Given a graph $A$ with nodes $p_{1},p_{2},...,p_{n}$, let $\{A^{*}\}$ be the set of all graphs isomorphic to $A$. Consider a graph $Q$ with vertices $a,b,$ and $c$ where there are three edges between $a$ and $b$, one edge between $a$ and $c$, and no edges between $b$ and $c$. Then $Q^{*}\in \{Q^{*}\}$ if and only if it has vertices $d,e,$ and $f$ where there are three edges between two of these nodes, one edge between one of the previous two nodes and the third, and zero edges between the other one of the first two nodes and the third node.
Now, consider the adjacency matrix for a graph $P$. Call it $M(P)$. Then, what conditions may we impose on the relationship between $M(P)$ and $M(A)$ which are necessary and sufficient for saying that $P\in \{A^{*}\}$? Are there results pertaining to this issue? If there are, what are they?