On local spectra of classical distance-regular graphs with $b = 2$

Abstract

Using a combination of spectral techniques and combinatorial counts, we prove a number of non-existence results for putative distance-regular graphs with classical parameters, focusing on the case $b=2$. Among those, the case $(d,b,α,β)=(d,2,1,2^d-2)$ passed all known feasibility conditions and required a deeper combinatorial analysis, from which nonexistence has been derived whenever $d \ge 9$. In doing so, we constructed a new family of regular graphs with four distinct eigenvalues which would have been local graphs in such a distance-regular graph.