**Speaker**: Alexander Gavrilyuk, Ural Branch of the Russian Academy of Sciences

**Date**: Sat, Dec 12, 2015

**Time**: 16:40 - 17:40

**Venue**: Middle Meeting Room

**Title**: On Q-polynomial distance-regular graphs of type 2

**Abstract**:

Bannai and Ito's interpretation of the Leonard theorem says that the intersection numbers of a Q-polynomial distance-regular graph have at least one of seven possible types: 1, 1A, 2, 2A, 2B, 2C, 3.

In 1986 Terwilliger [1] classified all Q-polynomial distance-regular graphs of type 2 with diameter at least 14. We improve this diameter bound. In fact, if the diameter of a Q-polynomial distance-regular graph $\Gamma$ of type 2 is less than 14, then some of arguments of the proof from [1] are not valid. Assuming that the diameter is at least 8 and studying possible locations of the roots of the Terwilliger polynomial of $\Gamma$ (recently calculated in [2]), we replace these arguments with their slightly extended version.

This is joint work with Jack Koolen.

**Slides**:
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**References**:

[1] P. Terwilliger, A class of distance-regular graphs that are Q-polynomial, J. Combin. Theory Ser. B, 40(2):213-223, 1986.

[2] A.L. Gavrilyuk, J.H. Koolen, The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to pseudo-partition graphs, Linear Algebra Appl., 466:117-140, 2015.