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The spectral excess theorem for graphs with few eigenvalues whose distance- 2 or distance-1-or-2 graph is strongly regular Dalfó, Cristina Fiol Mora, Miguel Ángel Koolen, Jack Distance-regular graph Distance-2 graph Spectrum Predistance polynomials We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d+1 is the number of different eigenvalues of Γ. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular. Research of C. Dalfó and M. A. Fiol is partially supported by Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR) under project 2017SGR1087. Research of J. Koolen is partially supported by the National Natural Science Foundation of China under project No. 11471009, and the Chinese Academy of Sciences under its ‘100 talent’ programme. The research of C. Dalfó has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922. 2019-11-11T11:45:09Z 2019-11-11T11:45:09Z 2018-07-13 2019-11-11T11:45:09Z info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion Versió postprint del document publicat a: https://doi.org/10.1080/03081087.2018.1491944 Linear & Multilinear Algebra, 2019, vol. 67, num. 12, p. 2373-2381 info:eu-repo/grantAgreement/EC/H2020/734922/EU/CONNECT (c) Taylor & Francis, 2019 info:eu-repo/semantics/openAccess Taylor & Francis