Final answer:
The tree-width of a finite graph is at least its minimum degree. However, this is not true for infinite graphs.
Step-by-step explanation:
The tree-width of a finite graph is defined as the minimum size of a tree decomposition of the graph. A tree decomposition is a collection of subsets of vertices in the graph, called bags, with certain properties. The width of the tree decomposition is the maximum size of any bag minus one.
Now, let's show that the tree-width of a finite graph is at least its minimum degree. Let G be a finite graph with a minimum degree of k. Consider a tree decomposition of G with bags B_1, B_2, ..., B_n.
Since each vertex in G has a minimum degree of k, each bag in the tree decomposition must contain at least k+1 vertices. Therefore, the width of the tree decomposition is at least k, which means the tree-width of the graph G is at least k.
For infinite graphs, this statement is not necessarily true. In infinite graphs, the degree of a vertex can be arbitrarily large, even if the graph has finite tree-width. Therefore, we cannot conclude that the tree-width of an infinite graph is always at least its minimum degree.