Some Results on Distance Regular Graphs, or: How I Spent My Spring Break Emelie Curl April 7,2017 Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 1 / 1 Definitions Lemmas/Theorems/etc. A Proof Ending comments Works Cited Overview Background Information (So much background information including:) Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Lemmas/Theorems/etc. A Proof Ending comments Works Cited Overview Background Information (So much background information including:) Definitions Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 A Proof Ending comments Works Cited Overview Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Ending comments Works Cited Overview Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. A Proof Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Works Cited Overview Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. A Proof Ending comments Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Overview Background Information (So much background information including:) Definitions Lemmas/Theorems/etc. A Proof Ending comments Works Cited Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 2 / 1 Definitions Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions For those of you who have taken Design Theory (Math 605), most of this presentation will be review, so you should feel free to \check out" mentally. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions All of the graphs considered in this presentation are finite, undirected, and simple. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions Definition Suppose G is a connected graph with vertex set V (G) and edge set E (G) where E (G) consists of unordered pairs of two adjacent vertices. The distance d(x, y) between any two vertices x and y of G is the length of a shortest path connecting x and y in G. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions Definition We denote v as the number of vertices of G and define the diameter D of G as the maximum distance in G. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Definitions Definition For any vertex x 2 V (G), define Gi (x) to be the graph which consists of vertices that are at distance precisely i from x where 0 ≤ i ≤ D. In addition, we note that G−1(x) = GD+1(x) = Æ. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 3 / 1 Boring: More Definitions! Definition A connected graph G with diameter D is called distance − regular if there are integers ai , bi , ci for 0 ≤ i ≤ D such that for any two vertices x, y 2 V (G) with d(x, y) = i: (i) there are ci neighbors of y in Gi−1(x) i.e. there are ci vertices of distance i − 1 from x and distance 1 from y in G; (ii) there are bi neighbors of y in Gi+1(x) i.e. there are bi vertices of distance i + 1 from x and distance 1 from y in G; (iii) there are ai neighbors of y in Gi (x) i.e. there are ai vertices of distance i from x and distance 1 from y in G. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 4 / 1 Boring: More Definitions! Definition The parameters ai , bi−1, and ci (1 ≤ i ≤ D) are called the intersection numbers. Note that a0 = bD = c0 = 0, and c1 = 1 and any distance-regular graph is k−regular such that k = b0. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 4 / 1 Example Now, here's an example of such a graph. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 Example Note that a connected strongly regular graph is just a distance-regular graph with diameter two. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 5 / 1 A bunch of parameters and relations Now, here's a ton of parameters and relations related to distance-regular graphs! Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations (i) Lemma Note that ai = k − bi − ci where ai = jG(y) \ Gi (x)j holds for any two vertices x, y with d(x, y) = i for 0 ≤ i ≤ D. I submitted verifying this lemma as a question for the exam. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations (ii) Definition For a distance-regular graph G and a vertex x 2 V (G), we let ki = jGi (x)j be the ith valency of G. We have k0 = 1,k1 = k, and ki 6= 0 for all 0 ≤ i ≤ D. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations (iii) Lemma The total number of vertices is v = 1 + k1 + ... + ki + ... + kD . Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations (iv) Definition i Let pjh = jfwjw 2 Gj (x) \ Gh(y)gj for any y 2 Gi (x) with 0 ≤ i, j, h ≤ D (also called intersection numbers). That is, d(x, y) = i, d(x, w) = j, d(y, w) = h. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations (v) Lemma i i i Also, ci = pi−1,1 for 1 ≤ i ≤ D, ai = pi,1 for 0 ≤ i ≤ D, bi = pi+1,1 for 0 ≤ i ≤ D − 1. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 A bunch of parameters and relations Definition The array fb0, b1, ... , bD−1; c1, c2, ... , cD g is called the intersection array of G. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 6 / 1 Another Question?! Also, Possibly a good (easy) question for the final is to show the Petersen graph is determined as a distance-regular graph by its intersection array. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 7 / 1 Still More Definitions Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Still More Definitions Definition Suppose that G is a distance-regular graph with valency k ≥ 2 and diameter D ≥ 2. Then, we let Ai be the matrix of G such that the rows and the columns of Ai are indexed by the vertices of G and the (x, y)−entry is 1 whenever x and y are at a distance i and 0 otherwise. We will denote the adjacency matrix of G as A instead of A1. The eigenvalues of the graph G are the eigenvalues of A. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Still More Definitions Definition Suppose that G is a distance-regular graph with valency k ≥ 2 and diameter D ≥ 2. Then, we let Ai be the matrix of G such that the rows and the columns of Ai are indexed by the vertices of G and the (x, y)−entry is 1 whenever x and y are at a distance i and 0 otherwise. We will denote the adjacency matrix of G as A instead of A1. The eigenvalues of the graph G are the eigenvalues of A. Duh. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Still More Definitions Definition Note that the matrices Ai satisfy the relations: (i) A0 = I , A1 = A; (ii) AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for i = 0, ... , D; (iii) A0 + A1 + ... + AD = J; (iv) Also, we have A−1 = AD+1 = 0. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 8 / 1 Proof Sketch To see that AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for 1 ≤ i ≤ D : Fix vertices x, y 2 G. Emelie Curl Some Results on Distance Regular Graphs, or: How I Spent My SpringApril Break7,2017 9 / 1 Proof Sketch To see that AAi = ci+1Ai+1 + ai Ai + bi−1Ai−1 for 1 ≤ i ≤ D : Fix vertices x, y 2 G.