# Visual analysis of adjacency matrices

Given a simple graph $$G$$ and its adjacency matrix $$A$$, one may naturally consider the following graph $$A_G$$:

$$V(A_G)=\{(i,j):i

$$E(A_G)=\{\{(i,j),(k,l)\}\in2^{V(A_G)}:(|i-k|=1\text{ and }j=l)\text{ or }(i=k\text{ and }|j-l|=1)\}$$

Here's an example showing how $$A_G$$ can be used. Let $$G$$ be a tree and $$V(G)=\{1,...,n\}$$. If $$A_G$$ is a path, then one may easily show that the set $$\{|i-j|:\{i,j\}\in E(G)\}$$ has $$n-1$$ elements. The latter implies $$G$$ is graceful. On the other hand, we can prove that $$G$$ is a caterpillar iff we can write $$A$$ in such way that $$A_G$$ becomes a path. This gives us another proof of the well-known result that all caterpillars are graceful.

Informally, $$A_G$$ is a picture prescribed to $$G$$, and one may wish to study its symmetries (that is $$\text{Aut} A_G$$). My question is whether it was studied before how properties of $$G$$ correlate with properties of $$A_G$$. I'll be also glad if you point out some other examples.