Are two graphs isomorphic?

Question

Are the two graphs isomorphic?

$$G_1=\begin{bmatrix} a & b & c & d & e & f \\ b & a & a & c & d & a \\ c & c & b & e & f & d \\ f & & d & f & & e \end{bmatrix}\ \quad\quad G_2=\begin{bmatrix} u & v & w & x & y & z \\ v & u & v & u & x & u \\ x & w & x & w & z & w \\ z & & z & y & & y \end{bmatrix}$$

$a,b,c$ creates a triangle in $G_1$, but no triangle is created in $G_2$.

Is that enough to claim that they are not isomorphic?

Answer 1

This is a good example of two graphs with the same degree sequence $(3,3,3,3,2,2)$ which are not isomorphic. As the question states, the existence of the two triangles in $G_1$ and their absence in $G_2$ is proof that the two graphs are not isomorphic.

Here they are in graphical form:

Answer 2

"Isomorphic" does not only mean that the (sorted) degree-sequences must coincide.

For every set of vertices in one graph there must be a corresponding set of vertices in the other graph with the property that two vertices in one set are connected if and only if the corresponding vertices in the other set are connected.

In short, every subgraph must occur also in the other graph, if they are isomorphic.

So, if one graph contains a triangle, quadrilateral, pentagonal and so on, but not the other, they cannot be isomorphic.