Distance-regular graph

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In mathematics, a distance-regular graph is a regular graph such that, given any two vertices v and w at any distance i, the number of vertices adjacent to w and at distance j from v depends only on i and j, not on the particular pair of vertices. Every distance-transitive graph is distance regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.

Alternatively, a distance-regular graph is a graph for which there exist integers bi,ci,i=0,...,d such that for any two vertices x,y in G and distance i=d(x,y), there are exactly ci neighbors of y in Gi-1(x) and bi neighbors of y in Gi+1(x), where Gi(x) is the set of vertices y of G with d(x,y)=i (Brouwer et al. 1989, p. 434). The array of integers characterizing a distance-regular graph is known as its intersection array. examples:http://mathworld.wolfram.com/images/eps-gif/DistanceRegularCubic_800.gif

A distance-regular graph with diameter 2 is strongly regular, and conversely (unless the graph is disconnected).

Contents

[edit] Intersection numbers

It is usual to use the following notation for a distance-regular graph G. The number of vertices is n. The number of neighbors of w (that is, vertices adjacent to w) whose distance from v is i, i + 1, and i − 1 is denoted by ai, bi, and ci, respectively; these are the intersection numbers of G. Obviously, a0 = 1, c0 = 0, and b0 equals k, the degree of any vertex. If G has finite diameter, then d denotes the diameter and we have bd = 0.

The numbers ai, bi, and ci are often displayed in a three-line array

\left\{\begin{matrix} - & c_1 & \cdots & c_{d-1} & c_d \\ a_0 & a_1 & \cdots & a_{d-1} & a_d \\ b_0 & b_1 & \cdots & b_{d-1} & - \end{matrix}\right\},

called the intersection array of G. They may also be formed into a tridiagonal matrix

B:= \begin{pmatrix} a_0 & b_0 & 0 & \cdots & 0 & 0 \\ c_1 & a_1 & b_1 & \cdots & 0 & 0 \\ 0 & c_2 & a_2 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{d-1} & b_{d-1} \\ 0 & 0 & 0 & \cdots & c_d & a_d \end{pmatrix} ,

called the intersection matrix.

[edit] Distance adjacency matrices

Suppose G is a connected distance-regular graph. For each distance i = 1, ..., d, we can form a graph Gi in which vertices are adjacent if their distance in G equals i. Let Ai be the adjacency matrix of Gi. For instance, A1 is the adjacency matrix A of G. Also, let A0 = I, the identity matrix. This gives us d + 1 matrices A0, A1, ..., Ad, called the distance matrices of G. Their sum is the matrix J in which every entry is 1. There is an important product formula:

AAi = aiAi + biAi + 1 + ciAi − 1.

From this formula it follows that each Ai is a polynomial function of A, of degree i, and that A satisfies a polynomial of degree d + 1. Furthermore, A has exactly d + 1 distinct eigenvalues, of which the largest is k, the degree.

The distance matrices span a vector subspace of the vector space of all n × n real matrices. It is a remarkable fact that the product Ai Aj of any two distance matrices is a linear combination of the distance matrices:

A_i A_j = \sum_{k=0}^d p_{ij}^k A_k .

This means that the distance matrices generate an association scheme. The theory of association schemes is central to the study of distance-regular graphs. For instance, the fact that Ai is a polynomial function of A is a fact about association schemes.

[edit] Examples

[edit] Classification

There seems to be no hope of classifying all distance-regular graphs, desirable as that would be.

[edit] References

A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5