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.

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

Contents 1 Intersection numbers 2 Distance adjacency matrices 3 Examples 4 Classification 5 References

[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 a_i, b_i, and c_i, 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 b_d = 0.

The numbers a_i, b_i, and c_i are often displayed in a three-line array

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

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 G_i in which vertices are adjacent if their distance in G equals i. Let A_i be the adjacency matrix of G_i. 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, ..., A_d, called the distance matrices of G. Their sum is the matrix J in which every entry is 1. There is an important product formula:

AA_i = a_iA_i + b_iA_{i + 1} + c_iA_{i − 1}.

From this formula it follows that each A_i 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 A_i A_j of any two distance matrices is a linear combination of the distance matrices:

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 A_i is a polynomial function of A is a fact about association schemes.

[edit] Examples

• Complete graphs are distance regular with diameter 1 and degree v−1.
• Cycles C_2d+1 of odd length are distance regular with k = 2 and diameter d. The intersection numbers a_i = 0, b_i = 1, and c_i = 1, except for the usual special cases (see above) and c_d = 2.
• All Moore graphs, in particular the Petersen graph and the Hoffman-Singleton graph, are distance regular.
• Strongly regular graphs are distance regular.

[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