Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.06337 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Let $G=(V,E)$ be a simple and connected graph. A $h$-order invariant of $G$ based on the path sequence is defined from a set of real numbers ${f(x_{0},x_{1},\cdots,x_{h})}$ as $^{h}I_f(G)=\sum\limits_{v_{0}v_{1}v_{2}\cdots v_{h}}f\left(d_{0},d_{1},\cdots,d_{h}\right)$, where the sum runs over all paths $v_{0}v_{1}v_{2}\cdots v_{h}$ of length $h$ and $d_{i}$ is the degree of vertex $v_i$ in $G$. In this paper, we first show that the $h$-order invariant of a starlike tree $S_{n}$ can be determined completely by its branches whose length does not exceed $h$. And then we find conditions on the function $f$ for some graph families $\mathcal{G}$ such that any graph $G\in\mathcal{G}$ can be determined by the higher order invariants $^{h}I_f(G)$ for $0\leqslant h\leqslant ρ$, where $ρ$ is the length of a longest path in $G$.