On the Calculation of Multiplicities for P-Polynomial C-Algebras

Abstract

An alternative rational function, with polynomial components of smaller degree, is constructed to compute multiplicities for a P-polynomial C-algebra whose generating tri-diagonal matrix has a set of repeated column entries. As a consequence, some upper bounds are derived for the diameter of the algebra. The bound in the case of an integral table algebra generalizes a well known result of Bannai and Ito for distance-regular graphs.

Key Words:

• C-algebra
• Distance-regular graph
• Multiplicities
• Polynomials

2010 Mathematics Subject Classification:

• 16K20
• 13P05

ACKNOWLEDGMENTS

Much of the content of this article appears in the second author's doctoral dissertation [10], which was written under the supervision of the first author and submitted to Northern Illinois University in 2007. Both authors thank the anonymous referee, whose comments included a deeper version of our original Proposition 2.1, leading to shortened proofs of both Propositions 2.1 and 2.5.

Notes

Communicated by M. Cohen.