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

Abstract

Let $P(n)$ denote the set of all partitions of $n$, whose elements are nondecreasing sequences of positive integers whose sum is $n$. For ${\bf a}=( n_{1}, n_{2},\ldots, n_{d}) \in P(n)$, let $G({\bf a},v)$ denote the graph obtained from connected graph $G$ appending $d$ paths with lengths $n_{1},n_{2},\ldots,n_{d}$ on vertex $v$ of $G$. We show that the ordering of graphs in $G_{n}(v)=\{ G({\bf a},v) \mid {\bf a} \in P(n) \}$ by $A_\alpha$-spectral radii coincides with the shortlex ordering of $P(n)$.