Are these graphs Isomorphic

Question

please consider this two graphes.

G1:

G2:

Are they Isomorphic?

Is G1 a planer graph? It contains a K 3,3 or k5?

thanks alot

Answer 1

These two graphs are not co-spectral. The characteristic polynomial of the adjacency matrix of $G_1$ is $$f(x)=x^8-18x^6-24x^5+43x^4+64x^3-38x^2-40x+21.$$ However, the characteristic polynomial of the adjacency matrix for $G_2$ is $$g(x)=x^8-17x^6-14x^5+32x^4+10x^3-7x^2.$$ Therefore, they cannot be isomorphic.

Both of these graphs contain $K_5$ as a minor. I do not know the best way to describe this, so I will describe the steps needed to see the minor. For $G_1$, label the top vertex with degree 5 $v_1$, and label the rest $v_2$,...,$v_8$ going clockwise from $v_1$. If you contract $v_2$ and $v_4$, then contract $v_3$ and $v_5$, and finally contract $v_6$ and $v_7$, then you should see a $K_5$ minor.

For $G_2$, you just need to remove some edges. Let me describe the star in the $K_5$ minor. Again label the vertices in the same manner: start at the top with $v_1$ and move clockwise until you label the last vertex $v_8$. A star in $G_2$ is the following cycle: $v_1, v_4, v_8, v_3, v_6, v_1$. If you just draw this cycle and the cycle $v_1,...,v_8,v_1$, then you should see the $K_5$ minor.

Answer 2

The answer is no.

In $G_1$ every vertex of degree $4$ is connected with another vertex of degree $4$. In $G_2$ you can find two vertices of degree $4$ which are only connected to vertices of degree $5$.

From here it is easy to conclude that the graphs are not isomorphic, let me know if you need more details.