Are two graphs isomorphic? 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?

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:

