Summary
An operation loosely described as a type of composition of graphs is studied. Under rather flexible conditions, the resulting composite graphs must be cospectral. This operation is sufficiently powerful to generate eighty‐one cospectral pairs with at most nine vertices. These pairs include the unique smallest cospectral pair, the smallest cospectral connected pair, and one pair of trees with nine vertices. It is felt that this operation provides a unified explanation of cospectrality in several cases that were previously viewed as coincidental.