A Questionable Distance-Regular Graph Rebecca Ross Abstract In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection numbers. We discuss results following from the de¯nition of the intersection algebra. We investigate two examples of distance-regular graphs and show how these results apply. Finally, we introduce parameters that determine intersection numbers. We investigate if these intersection numbers are nonnegative and feasible. 1. Introduction In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection numbers. We discuss results following from the de¯nition of the intersection algebra. We investigate two examples of distance- regular graphs and show how these results apply. Finally, we introduce parameters that determine intersection numbers. We investigate if these intersection numbers are nonnegative and feasible. 2. Preliminaries In this section we ¯x notation and review the basic de¯nitions regarding graphs, distance- regularity graphs, and intersection numbers. For more information concerning graphs, see West [1] or Biggs [3]. 2.1 Graphs A graph ¡ consists of a ¯nite set V , whose elements are called vertices, another ¯nite set E, whose elements are called edges, and a subset I of V x E, called the incidence relation such that each edge is incident with exactly two vertices, and no pair of edges is incident with the same pair of vertices. Two distinct vertices are said to be adjacent if there exists an edge incident to them both. A walk is a sequence of vertices such that consecutive vertices in the sequence are adjacent. A path is a walk with distinct vertices. A graph is connected if for each pair of distinct vertices there exists a path containing them. The length of a walk with n vertices is n ¡ 1. The distance between vertices u and v, denoted by @(u; v), is the length of the shortest path containing the vertices. The diameter, d, of a graph is the maximum distance between any two vertices in V (¡). A graph automorphism is a permutation of the vertex set that preserves the adjacency relation. The group of automorphisms of a graph, ¡, is denoted by Aut(¡). A graph ¡ is vertex-transitive if for every u; v 2 V (¡), there exists an automorphism mapping u to 1 v. A connected graph ¡ is distance-transitive if for any vertices u; v; x; y 2 V (¡) such that @(u; v) = @(x; y) there exists an automorphism g 2 Aut(¡) that maps u to x and v to y. 2.2 Distance-regular graphs Let ¡ be a connected graph with diameter d. For each u 2 V (¡) and each integer i, de¯ne ¡i(u) := fv 2 V (¡) j @(u; v) = ig: If k is a nonnegative integer, we say ¡ is regular, with valency k, if for all u 2 V (¡), h j¡1(u)j = k. We say ¡ is distance-regular, with intersection numbers pij (0 · h; i; j · d), whenever for all integers h; i; j (0 · h; i; j · d) and all u; v 2 V (¡) with @(u; v) = h, h j¡i(u) \ ¡j(v)j = pij: (1) These intersection numbers must be nonnegative and satisfy a triangle inequality, in that h pij = 0 if one of h, i, j is greater than the sum of the other two. Necessary conditions such as these on the intersection numbers are often called feasibility conditions. Unless a set of parameters meets these criteria, it is impossible (infeasible) to have a distance-regular graph with these intersection numbers. For more information on feasibility conditions, see [2] For convenience, we de¯ne i ci := pi¡1;1 (1 · i · d); i bi := pi+1;1 (0 · i · d ¡ 1); i ai := pi1 (1 · i · d) and de¯ne c0 = 0 and bd = 0. Note that a0 = 0 and c1 = 1. Also, for ¡ regular with valency k, we have k = b0. We can also observe that ci + ai + bi = k (0 · i · d): (2) For any vertex v of a distance-regular graph ¡ with diameter d, de¯ne ki = j¡i(v)j for 1 · i · d. It follows then that there are ki¡1 vertices in ¡i¡1(v) and each of those vertices is adjacent to bi¡1 vertices in ¡i(v). Also, there ki vertices in ¡i(v) that are adjacent to ci vertices in ¡i¡1(v). So counting the number of edges incident with both ¡i¡1(v) and ¡i(v) is ki¡1bi¡1 = kici. Thus, kb1b2 : : : bi¡1 ki = (3) c2c3 : : : ci For the remainder of this paper, we will assume ¡ is distance-regular with diameter d ¸ 3. 