Geodesic
The spectral excess theorem for distance-regular graphs having distance-$d$ graph with fewer distinct eigenvalues
Journal of Algebraic Combinatorics, Tome 43 (2016) no. 4, pp. 827-836.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $\Gamma $ be a distance-regular graph with diameter $d$ and Kneser graph $K=\Gamma _d$, the distance-$d$ graph of $\Gamma $. We say that $\Gamma $ is partially antipodal when $K$ has fewer distinct eigenvalues than $\Gamma $. In particular, this is the case of antipodal distance-regular graphs ($K$ with only two distinct eigenvalues) and the so-called half-antipodal distance-regular graphs ($K$ with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with $d+1$ distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance $d$ from every vertex. This can be seen as a more general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.
Classification : 05C12, 05C50
Keywords: distance-regular graph, Kneser graph, partial antipodality, spectrum, predistance polynomials 
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     author = {Fiol, M. A.},
     title = {The spectral excess theorem for distance-regular graphs having distance-$d$ graph with fewer distinct eigenvalues},
     journal = {Journal of Algebraic Combinatorics},
     pages = {827--836},
     publisher = {mathdoc},
     volume = {43},
     number = {4},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2016__43_4_a4/}
}
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Fiol, M. A. The spectral excess theorem for distance-regular graphs having distance-$d$ graph with fewer distinct eigenvalues. Journal of Algebraic Combinatorics, Tome 43 (2016) no. 4, pp. 827-836. http://geodesic.mathdoc.fr/item/JAC_2016__43_4_a4/