Does the order matter in the adjacency matrix?
Let $f$ be an isomorphism from graph $G$ to graph $G'$ which takes vertex $v_i$ to vertex $v_i'$.  Let $A$ and $A'$ be the adjacency matrix of $G$ and $G'$, respectively.  If the rows and columns of $A$ are in the order $v_1,\ldots,v_n$, and the rows and columns of $A'$ follow the same order $v_1', \ldots, v_n'$, then it can be seen that $A=A'$.  

If $f$ is an isomorphism from $G$ to itself, then $f$ is called an automorphism of $G$.  Thus, $f$ is a permutation of $V(G)$ which maps edges to edges and nonedges to nonedges.  If we permute the rows of the adjacency matrix $A$ by $f$, and if we apply the same permutation to the columns of $A$, we get back the same matrix $A$. Equivalently, a permutation $f$ of $V(G)$ is an automorphism of $G$ if and only if the permutation matrix $P$ corresponding to $f$ satisfies the equation $P^{-1}AP=A$.  Recall that premultiplying a matrix $A$ effects row operations/permutations and postmultiplication effects column permutations.

So, graphs with the same structure (i.e. isomorphic graphs) have the same adjacency matrix (up to a permutation of the rows and columns - here, the same permutation must be applied to both the rows and the columns).  The graph invariants which we study are the same for isomorphic graphs and so the ordering of vertices you choose for the adjacency matrix shouldn't matter.