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We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graphis strongly regular. We provide a characterization of such graphs $\Gamma$(among regular graphs with few distinct eigenvalues) in terms of the spectrumand the mean number of vertices at maximal distance $d$ from every vertex,where $d+1$ is the number of different eigenvalues of $\Gamma$. This can beseen as a another version of the so-called spectral excess theorem, whichcharacterizes in a similar way those regular graphs that are distance-regular.

The spectral excess theorem for graphs with few eigenvalues whose distance-$2$ or distance-$1$-or-$2$ graph is strongly regular