On the relationship between shortlex order and $A_{\alpha }$-spectral radii of graphs with starlike branch tree

Let $P(n)$ denote the set of all partitions of $n$, whose elements are nondecreasing sequences of positive integers whose sum is $n$. For $\mathbf{a}=(n_1,n_2,\dots ,n_d)\in P(n)$, let $G(\mathbf{a},v)$ denote the graph obtained from connected graph $G$ appending $d$ paths with lengths $n_1,n_2,\dots ,n_d$ on vertex $v$ of $G$. We show that the ordering of graphs in $G_n(v)=\{G(\mathbf{a},v)\mid \mathbf{a}\in P(n)\}$ by $A_{\alpha }$-spectral radii coincides with the shortlex ordering of $P(n)$.