Strongly Cospectral Vertices


Another fact about perfect state transfer is that it can only take place between strongly cospectral vertices, which we will define soon. Suppose there is perfect state transfer between vertex $u$ and vertex $v$ at time $\tau$ with respect to the matrix $H$. Then \[exp(i\tau H) e_u = \gamma e_v\] for some phase factor $\gamma$. Using spectral decomposition, \[\sum_r e^{i\tau \theta_r}E_r e_u = \gamma e_v.\] Premutiply both sides by $E_r$ and we have \[e^{i\tau \theta_r} E_r e_u = \gamma E_r e_v\] for each eigenvalue $\theta_r$. Since both $E_r$ and $e_v$ are real and $|\gamma|=1$, moving phase factors to one side yields \[E_r e_u = \pm E_r e_v\] for all $r$. Formally, if two vertices satisfy the above equation for all eigenvalues, then we say they are strongly cospectral to each other.

If $u$ and $v$ are strongly cospectral, then they are cospectral. This is because for all $r$, \[(E_r)_{u,u} = e_u^T E_r e_u = (\pm) e_u^T E_r e_v = (\pm) e_v^T E_r e_u = (\pm) (\pm) e_v^T E_r e_u = (E_r)_{v,v}.\] However, being strongly cospectral is more than just being cospectral. It also requires that for each idempotent $E_r$, the $u$-th column is a scalar multiple of the $v$-th column. We say two vertices $u$ and $v$ are parallel if \[E_r e_u = \alpha_r E_r e_v\] for all $r$. It is not hard to prove the following.

(Characterizing strongly cospectrality) Two vertices $u$ and $v$ are stronly cospectral if and only if they are cospectral and parallel.

We consider strongly cospectral vertices in graphs and trees. Here is a list of our computational results, with respect to the adjacency matrix only. The data is in .sobj format.

  1. Cospectral orbits, for trees on up to 15 vertices. data
  2. Cospectral orbits which are also cospectral in the complements, for trees on up to 15 vertices. data
  3. Strongly cospectral vertices, for trees on up to 14 vertices. data
  4. Vertices that are both strongly cospectral and periodic, for trees on up to 15 vertices. data   figures
  5. Lists of at least three vertices that are strongly cospectral to each other, for connected graphs on up to 8 vertices. data

The following questions remain unanswered.

  1. Is there a tree with at least three strongly cospectral vertices?