Problem

Solutions For Applied Combinatorics Chapter 1.2 Problem 1E

Step-by-Step Solution

Solution 1

To list all nonisomorphic undirected graphs with 4 vertices, do the following. First, note that an undirected graph with 4 vertices can have at most 6 edges. Now consider the cases where number of edges range from 0 to 6. For each case, draw all possible graphs, and check if any are isomorphic to any other. Keep only one graph from each set of isomorphic graphs.

Step #2 of 6

Case 1: 0 edges. Clearly only one graph is possible, as there are no edges. Hence the graph is

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-1.png$

Case 2: 1 edge. Again, this one edge only creates one possible graph. The graph is

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-2.png$

Step #3 of 6

Case 3: 2 edges. In this case, either the edges can share a vertex, or not. This create two cases. They are:

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-3.png$

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-4.png$

Step #4 of 6

Case 4: 3 edges. In this case, either they form a triangle. Or, they form a path. These are the only two possible cases. They are:

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-6.png$

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-5.png$

Step #5 of 6

Case 5: 4 edges. This case is similar to Case 3, as the 2 non-edges can be thought of as edges from Case 3. Thus, there are two possible graphs. They are:

$D:\Dropbox\Chegg TBS\514969-Applied Combinatorics\Solutions\1.2-1E-8.png$