# Word-representability of Toeplitz graphs

## TL;DR

This paper explores the word-representability of Toeplitz graphs, a class of Riordan graphs, establishing conditions for their representability, constructing examples, and pioneering the study of infinite word-representable graphs.

## Contribution

It introduces the concept of word-representability for Toeplitz graphs, merges Riordan matrix theory with graph representation, and initiates the study of infinite word-representable graphs.

## Key findings

- Several classes of Toeplitz graphs are proven to be word-representable.
- A method for constructing non-word-representable Toeplitz graphs is provided.
- First examples and discussion of infinite word-representable graphs are presented.

## Abstract

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.