Theory Sublime

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Introduction to graph products

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A directed graph G is a tuple G=(V(G), E(G)) where V is a finite nonempty set called the set of vertices of G and E is a finite subset of V \times V called the set of arcs of G. A graph is called undirected, if its arcs are not ordered pairs but two-element sets of vertices.

For undirected graphs G and H, define G \boxtimes H as the graph with V(G \boxtimes H) = V(G) \times V(H) and edge set defined as follows: (uv, wx) is an edge in G \boxtimes H iff u=w or (u,w) is an edge in G, and v=x or (v,x) is an edge in H. This operation is called the Strong Product (aka Shannon product) of graphs.

Strong product of two edges

Strong product of two edges

For undirected graphs G and H, define G \times H as the graph with vertex set V(G \times H) = V(G) \times V(H) and edge set defined as follows: (uv, wx) is an edge in G \times H iff (u,w) is an edge in G, and (v,w) is an edge in H. This operation is called the Direct Product (aka Categorical product) of graphs.

Direct product of two arcs, each with loops on both its vertices

Direct product of two arcs, each with loops on both its vertices

For directed graphs G and H, define G \circ H as the graph with the vertex set V( G \circ H ) = V(G) \circ V(H) and edge set defined as follows: (uv, wx) is an edge in G \circ H iff either (u,w) is an edge in G or (v,x) is an edge in H. This is called the OR Product (aka Sperner product) of graphs.

For undirected graphs G and H, define G \Box H as the graph with vertex set V(G \Box H) = V(G) \times V(H) and edge set defined as follows: (uv,wx) is an edge in G \Box H iff either u=w and (v,x) is an edge in H or (u,w) is an edge in G and v=x. This is known as the Cartesian Product of graphs.

I will discuss more about these interesting products later.

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Written by Vanessa

July 4, 2009 at 3:22 am

Posted in Graph theory

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