### Abstract

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.

Language | English |
---|---|

Pages | 1-18 |

Number of pages | 18 |

Journal | Discrete Applied Mathematics |

Publication status | Accepted/In press - 21 Jul 2019 |

### Fingerprint

### Keywords

- Toeplitz graph
- word-representable graph
- Riordan graph
- pattern

### Cite this

*Discrete Applied Mathematics*, 1-18.

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*Discrete Applied Mathematics*, pp. 1-18.

**Word-representability of Toeplitz graphs.** / Cheon, Gi-Sang; Kitaev, Sergey; Kim, Jinha; Kim, Minki.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Word-representability of Toeplitz graphs

AU - Cheon, Gi-Sang

AU - Kitaev, Sergey

AU - Kim, Jinha

AU - Kim, Minki

PY - 2019/7/21

Y1 - 2019/7/21

N2 - 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... (of even or odd length) or a word of the form yxyx... (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.

AB - 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... (of even or odd length) or a word of the form yxyx... (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.

KW - Toeplitz graph

KW - word-representable graph

KW - Riordan graph

KW - pattern

UR - https://www.sciencedirect.com/journal/discrete-applied-mathematics

M3 - Article

SP - 1

EP - 18

JO - Discrete Applied Mathematics

T2 - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -