An expander graph belongs to a family of graphs of uniformly bounded maximum degree, with vertex
counts tending to infinity, whose edge
expansion constants are bounded below by a positive constant independent of the
vertex count. For a family of 
-regular graphs, a two-sided
spectral expansion condition is 
for every eigenvalue 
of the normalized adjacency
matrix restricted to the subspace orthogonal to
the constant vector, with a fixed 
(Kharlampovich and Vdovina 2026).
Kharlampovich and Vdovina (2026) report that the Aleshin graphs are two-sided expanders with normalized spectral
gap at least 
.
Their argument uses a recent result for the Bellaterra
graphs; the complete chain of dependencies has not been independently verified
here.