2 3. The Adjacency Algebra, A Given a graph, ¡ with vertex set fv1; v2; :::; vng, the adjacency matrix of ¡ is the n £ n matrix, A, whose entry (A)ij is the number of edges in ¡ with endpoints fvi; vjg. We notice that by our de¯nition of graph, A is a symmetric 0¡1 matrix. It also follows then that if ¸ is an eigenvalue of A, then ¸ is real and that the multiplicity of ¸ as a root of det(¸I ¡ A) = 0 is equal to the dimension of the space generated by the eigenvector corresponding to ¸. We will usually refer to the eigenvalues of A as the eigenvalues of ¡. De¯nition 3.0.1 Let ¡ be a distance-regular graph with diameter d and adjacency matrix A. De¯ne the adjacency algebra of ¡ to be the algebra given by spanfA0; A; A2;:::g and denoted by A. Proposition 3.0.2 For any nonnegative integer k, the (i; j)th entry of Ak counts the number of vi; vj-walks of length k. Proof. This holds for k = 0 since A0 is the identity matrix, I. For k = 1, A1 = A is the adjacency matrix, which by de¯nition counts the number of walks of length 1 between any l two vertices. Suppose true for A for l < k. Observe that every walk of length k from vi to vj consists of a walk of length k ¡ 1 from vi to a vertex vl that is adjacent to vj. By the k¡1 induction hypothesis, (A )il is the number of walks of length k ¡ 1 with endpoints fvi; vlg for 0 · l < k. Thus k k¡1 (A )ij = (A A)ij Xn k¡1 = (A )il(A)lj l=1 k¡1 k¡1 k¡1 = (A )i1(A)1k + (A )i2(A)2k + ¢ ¢ ¢ + (A )in(A)nj which equals the number of walks of length k with endpoints fvi; vjg. ¤ Lemma 3.0.3 Let ¡ be a distance-regular graph with diameter d and adjacency algebra A. Then dim(A) ¸ d + 1. Proof. Pick any u; v 2 V (¡) and let @(u; v) = k. Then for all i < k the uv-entry of Ai will be zero since the shortest path connecting u to v is of length k and the uv-entry of Ak is nonzero. Since the diameter of ¡ is d, there exist vertices x; y 2 V (¡) such that @(x; y) = i for 0 · i · d. So for all 0 · i · d, Ai is not contained in the span of A0; ¢ ¢ ¢ ;Ai¡1. Thus, fA0; A; : : : ; Adg are linearly independent and dim A ¸ d + 1. ¤ Corollary 3.0.4 A connected graph with diameter d has at least d+1 distinct eigenvalues. Proof. Let ¡ be a connected graph with adjacency matrix, A. By Lemma 3.0.3, the dimen- sion of A is at least d + 1, thus the minimum polynomial of A has degree at least d + 1. It follows then that A has at least d + 1 distinct eigenvalues. ¤ 3 3.1 Bose-Mesner Algebra, M We next recall the Bose-Mesner algebra of ¡. For each integer i (0 · i · d), let Ai be the ith distance matrix, which is the n £ n matrix where each rs-entry is given by ½ 1 if @(vr; vs) = i; (Ai)rs = (vr; vs 2 V (¡)): 0 if @(vr; vs) 6= i We notice that A0 = I and that A1 is the adjacency matrix A of ¡. Also, A0 + A1 + ::: + Ad = J; t where J is the all-1 matrix of size n. It is also clear that Ai = Ai for (0 · i · d). Lemma 3.1.1 Let ¡ be a distance-regular graph with diameter d. For 1 · i; j · d, Xd h AiAj = pijAh: h=0 th Proof. Let ¡ be a distance-regular graph with diameter d and Ak the k distance matrix of ¡ for (0 · k · d). Pick any x; y 2 V (¡) and let t = @(x; y). Then X (AiAj)xy = (Ai)xz(Aj)zy z2V (¡) = jfz 2 V (¡) j @(x; z) = i and @(z; y) = jgj t = pij: Pd h On the other hand, the xy-entry of h=0 pijAh is equal to Xd h t pij(Ah)xy = pij(At)xy h=0 t = pij(1) t = pij; since (Ah)xy = 0 if h 6= t. ¤ Corollary 3.1.2 Let ¡ be a distance-regular graph. For 1 · i · d, AAi = bi¡1Ai¡1 + aiAi + ci+1Ai+1: Proof. By applying Lemma 3.1.1 and the de¯nition of bi; ai, and ci, the equation follows easily. ¤ It follows then that 1 Ai+1 = [AAj ¡ bj¡1Aj¡1 ¡ ajAj]; ci+1 4 which is a polyomial in A with degree j + 1. Now, the distance matrices are linearly independent (exactly one matrix has a nonzero entry in every position) and, by Lemma 3.1.1, closed under multiplication.