On Toeplitz graphs being line graphs

Abstract

A Toeplitz graph $T_n \langle t_1,t_2,\ldots,t_k\rangle$ is a simple graph with the vertex set $[n]$ such that two vertices $v$ and $w$ are adjacent if and only if $|v-w| = t_i$ for some $i \in [k]$. In this paper, we investigate line Toeplitz graphs, which are Toeplitz graphs that happen to be line graphs. We first show that for a sufficiently large $n$, the family of claw-free Toeplitz graphs of order $n$ is $T_n \langle t,2t,\ldots,kt\rangle$ for some nonnegative integers $t$ and $k$. Interestingly, this family consists of a union of Toeplitz graphs each of which is isomorphic to a $k$-tree the notion of which was introduced by Patil in 1986. Then we completely characterize $T_n \langle t,2t,\ldots,kt\rangle$ for any positive integer $n$ that is a line graph. Furthermore, we provide a comprehensive description of a line Toeplitz graph $T_n \langle t_1,t_2\rangle$ and $T_n \langle t_1,t_2,t_3\rangle$. In general, line Toeplitz graph seems very challenging to characterize completely. Even for $T_n \langle t_1,t_2,t_3\rangle$, it was not easy to do so. It is also worth mentioning that there is a line Toeplitz graph that is not in the form $T_n \langle t,2t,3t\rangle$.