The Adjacency Matrix and The nth Eigenvalue

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1 Spectral Graph Theory Lecture 3 The Adjacecy Matrix ad The th Eigevalue Daiel A. Spielma September 5, About these otes These otes are ot ecessarily a accurate represetatio of what happeed i class. The otes writte before class say what I thik I should say. The otes writte after class way what I wish I said. Be skeptical of all statemets i these otes that ca be made mathematically rigorous. 3.2 Overview I this lecture, I will discuss the adjacecy matrix of a graph, ad the meaig of its smallest eigevalue. This correspods to the largest eigevalue of the Laplacia, which we will examie as well. We will relate these to bouds o the chromatic umbers of graphs ad the sizes of idepedet sets of vertices i graphs. I particular, we will prove Ho ma s boud, ad some geeralizatios. Warig: I am goig to give a alterative approach to Ho ma s boud o the chromatic umber of a graph i which I use the Laplacia istead of the adjacecy matrix. I just worked this out last ight, so I still do t kow if it is a good idea or ot. But, I m goig to go with it. My proof of Ho ma s boud i the regular case will be much simpler tha the proof that I gave i The Adjacecy Matrix Let A be the adjacecy matrix of a (possibly weighted) graph G. As a operator, A acts o a vector x 2 IR V by (Ax )(u) = X w(u, v)x (v). (3.1) (u,v)2e We will deote the eigevalues of A by µ 1,...,µ. But, we order them i the opposite directio tha we did for the Laplacia: we assume µ 1 µ 2 µ. 3-1

2 Lecture 3: September 5, The reaso for this covetio is so that µ i correspods to the ith Laplacia eigevalue, is a d-regular graph, the D = I d, ad i. If G L = I d A, ad so i = d µ i. So, we see that the largest adjacecy eigevalue of a d-regular graph is d, ad its correspodig eigevector is the costat vector. We could also prove that the costat vector is a eigevector of eigevalue d by cosiderig the actio of A as a operator (3.1): if x (u) = 1 for all u, the (Ax )(v) =d for all v. 3.4 The Largest Eigevalue, µ 1 We ow examie µ 1 for graphs which are ot ecessarily regular. Let G be a graph, let d max be the maximum degree of a vertex i G, ad let d ave be the average degree of a vertex i G. Lemma d ave apple µ 1 apple d max. Proof. The lower boud follows by cosiderig the Rayleigh quotiet with the all-1s vector: P x T Ax 1 T P A1 µ 1 = max x x T x 1 T 1 = i,j A(i, j) i = d(i). To prove the upper boud, Let 1 be a eigevector of eigevalue µ 1. Let v be the vertex o which it takes its maximum value, so 1(v) 1(u) for all u, ad assume without loss of geerality that 1(v) 6= 0. We have µ 1 = (A 1)(v) 1(v) = P u v 1(u) = X 1(v) u v 1(u) 1(v) apple X 1 apple d(v) apple d max. (3.2) u v Lemma If G is coected ad µ 1 = d max, the G is d max -regular. Proof. If we have equality i (3.2), the it must be the case that d(v) =d max ad 1 (u) = 1 (v) for all (u, v) 2 E. Thus, we may apply the same argumet to every eighbor of v. As the graph is coected, we may keep applyig this argumet to eighbors of vertices to which it has already bee applied to show that 1(z) = 1 (v) ad d(z) =d max for all z 2 V.

3 Lecture 3: September 5, The Correspodig Eigevector The eigevector correspodig to the largest eigevalue of the adjacecy matrix of a graph is usually ot a costat vector. However, it is always a positive vector if the graph is coected. This follows from the Perro-Frobeius theory. I fact, the Perro-Frobeius theory says much more, ad it ca be applied to adjacecy matrices of strogly coected directed graphs. Note that these eed ot eve be diagoalizable! We will defer a discussio of the geeral theory util we discuss directed graphs, which will happe towards the ed of the semester. If you wat to see it ow, look at the third lecture from my otes from I the symmetric case, the theory is made much easier by both the spectral theory ad the characterizatio of eigevalues as extreme values of Rayleigh quotiets. Theorem [Perro-Frobeius, Symmetric Case] Let G be a coected weighted graph, let A be its adjacecy matrix, ad let µ 1 µ 2 µ be its eigevalues. The a. µ 1 µ,ad b. µ 1 >µ 2, c. The eigevalue µ 1 has a strictly positive eigevector. Before provig Theorem 3.5.1, we will prove a lemma that will be useful i the proof ad a few other places today. It says that o-egative eigevectors of o-egative adjacecy matrices of coected graphs must be strictly positive. Lemma Let G be a coected weighted graph (with o-egative edge weights), let A be its adjacecy matrix, ad assume that some o-egative vector is a eigevector of A. The, is strictly positive. Proof. Assume by way of cotradictio that is ot strictly positive. So, there is some vertex u for which (u) = 0. Thus, there must be some edge (u, v) for which (u) =0but (v) > 0. We would the (A )(u) = X w(u, z) (z) w(u, v) (v) > 0, (u,z)2e as all the terms w(u, z) ad (z) are o-egative. But, this must also equal µ (u) = 0, where µ is the eigevalue correspodig to. This is a cotradictio. So, we coclude that must be strictly positive. Proof of Theorem Let 1,..., be the eigevectors correspodig to µ 1,...,µ. We start with part c. Recall that µ 1 = max xx x T Ax x T x.

