For each pair of graphs show that they are not isomorphic by

For each pair of graphs, show that they are not isomorphic by showing that there is a property that is preserved under isomorphism which one graph has and the other does not.

Solution

In question (a) The number of edges in the left side graph is 7 but the right side graph contains 8 edges. Thus the necessary condition that the number of edges in these two graphs should be same is failed. Hence these two graphs are not isomorphic.

In question (b) The number of vertices in the left side graph is 5 but the right side graph contains 4 vertices. Thus the necessary condition that the number of vertices in these two graphs should be same is failed. Hence these two graphs are not isomorphic.

In question (c) the left side graph has a vertex (vertex-1) with degree 4. But there is no vertex in right side graph with degree 4. For the graphs to be isomorphic the necessary condition is their degree sequence must coincide. Since this conditionn failed we can conclude that these two graphs given in (c) are also not isomorphic.