The Bellaterra graph at level 
has the 
binary words of length 
as vertices, with edges defined
by three involutions 
.
These fix the empty word and satisfy 
, 
, 
, 
, and 
. All labels and edge
multiplicities are retained. A fixed word contributes
one to the corresponding diagonal entry of the adjacency
matrix, giving row sum three (Kharlampovich and Vdovina 2026).
Kharlampovich and Vdovina (2026) report a graph diameter bound 
,
improving the previous quadratic-in-
bound to the sharp order 
, where 
is big-Theta notation.
Their argument uses their recent Ramanujan result for this family, and the complete
proof dependencies have not been independently verified
here.
