Quasi-Spectral Characterization of Strongly Distance-Regular Graphs

M. A. Fiol

Abstract


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.


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