There are two isomorphic graphs for which I want to find their spectrum(their eigenvalues). I am confused that the spectrum will be same for both the graphs since they are isomorphic, but I am not sure.
I first wrote the adjacency matrix for both and they are also same, unless I did something wrong. The graphs have color coded nodes so when I did the adjacency matrices for both, I wrote the nodes in the color-coded manner (like mapping the nodes from first graph to another graph for row and column of adjacency matrix, and hence, the values had to be the same). But I also did the adjacency matrix for second graph according to the number-labels given to nodes, and then the answer had to be different because now the alignment is different.
I mean if they are isomorphic then their adjacency matrices will be same or not? If yes, which sounds obvious because of their property of isomorphism, then why is the question asking for solving adjacency and spectrum for both graphs?
(I watched youtube videos and read pages online but I still have doubts if I am solving this correctly.)
These concepts are new to me, so maybe I am missing on something important here. Please if someone can simplify this or point where I am doing wrong in this problem, that would be much appreciated.
For reference, here is an image of graphs->isomorphic graphs
Adjacency matrices for the graphs.solution
Best Answer
Let $G_1=(V_1,E_1)$ and $G_2=(V_2, E_2)$ be two isomorphic graphs.
If $V_1= \{ v_1,.., v_n\}$ and $V_2=\{w_1,..,w_n\}$ and $f: V_1 \to V_2$ is the isomorphism, you can define a permutation $\sigma$ via $$f(v_i)=w_{\sigma(i)}$$
Now, if $P:=P_\sigma$ is the corresponding permutation matrix, then you have $$A_1=PA_2 P^{-1}$$ from where you can deduce that $A_1,A_2$ have the same spectrum.