  A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$
is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$
around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said
to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and
Peled studied HR-graphs of level 2 and provided several methods to construct
families of graphs which are expanders "globally and locally". They ask whether
such HR-graphs of level 3 exist.
  In this paper we show how the theory of Coxeter groups, and abstract regular
polytopes and their generalisations, can lead to such graphs. Given a Coxeter
system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients
of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$,
which form an infinite family of expander graphs when $(W,S)$ is indefinite and
$\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in
this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion
stems from applying superapproximation to the congruence subgroups of the
linear group $W$.
  This machinery gives a rich collection of families of HR-graphs, with various
interesting properties, and in particular answers affirmatively the question
asked by Chapman, Linial and Peled.
