Structural properties of Toeplitz graphs

TL;DR

This paper investigates the structural properties of Toeplitz graphs, providing characterizations for chordality, perfection, and regularity, and computing key graph parameters.

Contribution

It offers new characterizations of Toeplitz graphs' chordality, perfection, and regularity, and computes their clique cover numbers.

Findings

Characterization of $K_q$-free Toeplitz graphs.

Conditions for Toeplitz graphs to be chordal and perfect.

Toeplitz graphs are regular if and only if they are circulant.

Abstract

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.