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Anantharaman and Le Masson proved that any family of eigenbases of theadjacency operators of a family of graphs is quantum ergodic (a form ofdelocalization) assuming the graphs satisfy conditions of expansion and highgirth. In this paper, we show that neither of these two conditions issufficient by itself to necessitate quantum ergodicity. We also show thathaving conditions of expansion and a specific relaxation of the high girthconstraint present in later papers on quantum ergodicity is not sufficient. Wedo so by proving new properties of the Cartesian product of two graphs whereone is infinite.

The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph