Periodic Vertices

Adjacency Matrix

Periodicity with respect to the adjacency matrix is trickier, in that the eigenvalue support of a periodic vertex may not be integral. In the second case of Theorem 1, the eigenvalues in $S(v)$ are of the form \[\theta_r = \frac{a+b_r\sqrt{\Delta}}{2}\] for some fixed integer $a\ne 0$ and some integer $b_r$. Thus, if $v$ is periodic, then either

1. the polynomial $\psi_v(t)$ splits into linear factors, or
2. the polynomial $\psi_v(t)$ has at most one linear factor, and the rest factors are quadratic with the same coefficient of $t$.

We did the calculation for connected graphs on up to eight vertices. Compared to the Laplacian case, there are a lot fewer examples.

Here are all the examples we have found. Again we store the full data in this file.