SPECIAL TOPICS IN GRAPH THEORY MICHELLE DELCOURT Abstract. This three part lecture series is based primarily on the book Elementary Number Theory, Group Theory, and Ramanujan Graphs by Davidoff, Sarnak, and Valette. I will discuss background information and the explicit construction of (p + 1)-regular Ramanujan graphs by Lubotzky-Phillips-Sarnak and Margulis, where p is an odd prime. 1. Basic definitions and results We start with some basic definitions and results in spectral graph theory. Let X = (V; E) be a graph. The adjacency matrix A of a finite graph X on n vertices is an n by n symmetric matrix. Therefore, A has n real eigenvalues counting multiplicities λ0 ≥ λ1 ≥ ::: ≥ λn−1: Lemma 1. Let X be a finite k-regular graph with n vertices. Then (1) λ0 = k (2) jλij ≤ k for 1 ≤ i ≤ (n − 1) (3) λ0 has multiplicity 1 if and only if X is connected. [Furthermore, the multiplicity is equal to the number of components.] Lemma 2. Let X be a connected, k-regular graph on n vertices. Then the following are equivalent (1) λn−1 = −k, (2) the spectrum of X is symmetrical about 0, (3) X is bipartite. Let X = (V; E) be a finite, connected graph on n vertices, and F ⊆ V . Definition 1. The boundary of F , denoted δF , is the set of edges required to disconnect F from any vertex in V − F . Date: October 31, November 2, and November 5, 2012. 2 Definition 2. The isoperimetric or expanding constant of X is jδF j n h(X) = min : F ⊆ V; 0 < jF j ≤ jF j 2 jδF j = inf : F ⊆ V; 0 < jF j < +1 : min fjF j; jV − F jg Think of X as a network that is transmitting information from vertex to vertex; the ex- panding constant measures the \quality" of X as a network in some sense. A large expanding constant means that information is able to propagate well. Consider the complete graph Kn jδF j k(n−k) versus the cycle Cn. For X = Kn, if jF j = k, then jδF j = k(n − k) so jF j = k = n − k n n and h(Kn) = n − 2 ∼ 2 ; Kn is highly connected and has a large expanding constant which grows proportionately with the number of vertices. For X = Cn, if jF j is a half cycle, 2 4 jδF j = 2 and h(Cn) = n ∼ n ; Cn is not highly connected and has a small expanding [ 2 ] constant that tends to 0 as the number of vertices increases. 2. Families of expanders Definition 3. If fXmgm≥1 is a family of finite, connected, k-regular graphs with jVmj ! +1 as m ! +1, then fXmgm≥1 is a family of expanders if there exists an > 0 such that h(Xm) ≥ for all m ≥ 1. Simply speaking, expander graphs are sparse yet highly connected k-regular graphs. Because of these nice properties, expander graphs have many applications in engineering and computer science from network design to cryptography. We would like to explicitly construct infinite family of expanders. A \good quality" expander has a large spectral gap λ0(Xm) − λ1(Xm) = k − λ1(Xm) (this will motivate the definition of a Ramanujan graph) as it measures \high connectedness". For an arbitrary graph X = (V; E), consider the functions f : V ! C and define Hilbert spaces, ( ) 2 X 2 l (V ) = f : V ! C : jf(v)j < +1 and v2V ( ) 2 X 2 l (E) = f : E ! C : jf(e)j < +1 : e2E Special Topics in Graph Theory Delcourt 3 Theorem 1 (1985 Alon-Milman; 1984 Dodziuk). Let X be a finite, connected k-regular graph. Then k − λ 1 ≤ h(X) ≤ p2k(k − λ ): 2 1 Proof. Let X = (V; E) be a finite, connected k-regular graph without loops. Randomly orient the edges; given an edge e 2 E, let e+ denote the head and e− denote the tail. The simplicial coboundary operator is d : l2(V ) ! l2(E) if for f 2 l2(V ) and e 2 E, df(e) = f(e+) − f(e−): Endow l2(V ) and l2(E) with hermitian scalar product; for example, X hf; gi = f(x)g(x): x2V Then, there is a unique continuous operator, the adjoint operator d∗ : l2(E) ! l2(V ) that is characterized by hdf; gi = hf; d∗gi for all f 2 l2(V ) and g 2 l2(E): Define a function δ : V × E ! {−1; 0; 1g by 8 >1; if x = e+ > <> δ(x; e) = −1; if x = e− > > :>0; otherwise, 2 