  Resolving parameters is a fundamental area of combinatorics with applications
not only to many branches of combinatorics but also to other sciences. In this
article, we construct a class of Toeplitz graphs, and will be denoted by
$T_{2n}(W)$, so that they are Cayley graphs. First, we review some of the
features of this class of graphs. In fact, this class of graphs are vertex
transitive, and by calculating the spectrum of the adjacency matrix related
with them, we show that this class of graphs cannot be edge transitive.
Moreover, we show that this class of graphs cannot be distance regular, and
since the computing resolving parameters of a class of graphs such that are not
distance regular is more difficult, then we regard this as justification for
our focus on some resolving parameters. In particular, we determine the minimal
resolving set, doubly resolving set and strong metric dimension for this class
of graphs.
