Global Parameters

Let G be a simple connected graph, and take 0<=i<=d(G), where d(G) is the graph diameter. Then G has global parameters c_i (respectively a_i, b_i) if the number of vertices at distance i-1 (respectively, i, i+1) from a given vertex v that are adjacent to a vertex w at distance i from v is the constant c_i (respectively a_i, b_i) depending only on i (i.e., not on v of w).

Global parameters may be computed by the GRAPE package in GAP using the function GlobalParameters(G), which returns a list of length d(G)+1 whose ith element is the list [c_(i-1),a_(i-1),b_(i-1)] (except that if some global parameter does not exist then -1 is put in its place). Note that G is a distance-regular graph iff this function returns no -1 in place of a global parameter.

A distance-regular graph with global parameters [[c_0,a_0,b_0],[c_1,a_1,b_1],[c_2,a_2,b_2],[c_3,a_3,b_3],[c_4,a_4,b_4]] has intersection array {b_0,b_1,b_2,b_3;c_1,c_2,c_3,c_4}.

See also

Distance-Regular Graph, Intersection Array

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Cite this as:

Weisstein, Eric W. "Global Parameters." From MathWorld--A Wolfram Web Resource.

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