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 $P(n)$.
Categories: math.CO