BooleanGraph

BooleanGraph[bfunc, g1, ..., gn] gives the Boolean graph defined by the Boolean function bfunc on the graphs g1, ..., gn.

Examples

Intersection of two graphs:

g1 = Graph[{1 -> 2, 2 -> 3}];
g2 = Graph[{2 -> 3, 3 -> 4}];
BooleanGraph[And, g1, g2]

Union of graphs:

BooleanGraph[Or, g1, g2]

Please visit the official Wolfram Language Reference for more details.