Very few Moore Graphs Anurag Bishnoi June 7, 2012 Abstract We prove here a well known result in graph theory, originally proved by Hoffman and Singleton, that any non-trivial Moore graph of diameter 2 is regular of degree k = 2; 3; 7 or 57. The existence (and uniqueness) of these graphs is known for k = 2; 3; 7 while it is still an open problem if there is a moore graph of degree 57 or not. 1 Basics We'll be talking about simple undirected connected graphs. There is a natural metric d on such graphs given by the shortest distance. Definition 1.1. Diameter d of a graph is the maximum of all shortest distances between pair of vertices. Definition 1.2. Girth g of a non-tree graph is the length of a shortest cycle. Theorem 1.1. (Moore's Inequality) For a graph which is not a tree g ≤ 2d + 1. Proof. We'll prove this by contradiction. Say g > 2d + 1. Let C be a cycle of length g. Then there are two vertices u; v on C which are connected by a path of length d + 1. Since d is the diameter, there must be a path of length at most d between u; v. But these two paths, being distinct, will form a cycle of length at most 2d + 1 which is less than g. This contradicts the fact that g is the girth. Graphs that achieve equality in this bound are known as Moore Graphs. Trivial examples are odd cycles and complete graphs. A non-trivial example is the Petersen Graph with d = 2 and g = 5. Theorem 1.2. Moore Graphs are regular. Proof. Note that in a Moore Graph any two vertices are joined by a unique shortest path and if they are joined by a path of length at most d then that is the unique shortest path. Now, let u; v be two vertices of a Moore Graph G at distance d from each other. Then we can deduce that deg(u) = deg(v) by exhibiting a bijection between N(u) and N(v) as follows. Let x 2 N(u). Then x is joined to v by a shortest path. Let the image of x be the neighbor of v on this path. It is straightforward to check that this is a bijection. To show that all vertices have the same degree, take a cycle of length 2d + 1. Any two adjacent vertices vertices on this cycle are distance d apart from a single vertex on the cycle and therefore have the same degree. So, all vertices on this cycle have same degree, say k. Now let x be any vertex not on this cycle. Then there exists a point y on the cycle distance d apart from x. Hence deg(x) = k. 1 Theorem 1.3. If G is a n − vertex Moore Graph of degree k and diameter d, then n = 1 + k + k(k − 1) + ::: + k(k − 1)d−1. Proof. Let u be a vertex of G. Then for all 0 ≤ i ≤ d and for each vertex x with d(u; x) = i there exists is a unique path from u to x. Hence, the number of vertices from u at distance i is k(k − 1)i−1 where i ≥ 1, which can be proved by induction. So the total number of vertices in the graph is 1 + k + k(k − 1) + ::: + k(k − 1)d−1 since d is the maximum distance a vertex can have from u. 2 Distance Regular Graphs Definition 2.1. Given a graph G with diameter d we define Ai, for 0 ≤ i ≤ d to be the n × n matrix with rows and columns indexed by V (G) such that Ai(x; y) = 1 if d(x; y) = i and 0 otherwise. Note that A0 is the identity matrix and A1 the adjacency matrix of G. We also have Pd i=0 Ai = J. Definition 2.2. A graph G with diameter d is called distance regular graph if there exists constants a0; : : : ; ad; b0; : : : ; bd−1; c1; : : : ; cd such that the following holds : Given any two vertices x; y at distance i, the number of neighbors of x at distance i − 1=i=i + 1 from y is ci=ai=bi. Note that a distance regular graph is k-regular with k = b0 and hence it is a stronger condition than regularity. Also by triangle inequality we see that if d(x; y) = i and z 2 N(x) then i − 1 ≤ d(z; y) ≤ i + 1. Therefore we have ai + bi + ci = k for all i, taking c0 = bd = 0. Theorem 2.1. Moore graphs are distance regular. Proof. It is easy to see that for k − regular Moore graphs with diameter d, c1 = c2 = ::: = cd = 1, a0 = ::: = ad−1 = 0, ad = k − 1, b0 = k, b1 = ::: = bd−1 = k − 1. Let Ai's be the matrices as defined before for a graph G. For distance regular graphs we can see that each Ai is a polynomial in A1 as follows : Lets calculare A1Ai. Then A1Ai(x; z) = #fy 2 v(G): d(x; y) = 1; d(y; z) = ig. Therefore we have A1Ai = bi−1Ai−1 + aiAi + ci+1Ai+1, 0 ≤ i ≤ d. Hence by finite induction on i we see that Theorem 2.2. There are polynomials pi, 0 ≤ i ≤ d s.t. Ai = pi(A1). They can be found by the following recurrence relation : p0(z) = 1; p1(z) = z and ci+1pi+1(z) = p1(z)pi(z) − bi−1pi−1(z) − aipi(z). P Let q(z) = pi(z). Then q(A1) = J. Therefore, Theorem 2.3. If A is the adjacency matrix of a distance regular graph G with diameter d and degree k then there is an explicitly computable polynomial q of degree d such that q(A) = J. And the eigenvalues of A are either k or some root of q(z). Proof. We see that since the graph is k−regular, k is an eigenvalue of A with eigenvector 1. Any other eigenvalue λ corresponds to an eigenvector x such that x is orthogonal to 1. So Jx = 0. Now, q(λ)x = q(A)x = Jx = 0 and x 6= 0. Therefore q(λ) = 0. 2 Let A be the adjacency matrix of a graph G of n vertices. Then A 2 Cn×n and A is self adjoint. Definition 2.3. The unital subring of Cn×n generated by A is called the adjacency algebra of G denoted by A. Clearly the adjacency algebra is a vector space over C. In fact it is a finite dimensional one since it is a subalgebra of Cn×n. For a distance regular graph we have the following result on this dimension. Theorem 2.4. Let G be a distance regular graph with diameter d and adjacency algebra A. Then dim(A) = d + 1. Proof. We first produce a linearly independent set of d + 1 elements in A to show that dim(A) ≥ d + 1 and then show that the powers of A upto d span A. Let A be the adjacency matrix of G. A0;A1;:::;Ad are polynomials in A by 2:2 and hence they belong to A. These matrices are 0 − 1 disjoint matrices and hence linearly independent. Now from 2:3 we know that A has at most d + 1 eigenvalues. Therefore the minimal polynomial of A has degree at most d+1. Hence Ad+1 is a polynomial in I; A; A2;:::;Ad. Hence by induction on m, Am is a polynomial in I; A; A2;:::;Ad for all m ≥ d + 1. Corollary 2.5. The polynomial q(z) defined in 2:3 has all distinct roots and all of them are eigenvalues of the adjacency matrix. So given an n − vertex distance regular graph G with diameter d, it has d + 1 distinct eigenvalues. Call them λ0; λ1; : : : ; λd and and let V0;V1;:::;Vd be the corresponding n 0 eigenspaces associated with them. Then we know that C = V0 ⊕ V1 ⊕ ::: ⊕ Vd. Let Eis be the projection matrices corresponding to the eigenspaces (after choosing some basis). 2 Then we have Ei = Ei and EiEj = 0 when i 6= j. 0 Theorem 2.6. We also have the following properties for these Eis : 1. E0;E1;:::;Ed is another basis of A. 2. tr(Ei) = multiplicity of λi. i Pd i 3. A = j=0 λjEj. Pd 4. Ai = j=0 pi(λj)Ej. The previous theorem gives us some idea of how to attack the problem of finding Moore graphs. Since we must have that the (d + 1) × (d + 1) matrix M = (pi(λj)) is invertible and tr(Ei) is an integer for all i. 3 Strongly Regular Graphs Definition 3.1. A strongly regular graph (srg) is a distance regular graph of diameter at most 2. If diameter is one then it will be the complete graph so we generally ignore that case. Therefore the parameters for srg are c1 = 1, c2, a0 = 0, a1, b0 = k, b1 = k − 1 − a1. The free parameters a1 and c2 are generally denoted as λ, µ and we have an equivalent definition : 3 Definition 3.2. An srg with parameters (v; k; λ, µ) is a regular graph of degree k on v vertices such that : 1. Any two adjacent vertices have exactly λ common neighbors. 2. Any two non adjacent vertices have exactly µ common neighbors. Theorem 3.1. If G is an srg with parameters (v; k; λ, µ) then its complement G¯ is also an srg.