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 $G$.
Categories: math.CO