I'm a computer science student working on a research project that deals with computational study of atomic clusters. I'm using a graph based representation of the clusters using a binary connectivity matrix. My question is, are the eigenvalues invariant to a transformation in particle indices? 


For example, if I have a 3 particle cluster with particle 1 attached to 2 and 2 attached to 3. The connectivity matrix will be different if I swapped the particle index even though it is the same structure. My question is will the eigenvalues be the same? I checked it using python and indeed, the values were same, but I'm trying to look for a formal established proof (or an article to cite). 


I really appreciate the help and I apologize if it is a stupid question. 



Question about eigenvalues of connectivity matrices for graphs
If $P$ is a *[permutation matrix](https://en.m.wikipedia.org/wiki/Permutation_matrix)*, i.e., $P$ is binary and doubly stochastic, then $P^\top = P^{-1}$. Thus, any matrix $A$ is co-spectral to $P^\top A P$. Furthermore, if the matrix $A$ is the incidence matrix of a graph, then a straightforward exercise shows that the latter is a relabeling of the indices of the graph.


