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arXiv · 1806.00789

On majorization of closed walks vector of trees with given degree sequences

Ya-Hong Chen·Daniel Gray·Ya-Lei Jin·Xiao-Dong Zhang
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Abstract

Let $C_{v}(k;T)$ be the number of the closed walks of length $k$ starting at vertex $v$ in a tree $T$. We prove that for a given tree degree sequence $π$, then for any tree with degree sequence $π$, the sequence $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence $C(k, T_π^*)\equiv C(k, T_π^*, v\in V(T^*))$, where $T_π^*$ is the greedy tree corresponding to $π$. In addition, for two trees degree sequences $π,~π'$, if $π$ is majorized by $π'$, then $C(k;T_π^*)$ is weakly majorized by $C(k;T_{π'}^*)$.

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Ya-Hong Chen, Daniel Gray, Ya-Lei Jin, Xiao-Dong Zhang. 2018-06-03. On majorization of closed walks vector of trees with given degree sequences. https://arxiv.org/abs/1806.00789

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