Abstract A segment of a tree T is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of T form its segment sequence, in analogy to the degree sequence. For a connected graph G = ( V ( G ) , E ( G ) ) , the cover cost (resp. reverse cover cost) of a vertex u in G is defined as C C G ( u ) = ∑ v ∈ V ( G ) H u v (resp. R C G ( u ) = ∑ v ∈ V ( G ) H v u ), where H u v is the expected hitting time for random walk beginning at u to visit v . In this paper, the unique tree with the minimum cover cost and minimum reverse cover cost among all trees with given segment sequence are characterized. Furthermore, the unique tree with the maximal reverse cover cost among all trees with given segment sequence are also identified.

中文翻译：

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给定段序列树的极值覆盖成本和反向覆盖成本

摘要 树T的一个段是一条路径，其末端顶点的度数为1或至少为3，而所有内部顶点的度数为2。T的所有段的长度构成它的段序列，类似于度数序列。对于连通图 G = ( V ( G ) , E ( G ) ) ，G 中顶点 u 的覆盖成本（相应的反向覆盖成本）定义为 CCG ( u ) = ∑ v ∈ V ( G ) H uv (resp. RCG ( u ) = ∑ v ∈ V ( G ) H vu )，其中 H uv 是从 u 开始的随机游走访问 v 的预期命中时间。本文对给定段序列的所有树中覆盖成本和反向覆盖成本最小的唯一树进行了表征。此外，还识别了具有给定段序列的所有树中具有最大反向覆盖成本的唯一树。