4 Lecture 3: September 5, Let 1 be a eigevector of µ 1, ad costruct the vector x such that x (u) = (u), for all u. We will show that x is a eigevector of eigevalue µ 1. We have x T x = T. Moreover, T 1 A 1 = X A(u, v) (u) (v) apple X A(u, v) (u) (v) = x T Ax. u,v u,v So, the Rayleigh quotiet of x is at least µ 1. As µ 1 is the maximum possible Rayleigh quotiet, the Rayleigh quotiet of x must be µ 1 ad x must be a eigevector of µ 1. So, we ow kow that A has a eigevector x that is o-egative. We ca the apply Lemma to show that x is strictly positive. To prove part b, let be the eigevector of µ ad let y be the vector for which y(u) = (u). I the spirit of the previous argumet, we ca agai show that µ = A apple X u,v A(u, v)y(u)y(v) apple µ 1 y T y = µ 1. To show that the multiplicity of µ 1 is 1 (that is, µ 2 <µ 1 ), cosider a eigevector 2. As 2 is orthogoal to 1, it must cotai both positive ad egative values. We ow costruct the vector y such that y(u) = 2(u) ad repeat the argumet that we used for x. We fid that µ 2 = T 2 A apple y T Ay y T y apple µ 1. From here, we divide the proof ito two cases. First, cosider the case i which y is ever zero. I this case, there must be some edge (u, v) for which 2 (u) < 0 < 2 (v). The the above iequality T must be strict because the edge (u, v) will make a egative cotributio to 2 A 2 ad a positive cotributio to y T Ay. We will argue by cotradictio i the case that y has a zero value. I this case, if µ 2 = µ 1 the y will be a eigevector of eigevalue µ 1. This is a cotradictio, as Lemma says that a o-egative eigevector caot have a zero value. So, if y has a zero value the y T Ay <µ 1 ad µ 2 <µ 1 as well. The followig characterizatio of bipartite graphs follows from similar ideas. Propositio If G is a coected graph, the µ = µ 1 if ad oly if G is bipartite. Proof. First, assume that G is bipartite. That is, we have a decompositio of V ito sets U ad W such that all edges go betwee U ad W. Let 1 be the eigevector of µ 1.Defie ( x (u) = 1(u) if u 2 U,ad 1(u) if u 2 W.

5 Lecture 3: September 5, For u 2 U, wehave (Ax )(u) = X x (v) = (u,v)2e X (u,v)2e (v) = µ 1 (u) = µ 1 x (u). Usig a similar argumet for u 62 U, we ca show that x is a eigevector of eigevalue µ 1. To go the other directio, assume that µ = ad agai observe µ 1. We the costruct y as i the previous proof, µ = A = X u,v A(u, v) (u) (v) apple X u,v A(u, v)y(u)y(v) apple µ 1 y T y = µ 1. For this to be a equality, it must be the case that y is a eigevalue of µ 1, ad so y = 1. For the first iequality above to be a equality, it must also be the case that all the terms (u) (v) have the same sig. I this case that sig must be egative. So, we every edge goes betwee a vertex for which (u) is positive ad a vertex for which (v) is egative. Thus, the sigs of give the bi-partitio. The th eigevalue, which is the most egative i the case of the adjacecy matrix ad is the largest i the case of the Laplacia, correspods to the highest frequecy vibratio i a graph. Its correspodig eigevector tries to assig as di eret as possible values to eighborig vertices. This is, it tries to assig a colorig. I fact, there are heuristics for fidig k colorigs by usig the k 1 largest eigevectors [AK97]. 3.6 Graph Colorig ad Idepedet Sets A colorig of a graph is a assigmet of oe color to every vertex i a graph so that each edge attaches vertices of di eret colors. We are iterested i colorig graphs while usig as few colors as possible. Formally, a k-colorig of a graph is a fuctio c : V!{1,...,k} so that for all (u, v) 2 V, c(u) 6= c(v). A graph is k-colorable if it has a k-colorig. The chromatic umber of a graph, writte G, is the least k for which G is k-colorable. A graph G is 2-colorable if ad oly if it is bipartite. Determiig whether or ot a graph is 3-colorable is a NP-complete problem. The famous 4-Color Theorem [AH77a, AH77b] says that every plaar graph is 4-colorable. A set of vertices S is idepedet if there are o edges betwee vertices i S. I particular, each color class i a colorig is a idepedet set. The problem of fidig large idepedet sets i a graph is NP-Complete, ad it is very di cult to eve approximate the size of the largest idepedet set i a graph. However, for some carefully chose graphs oe ca obtai very good bouds o the sizes of idepedet sets by usig spectral graph theory. We may later see some uses of this theory i the aalysis of error-correctig codes ad sphere packigs.

