4.5. Question 4

Try these first

1. 1.

   What is the vertex set of the Cartesian product if G1 has vertices {A, B} and G2 has vertices {1, 2}?

   Hint

   Consider pairwise combinations of vertices.

2. 2.

   Define the degree of vertex (A, 2) if vertex A in G1 has degree 2 and vertex 2 in G2 has degree 3.

   Hint

   Add the degrees of A and vertex 2.

3. 3.

   What does the Cartesian product of two graphs produce?

   • A single graph with no edges
   • Ordered pairs of vertices
   • Two separate vertex sets

   Hint

   Think about how we structure new vertices from existing ones.

4. 4.

   True or False: The edges of the Cartesian product are solely based on edges from the first graph.

   • True
   • False

   Hint

   Remember the conditions for forming edges.

5. 5.

   Given two graphs where G1 has 4 vertices and 2 edges, G2 has 3 vertices and 2 edges, find the Cartesian product of the two graphs including the vertex and edge counts.

   Hint

   Apply the previous edge counting formulas carefully to both graphs.

6. 6.

   Create a unique Cartesian product for G1 with a tree structure and G2 as a cycle graph. Discuss the resulting graph structure and edge counts.

   Hint

   Explore specific characteristics of tree and cycle interactions.