Problem

Solutions For Applied Combinatorics Chapter 1.2 Problem 10E

Step-by-Step Solution

Solution 1

Step #1 of 3

Two graphs must have the same number of vertices and the same number of edge.

The objective is to determine pairs of graph are isomorphic or not.

Step #2 of 3

If consider the direction of these two graphs then determine pairs of graph are isomorphic or not.

$C:\Users\Admin\AppData\Local\Microsoft\Windows\INetCache\Content.Word\3.png$

Two graphs must have the same number of vertices and the same number of edge. The above two graphs pass this initial test. Both graph have vertices and edges.

Let us examine the degrees of the different vertices.

From above figure, the degree of and degree and the degree of and degree .

Then the two graphs have the same number of vertices of degree and degree .

Now, start with vertex in the left graph. By rotational symmetry, match to any vertex in the right graph (that is, if the two graphs are isomorphic, there will exist an isomorphism with matched to any vertex in the right graph).

Let use the match

The set of neighbors of (vertices adjacent to ) must be matched with the set of neighbors of .

So, the set of neighbors of and is and respectively.

Similarly, the set of neighbors and is and . At vertex in-degree and out-degree arrow are present but ay vertex only and in-degree arrow are present

Hence, pairs of graph are not isomorphic.

Step #3 of 3

The objective is to determine pairs of graph are isomorphic or not.

$C:\Users\Admin\AppData\Local\Microsoft\Windows\INetCache\Content.Word\4.png$

Two graphs must have the same number of vertices and the same number of edge. The above two graphs pass this initial test. Both graph have vertices and edges.

Let us examine the degrees of the different vertices.

From above figure, the degree of and degree , but in the left graph the in-degree and the out-degree vertex are alternate but in the right graph the in-degree and the out-degree vertex are in continuity.