When we want to compute the multiplicity of an eigenvalue of a 0-1 symmetric matrix (viewed as the adjacency matrix of an undirected regular graph), we commonly resort to the know lemma of Feit and Higman which states the following:

Let $\theta$ be a simple root of the polynomial $p(x)$, and set $p_{\theta}=p(x)/(x-\theta)$. If $M$ is a matrix satisfying $p(M)=0$, then the multiplicity of $\theta$ as an eigenvalue of $M$ is given by $\frac{tr(p_\theta(M))}{p_\theta(\theta)}$, where $tr$ gives the trace.

The complication with this approach is the computation of the traces, especially when the degree of $p(x)$ is much higher than the girth of the corresponding graph. 

If the graph is distance-regular then an alternative (and simpler) approach could be used.

My question is the following:

Is there any other way to compute the multiplicity of an eigenvalue for such a matrix if we know the girth, diameter and degree of the corresponding graph (and possibly other properties)? 

I would appreciate any help I could get in this regard...

Thanks in advance, and regards,
Guillermo