  We are concerned with sequences of graphs having a grid geometry, with a
uniform local structure in a bounded domain $\Omega\subset {\mathbb R}^d$,
$d\ge 1$. We assume $\Omega$ to be Lebesgue measurable with regular boundary
and contained, for convenience, in the cube $[0,1]^d$. When $\Omega=[0,1]$,
such graphs include the standard Toeplitz graphs and, for $\Omega=[0,1]^d$, the
considered class includes $d$-level Toeplitz graphs. In the general case, the
underlying sequence of adjacency matrices has a canonical eigenvalue
distribution, in the Weyl sense, and we show that we can associate to it a
symbol $f$. The knowledge of the symbol and of its basic analytical features
provide many informations on the eigenvalue structure, of localization,
spectral gap, clustering, and distribution type. Few generalizations are also
considered in connection with the notion of generalized locally Toeplitz
sequences and applications are discussed, stemming e.g. from the approximation
of differential operators via numerical schemes.