P 2 then for e 2 E and f 2 l (V ) df(e) = x2V δ(x; e)f(x) and for x 2 V and g 2 l (E) ∗ P d g(x) = e2E δ(x; e)g(e). Let the combinatorial Laplacian operator be ∆ = d∗d : l2(V ) ! l2(V ): In other words, ∆ = k · Id − A: The combinatorial Laplacian operator has a number of nice properties. It does not depend on orientation. If f is a function on the vertex set, and P x2V f(x) = 0, 2 2 kdfk2 = hdf; dfi = h∆f; fi ≥ (k − λ1)kfk2: Consider the following special function 8 <>jV − F j; if x 2 F f(x) = :>−|F j; if x 2 V − F: Special Topics in Graph Theory Delcourt 4 P Then x2V f(x) = 0 so 2 2 2 kfk2 = jF jjV − F j + jV − F jjF j = jF jjV − F j(jV − F j + jF j) = jF jjV − F jjV j; and 8 <>0; if e is not a cross edge between F and V − F df(x) = :>±|V j; if e is a cross edge: 2 2 2 Because kdfk2 = jδF jjV j + 0 = jδF jjV j , 2 jV j jδF j ≥ (k − λ1)jF jjV − F jjV j and jδF j jV − F j ≥ (k − λ ) : jF j 1 jV j jV j jδF j k−λ1 k−λ1 If jF j ≤ 2 , F ≥ 2 , and then h(X) ≥ 2 . The second inequality is much more complicated see pages 20-23 [4]. A family of k-regular graphs is a family of expanders if and only if the spectral gap is bounded away from zero. The bigger the spectral gap, the better the \the quality" of the expander. Theorem 2. Let fXmgm≥1 be a family of finite, connected k-regular graphs without loops, such that jVmj ! +1 as m ! +1. The family fXmgm≥1 is a family of expanders if and only if there exists > 0 such that k − λ1(Xm) ≥ for every m ≥ 1. For many years, constructing large families of expanders has been an important prob- lem. Motivated by problems in network theory, in 1972-1973, Pinsker and Margulis worked on constructions. This work, however, gave no measure of the quality of the expanders. More recent work does (Gabber-Galil, Widgerson-Zuckerman, etc). Historically, isoperimet- ric inequalities have been studied in the Riemannian geometry setting and sometimes are called the Cheeger-Buser inequalities. Special Topics in Graph Theory Delcourt 5 3. Ramanujan Graphs Theorem 3 (Alon-Boppana). Let fXmgm≥1 be a family of finite, connected, k-regular graphs with jVmj ! +1 as m ! +1. Then p lim inf λ1(Xm) ≥ 2 k − 1: m!+1 Proof. The inequality is actually from the fact that the number of paths of length m from a vertex v to v in a k-regular graph is at least number of paths from a vertex v to v in a k-regular tree. Let X = (V; E) be a k-regular simple graph with jV j possibly infinite. A path of length r without backtracking is a sequence e = (x0; x1; : : : ; xr) of vertices in V such that xi is adjacent to xi+1 for i = 0; 1; : : : ; r − 1 and xi+1 6= xi−1 for i = 1; 2; : : : r − 1. The origin of e is x0, and the extremity of e is xr. For r 2 N, matrix Ar is an n by n matrix indexed by V × V with (Ar)xy = the number of paths of length r with origin x and extremity y without backtracking. Define A0 = Id and note that A1 = A, the adjacency matrix of X. Lemma 3. Both of the following equalities hold: 2 (1) A1 = A2 + k · A0 (2) A1Ar = ArA1 = Ar+1 + (k − 1)Ar−1; for r ≥ 2: Proof. 2 (1) For x 6= y 2 V ,(A1)x;y counts the number of paths of length 2; there can be no 2 2 backtracking. Thus, (A1)x;y = (A2)x;y. If x = y, then (A1)x;y = k as the degree of x 2 is k; however, (A2)x;y = 0. Thus (A1)x;y = (A2)x;y + k. (2)( ArA1)x;y is the number of paths (x0 = x; x1; : : : ; xr; xr+1 = y) without backtracking except possibly on the last step. If xr−1 6= y then (x0 = x; x1; : : : ; xr; xr+1 = y) has no backtracking and there are (Ar+1)x;y such paths. If xr−1 = y then there was backtracking at the last step; there are (k − 1)(Ar−1)x;y such paths. Thus, ArA1 = Ar+1 + (k − 1)Ar−1; A1Ar = Ar+1 + (k − 1)Ar−1 is left as an exercise. Special Topics in Graph Theory Delcourt 6 Because 1 ! X r 2 2 Art (A0 − A1t + (k − 1)t A0) = (1 − t )A0; r=0 the generating function for Ar is 1 X (1 − t2) A tr = r 1 − A t + (k − 1)t2 r=0 1 We would like to eliminate (1 − t2) on the Right Hand Side.