On average connectivity of the strong product of graphs ?
The average connectivity ?  ( G ) of a graph G G is the average, over all pairs of vertices, of the maximum number of internally disjoint paths connecting these vertices. The connectivity ?(G) ? ( G ) can be seen as the minimum, over all pairs of vertices, of the maximum number of internally disjoint paths connecting these vertices. The connectivity and the average connectivity are upper bounded by the minimum degree ?(G) ? ( G ) and the average degree d  ( G ) of G G , respectively. In this paper the average connectivity of the strong product G1?G2 G 1 ? G 2 of two connected graphs G1 G 1 and G2 G 2 is studied. A sharp lower bound for this parameter is obtained. As a consequence, we prove that ?  ( G 1 ? G 2 ) = d  ( G 1 ? G 2 ) if ?  ( G i ) = d  ( G i ) , i=1,2 i = 1 , 2 . Also we deduce that ?(G1?G2)=?(G1?G2) ? ( G 1 ? G 2 ) = ? ( G 1 ? G 2 ) if ?(Gi)=?(Gi) ? ( G i ) = ? ( G i ) , i=1,2 i = 1 , 2 .
average connectivity strong product of graphs average degree maximally connected graphs