# A characterization of $Q$-polynomial distance-regular graphs using the intersection numbers

## TL;DR

This paper characterizes $Q$-polynomial distance-regular graphs with diameter at least 3 by using intersection numbers to identify a positive semidefinite matrix with integer entries, where the matrix's determinant zero condition characterizes the $Q$-polynomial property.

## Contribution

It introduces a new characterization of $Q$-polynomial distance-regular graphs via a specific positive semidefinite matrix derived from intersection numbers.

## Key findings

- Matrix $G$ is positive semidefinite with integer entries.
- Determinant of $G$ is zero if and only if the graph is $Q$-polynomial.
- Provides a new algebraic criterion for $Q$-polynomial property.

## Abstract

We consider a primitive distance-regular graph $\Gamma$ with diameter at least $3$. We use the intersection numbers of $\Gamma$ to find a positive semidefinite matrix $G$ with integer entries. We show that $G$ has determinant zero if and only if $\Gamma$ is $Q$-polynomial.