On Toeplitz graphs being line graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913404981084160 |
|---|---|
| author | Cheon, Gi-Sang Kang, Bumtle Kim, Suh-Ryung Mojallal, Seyed Ahmad Ryu, Homoon |
| author_facet | Cheon, Gi-Sang Kang, Bumtle Kim, Suh-Ryung Mojallal, Seyed Ahmad Ryu, Homoon |
| contents | 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 $T_n \langle t,2t,3t\rangle$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_05317 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Toeplitz graphs being line graphs Cheon, Gi-Sang Kang, Bumtle Kim, Suh-Ryung Mojallal, Seyed Ahmad Ryu, Homoon Combinatorics 05C75, 05C76 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 $T_n \langle t,2t,3t\rangle$. |
| title | On Toeplitz graphs being line graphs |
| topic | Combinatorics 05C75, 05C76 |
| url | https://arxiv.org/abs/2201.05317 |