drawing non-isomorphic graphs

Question

I do understand that isomorphic means that they must have the same edges, vertices and adjacency must preserve. Can anyone please just draw a simple example with an explanation. Thanks

Answer 1

Take two disjoint triangles. Connect each endpoint on the triangle to exactly one distinct endpoint on another triangle. I.e., have a bijection on the vertices of these triangles.

Another such graph is $K_{3, 3}$.

The first graph is planar. The second isn't.

Answer 2

Draw a hexagon and label the vertices 1,2,3,4,5,6 clockwise. To make the graph 3-regular, we have to fill in some diagonals. First connect each vertex to the opposite vertex (1 to 4, 2 to 5, 3 to 6). It's easy to check that this graph does not contain a triangle (no vertex has two neighbors which are connected to each other).

Now draw another hexagon and add diagonals any other way (for example 1 to 4, 2 to 6, 3 to 5). This graph contains triangles (for example 1,2,6). So the two graphs can't be isomorphic.