  Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$
all letters but the copies of $x$ and $y$ we either obtain a word of the form
$xyxy\cdots$ (of even or odd length) or a word of the form $yxyx\cdots$ (of
even or odd length). A graph $G=(V,E)$ is word-representable if there exists a
word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$
if and only if $xy$ is an edge in $E$.
  In this paper we initiate the study of word-representable Toeplitz graphs,
which are Riordan graphs of the Appell type. We prove that several general
classes of Toeplitz graphs are word-representable, and we also provide a way to
construct non-word-representable Toeplitz graphs. Our work not only merges the
theories of Riordan matrices and word-representable graphs via the notion of a
Riordan graph, but also it provides the first systematic study of
word-representability of graphs defined via patterns in adjacency matrices.
Moreover, our paper introduces the notion of an infinite word-representable
Riordan graph and gives several general examples of such graphs. It is the
first time in the literature when the word-representability of infinite graphs
is discussed.
