  In this paper, we study structural properties of Toeplitz graphs. We
characterize $K_q$-free Toeplitz graphs for an integer $q \ge 3$ and give
equivalent conditions for a Toeplitz graph $G_n\langle t_1, t_2,\ldots,
t_k\rangle$ with $t_1<\cdots<t_k$ and $n \ge t_{k-1}+t_{k}$ being chordal and
equivalent conditions for a Toeplitz graph $G_n\langle t_1,t_2 \rangle$ being
perfect. Then we compute the edge clique cover number and the vertex clique
cover number of a chordal Toeplitz graph. Finally, we characterize the degree
sequence $(d_1,d_2,\ldots,d_n)$ of a Toeplitz graph with $n$ vertices and show
that a Toeplitz graph is a regular graph if and only if it is a circulant
graph.
