Cuts in Cartesian Products of Graphs

The k-fold Cartesian product of a graph G is defined as a graph on tuples (x1, . . . , xk) where two tuples are connected if they form an edge in one of the positions and are equal in the rest. Starting with G as a single edge gives G as a k-dimensional hypercube. We study the distributions of edges crossed by a cut in G across the copies of G in different positions. This is a generalization of the notion of influences for cuts on the hypercube. We show the analogues of the statements of Kahn, Kalai and Linial [11] and that of Friedgut [8], for the setting of Cartesian products of arbitrary graphs. We also extends the work on studying isoperimetric constants for these graphs [3] to the value of semidefinite relaxations for expansion. We connect the optimal values of the relaxations for computing expansion, given by various semidefinite hierarchies, for G and G.