Can we say anything about the order of the second largest eigen-value?

Question

Suppose we have a vertex-transitive graph ($G$) with degree $n$ and the number of vertices $N$.

Is it possible to say anything about the exact order of $\frac{1}{n-\lambda _2}$ in terms of $N$ and $n$? where $\lambda_2$ is the second largest eigen value of the adjacency matrix of $G$.

Thanks

Answer 1

What do you mean by exact order?

You cannot expect to get the exact value of your expression from $n$ and $N$ alone. Indeed there are cubic vertex transitive graphs of order $10$ that have a differ value of your expression.

Do you have any other structural properties at hand?

For Cayley graphs (and in general for vertex transitive graphs) there are formulas for their eigenvalues so perhaps it makes sense for you to look in this direction.