Diameters and Eigenvalues

We derive a new upper bound for the diameter of a -regular graph as a function of the eigenvalues of the adjacency matrix. Namely, suppose the adjacency matrix of has eigenvalues <!-- MATH ${\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}$ --> with <!-- MATH $\left| {{\lambda _1}} \right| \geq \left| {{\lambda _2}} \right| \geq \cdots \geq \left| {{\lambda _n}} \right|$ --> where <!-- MATH ${\lambda _1} = k$ --> , <!-- MATH $\lambda = \left| {{\lambda _2}} \right|$ --> . Then the diameter must satisfy <!-- MATH \begin{displaymath} D(G) \leq \left\lceil {\log (n - 1)/{\text{log}}(k/\lambda )} \right\rceil \end{displaymath} -->