished this paper without his advice ad isights o the topic. Refereces [] Paul R. Halmos. Fiite-Dimesioal Vector Spaces. Spriger-Verlag New York, Ic.974. [2] Sheldo Axler. Liear Algebra Doe Right, Secod Editio. Spriger-Verlag New York, Ic.997. [3] Reihard Diestel. Graph Theory. Spriger-Verlag New York, Ic [4] Da Spielma. Spectral Graph Theory ad its Applicatios. Olie lecture otes. 9 July 202. [5] Adre Wibisoo (Scribe), Joatha Keler (Lecturer). Olie lecture otes for course Algorithmist s Toolkit July 202.