Upper bounds given by equitable partitions of a primitive association scheme

Suppose that ( X , G ) is a primitive association scheme with | G | ≥ 3 and π is an equitable partition of ( X , G ) with | π | < | X | . We put π ∗ ≔ { C ∈ π ∣ | C | > 1 } and supp ( π ) ≔ ⋃ C ∈ π ∗ C . In this article we prove that | G | ≤ | supp ( π ) | − | π ∗ | + 1 , and show a necessary and sufficient condition for the equality to hold.