I was thinking about the definition for two graphs being isomorphic:
Basically considering two graphs $\mathcal{G}, \, \mathcal{S}$ they are known to be isomorphic if they are essentially identical in terms of their structure. In other words, denoting $A_{\mathcal{G}}, \, A_{\mathcal{S}}$ we have that
$$A_{\mathcal{S}} = PA_{\mathcal{G}}P^T$$
for some permutation matrix $P$.
This is essentially clear but I don't understand well why we would express the above thing that way, what is the meaning of applying the operation $P (\cdot)P^T$. What is the sense of applying $P$ to $A_{\mathcal{G}}$ and then applying again $P^T$?
Thank you!
Note that $P$ represents a permutation $\pi\colon \{1,\ldots,n\}\to \{1,\ldots,n\} $ and also that $P^T=P^{-1}$. If $\pi$ is the permutation that corresponds to how the vertices of $\mathcal G$ are identified with those of $\mathcal S$, then $PA_{\mathcal G}P^{T}$ maps some base vector $e_{\pi(k)}$ to $e_k$, then to the sum of all neighbours $e_{j_1},\ldots,e_{j_d}$ of $e_k$, and finally to the sum of all neigbours $e_{\pi(j_1)},\ldots,e_{\pi(j_d)}$ of $e_{\pi(k)}$, as desired.