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{\bf M.A. Fiol}
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{\bf Quasi-Spectral Characterization of Strongly \hfil\break
Distance-Regular Graphs }
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A graph $\Gamma$ with diameter $d$ is strongly distance-regular if $\Gamma$
is distance-regular and its distance-$d$ graph $\Gamma _d$ is strongly
regular. The known examples are all the connected
strongly regular graphs (i.e. $d=2$), all the antipodal
distance-regular graphs, and some distance-regular graphs with
diameter $d=3$.
The main result in this paper is a
characterization of these graphs (among regular graphs with $d$
distinct eigenvalues), in terms of the eigenvalues, the sum of the
multiplicities corresponding to the eigenvalues with (non-zero)
even subindex, and the harmonic mean of the degrees of the
distance-$d$ graph.
\bye