Row graphs of Toeplitz matrices

TL;DR

This paper investigates the structure of row graphs of Toeplitz matrices, especially focusing on triangle-free cases and (0,1)-Toeplitz matrices with specific row graph components, providing structural characterizations and classifications.

Contribution

It characterizes the row graphs of Toeplitz matrices with triangle-free structures and fully classifies (0,1)-Toeplitz matrices based on their row graph components.

Findings

Triangle-free row graphs have maximum row sum at most 2.

Such row graphs are disjoint unions of paths and cycles.

Complete characterization of (0,1)-Toeplitz matrices with cycle row graphs.

Abstract

In this paper, we study row graphs of Toeplitz matrices. The notion of row graphs was introduced by Greenberg et al. in 1984 and is closely related to the notion of competition graphs, which has been extensively studied since Cohen had introduced it in 1968. To understand the structure of the row graphs of Toeplitz matrices, which seem to be quite complicated, we have begun with Toeplitz matrices whose row graphs are triangle-free. We could show that if the row graph G of a Toeplitz matrix T is triangle-free, then T has the maximum row sum at most 2. Furthermore, it turns out that G is a disjoint union of paths and cycles whose lengths cannot vary that much in such a case. Then we study (0, 1)-Toeplitz matrices whose row graphs have only path components, only cycle components, and a cycle component of specific length, respectively. In particular, we completely characterize a (0, 1)-Toeplitz matrix whose row graph is a cycle.