We derive a new upper bound for the diameter of akk-regular graphGGas a function of the eigenvalues of the adjacency matrix. Namely, suppose the adjacency matrix ofGGhas eigenvaluesλ1,λ2,…,λn{\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}with|λ1|≥|λ2|≥⋯≥|λn|\left | {{\lambda _1}} \right | \geq \left | {{\lambda _2}} \right | \geq \cdots \geq \left | {{\lambda _n}} \right |whereλ1=k{\lambda _1} = k,λ=|λ2|\lambda = \left | {{\lambda _2}} \right |. Then the diameterD(G)D(G)must satisfy\[D(G)≤⌈log⁡(n−1)/log(k/λ)⌉D(G) \leq \left \lceil {\log (n - 1)/{\text {log}}(k/\lambda )} \right \rceil\]. We will consider families of graphs whose eigenvalues can be explicitly determined. These graphs are determined by sums or differences of vertex labels. Namely, the pair{i,j}\left \{ {i,j} \right \}being an edge depends only on the valuei+ji + j(ori−ji - jfor directed graphs). We will show that these graphs are expander graphs with small diameters by using an inequality on character sums, which was recently proved by N. M. Katz.