  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.
