A hierarchical graph is a kind of self-similar network, which is widely discussed and has a wide range of applications. In this paper, we introduce a class of weighted hierarchical graphs H (n, k), which is constructed using the hierarchical product of complete graphs. The weighted hierarchical graph depends on two parameters: the weighted factor r \((0<r\le 1)\) and the number k \((k\ge 3)\) of nodes of the complete graph \(K_k\). In addition, we also discuss the trapping problem and the average trapping time (ATT) for a weight-dependent walk on a weighted hierarchical graph. Based on the special structure of the weighted graph, we derived the exact expression of the ATT for a weighted hierarchical graph. The results show that the ATT obeys a power-law function with the exponent \(\frac{\mathrm{ln}(kr)}{\mathrm{ln}k}<1\), which is related to the factors k and r and indicate that for \(kr=k(r=1)\) and \(1<kr<k\), the ATT has a linear and sublinear relationship with the network order, respectively. Therefore, the smaller kr is, the more efficient the trapping process of the weighted hierarchical graph is. Compared with the unweighted hierarchical graph, the weighted hierarchical graph has a higher efficiency in the trapping process.

层次图是一种自相似网络，已经被广泛讨论并具有广泛的应用。在本文中，我们介绍了一类加权层次图H（n，  k），它是使用完整图的层次积构建的。权重分层图取决于两个参数：权重因子r \（（0 <r \ le 1）\）和完整图\（K_k \）的节点数k \（（k \ ge 3）\）。。此外，我们还讨论了加权层次图上与重量有关的步行的捕获问题和平均捕获时间（ATT）。基于加权图的特殊结构，我们得出了加权层次图的ATT精确表达式。结果表明，ATT服从幂指数函数，幂指数为\（\ frac {\ mathrm {ln}（kr）} {\ mathrm {ln} k} <1 \），与因子k和r表示对于\（kr = k（r = 1）\）和\（1 <kr <k \），ATT分别与网络顺序具有线性和亚线性关系。因此，kr较小是，加权层次图的陷印过程越有效。与未加权的层次图相比，加权的层次图在陷印过程中具有更高的效率。