2019/07/22 by Gi‐Sang Cheon, Jinha Kim, Cheon, Gi-Sang +5
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1907.09152
openalex publication_date 2019/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.