GraphProduct

<< DiscreteMath`Combinatorica`

This is the same as the issue with GraphJoin (see the GraphJoin notebook for more details) of not correctly inheriting graphics options during a graph operation.

g  =  
      SetGraphOptions[
        Cycle[4],  {{1,  2},  {2,  3},  EdgeStyle -> Thick,  
          EdgeColor -> Red},  
        VertexStyle -> Box[Large]]

h =  
      SetGraphOptions[
        Star[4],  {{1,  4},  {2, 4},  EdgeStyle -> Thick,  
          EdgeColor -> Green},  
        VertexStyle -> Disk[Large]]

ShowGraphArray[{g,  h}]

ShowGraph[GraphProduct[g,  h]]