6 Lecture 3: September 5, Ho ma s Boud Ho ma proved the followig upper boud o the size of a idepedet set i a graph G. Theorem Let G =(V,E) be a d-regular graph. The µ (G) apple. d µ I gave a overly-complicated proof of this theorem i Part of the complicatio was that I wrote my proof usig the adjacecy matrix. I will ow give a very simple proof usig the Laplacia matrix. I fact, I will prove a slightly stroger statemet that does ot require the graph to be regular. Theorem Let S be a idepedet set i G, ad let d ave (S) be the average degree of a vertex i S. The, d ave (S) S apple 1. To compare these two, observe that i the d-regular case d ave = d ad = d µ. So, we have 1 d ave (S) = d = µ d µ. Proof. Let S be a idepedet set of vertices ad let d(s) be the sum of the degrees of vertices i S. Recall that = max x x T Lx x T x. We also kow that the vector x that maximizes this quatity is, ad that is orthogoal to 1. So, we ca refie this expressio to As we did last class, we will cosider the vector where s = S /. As S is idepedet, we have x T Lx = max x?1 x T x. x = S s1, x T Lx = d(s) =d ave (S) S. We also computed the square of the orm of x last class, ad it comes out to x T x = (s s 2 ).

7 Lecture 3: September 5, So, we have Re-arragig terms, this gives which is equivalet to the claim of the theorem. d ave (S) S (s s 2 ) = d ave(s)s (s s 2 ) = d ave(s) 1 s. 1 d ave (S) Ho ma s boud usig the adjacecy matrix eigevalues does ot ecessarily hold for irregular graphs. However, the boud that oe would expect to get from it o the chromatic umber does. As a k-colorable graph must have a idepedet set of size at least /k, the expected theorem follows. s, Theorem (G) µ 1 µ µ =1+ µ 1 µ. Usig Theorem 3.7.2, we ca prove (ad you will do so o the first problem set) G d ave. These are the same i the regular case. I m ot sure which is better, or if they are eve comparable, i geeral. 3.8 Wilf s Theorem We did ot get to the followig material i today s lecture, but I assume that you will read it. While we may thik of µ 1 as beig a related to the average degree, it does behave di eretly. I particular, if we remove the vertex of smallest degree from a graph, the average degree ca icrease. O the other had, µ 1 ca oly decrease whe we remove a vertex. Let s prove that ow. Lemma Let A be a symmetric matrix with largest eigevalue 1. Let B be the matrix obtaied by removig the last row ad colum from A, ad let 1 be the largest eigevalue of B. The, 1 1. Proof. For ay vector y 2 IR 1,wehave y T By = T y y A. 0 0

8 Lecture 3: September 5, So, for y a eigevector of B of eigevalue 1, T y y A 1 = y T By y T y = 0 y 0 T y 0 0 apple max x 2IR x T Ax x T x. Of course, this holds regardless of which row ad colum we remove, as log as they are the same row ad colum. It is easy to show that every graph is (d max +1)-colorable. Assig colors to the vertices oe-by-oe. As each vertex has at most d max eighbors, there is always some color oe ca assig that vertex that is di eret that those assiged to its eighbors. The followig theorem of Wilf improves upo this boud. Theorem (G) applebµ 1 c +1. Proof. We prove this by iductio o the umber of vertices i the graph. To groud the iductio, cosider the graph with oe vertex ad o edges. It has chromatic umber 1 ad largest eigevalue zero 1. Now, assume the theorem is true for all graphs o 1 vertices, ad let G be a graph o vertices. By Lemma 3.4.1, G has a vertex of degree at most bµ 1 c. Let v be such a vertex ad let G {v} be the graph obtaied by removig this vertex. By Lemma ad our iductio hypothesis, G {v} has a colorig with at most bµ 1 c+1 colors. Let c be ay such colorig. We just eed to show that we ca exted c to v. As v has at most bµ 1 c eighbors, there is some color i {1,...,bµ 1 c +1} that does ot appear amog its eighbors, ad which it may be assiged. Thus, G has a colorig with bµ 1 c + 1 colors. For a example, cosider a path graph with at least 3 vertices. We have d max = 2, but 1 < 2. Refereces [AH77a] Keeth Appel ad Wolfgag Hake. Every plaar map is four colorable part i. dischargig. lliois Joural of Mathematics, 21: , [AH77b] Keeth Appel ad Wolfgag Hake. Every plaar map is four colorable part ii. reducibility. lliois Joural of Mathematics, 21: , [AK97] Alo ad Kahale. A spectral techique for colorig radom 3-colorable graphs. SICOMP: SIAM Joural o Computig, 26, If this makes you ucomfortable, you could use both graphs o two vertices