I look at sparsly populated matrices $W_1$ with entries $0$ and $1$ only. $W_1$ has a graph-theoretical background. I'll give the Definition 2.2:
We define the 0,1 edge matrix $W_1$ by orienting the $m$ edges of [the graph] $X$ and labeling them as in formula (2.1). Then $W_1$ is the $2m×2m$ matrix with $ij$ entry $1$ if edge $e_i$ feeds into $e_j$ provided that $e_j\neq e^{−1}_i$, and $ij$ entry $0$ otherwise. By “$a$ feeds into $b$,” we mean that the terminal vertex of edge $a$is the same as the initial vertex of edge $b$.
My graphs under inspection have $n$ vertices and are bicubic, so $W_1$ will have two entries per row. I gave a procedure to calculate $W_1$ here and now I'm trying to simplify the determination of the eigenvalues of $W_1$.
From numerical experiments, I assume that all my eigenvalues $\lambda_k$ are:
- $0$ and $\pm1$ or
- $\pm\sqrt2\exp(\pm i2\pi c)$ or
- $\pm\sqrt2\exp(\pm 2\pi c)$,
where $|c|\leq \frac{\log2}{4\pi}$. Can one make use of these facts? Maybe like a symmetry? Can we, for example group all complex eigenvalues on the unit circle, such that it gives a scaled unitary block diagonal structure?
Further, all four possible values of the second class are always available. This is due to the fact that the characteristic polynomial is real and of even degree, as Paul pointed out in a comment.
Fun fact(?): The eigenvalues of the third class, only appeared in example graph s, that were not ramanujan.
Since the number of eigenvalues $+1$ and $-1$ are at least $k_1=r(G)-1$ (it is the inverse Ihara $\zeta$ function), I know that the characteristic polynomial will be of the form: $$ {x^{k_0}} {(x^2-1)^{k_1}} \prod_\color{cyan}{\text{r}}(x^4-4x^2\cos(4\pi c_\color{cyan}{\text{r}}) +4) \underbrace{\prod_\color{red}{\text{n}}(x^4-4x^2\cosh(4\pi c_\color{red}{\text{n}})+4)}_{\text{only for $\color{red}{\text{n}}$on $\color{cyan}{\text{r}}$amanujan graphs?}}, $$
with $k_0=n^2-2m=n(n-3)$.
Any help or hint appreciated...
