Higher order invariants of a graph based on the path sequence

TL;DR

This paper introduces higher order invariants of graphs based on path sequences, showing they can uniquely determine certain graph families like starlike trees and potentially distinguish graphs within those families.

Contribution

It defines higher order invariants based on path sequences and establishes conditions under which these invariants uniquely identify graphs within specific families.

Findings

Higher order invariants of starlike trees depend on branch lengths up to h

Conditions are found for invariants to uniquely determine graphs in certain families

Invariants can distinguish graphs based on longest path length

Abstract

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 $0\leqslant h\leqslant \rho$, where $\rho$ is the length of a longest path in $G$.