Department of Mathematics Elementary Graph Theory Lecture 4
4. Cartesian product
The cartesian product G1 G 2 of two graphs G1 and G2 is a graph whose
vertex set V(G1 G 2 ) V1 V2 and two vertices u (u1 , v1 ) and v (u 2 , v 2 )
are adjacent in G1 G 2 if either [ u1 adjacent with u 2 in G1 and v1 v 2 in G 2
] or [ u1 u 2 in G1 and v1 adjacent with v 2 in G 2 ].
Some properties of G1 G 2
1. p(G1 G 2 ) p1p 2 .
2. q(G1 G 2 ) p1q 2 p 2q1 .
3. (G1 G 2 ) 1 2 .
4. (G1 G 2 ) 1 2 .
5. G1 G 2 G 2 G1 .
Problem: Prove that q(G1 G 2 ) p1q 2 p 2q1 .
Proof: Let w (u, v) be any vertex in G1 G 2 , then
5. Tensor product ( Kronecker product )
The tensor product G1 G 2 of two graphs G1 and G2 is a graph whose
vertex set V(G1 G 2 ) V1 V2 and two vertices u (u1 , v1 ) and v (u 2 , v 2 )
are adjacent in G1 G 2 if [ u1 adjacent with u 2 in G1 and v1 adjacent with v 2
in G 2 ].
Some properties of G1 G 2
1 p(G1 G 2 ) p1p 2 . 2 q(G1 G 2 ) 2q1q 2
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