A k-plex is a hypergraph with the property that each subset of a hyperedge is also a hyperedge and each hyperedge contains at most \(k+1\) vertices. We introduce a new concept called the k-Wiener index of a k-plex as the summation of k-distances between every two hyperedges of cardinality k of the k-plex. The concept is different from the Wiener index of a hypergraph which is the sum of distances between every two vertices of the hypergraph. We provide basic properties for the k-Wiener index of a k-plex. Similarly to the fact that trees are the fundamental 1-dimensional graphs, k-trees form an important class of k-plexes and have many properties parallel to those of trees. We provide a recursive formula for the k-Wiener index of a k-tree using its property of a perfect elimination ordering. We show that the k-Wiener index of a k-tree of order n is bounded below by \(2 {1+(n-k)k \atopwithdelims ()2} - (n-k) {k+1 \atopwithdelims ()2} \) and above by \(k^2 {n-k+2 \atopwithdelims ()3} - (n-k){k \atopwithdelims ()2}\). The bounds are attained only when the k-tree is a k-star and a k-th power of path, respectively. Our results generalize the well-known results that the Wiener index of a tree of order n is bounded between \((n-1)^2\) and \({n+1 \atopwithdelims ()3}\), and the lower bound (resp., the upper bound) is attained only when the tree is a star (resp., a path) from 1-dimensional trees to k-dimensional trees.

甲ķ -plex是与属性超图，一个超边的每个子集也是一个超边，并且每个超边含有至多\（K + 1 \）的顶点。我们引入了一个名为新概念ķ的-Wiener指数ķ -plex作为的总和ķ基数的每两个超边之间-distances ķ的ķ -plex。该概念不同于超图的维纳指数，后者是超图的每两个顶点之间的距离之和。我们提供的基本属性ķ的-Wiener指数ķ -plex。与树木是基本一维图的事实类似，k-树形成k -plex的重要一类，并具有许多与树平行的属性。我们利用k-树的k-维纳指数的完全消除顺序的性质为其提供了递归公式。我们显示n阶k树的k维纳指数由\（2 {1+（nk）k \ atopwithdelims（）2}-（nk）{k + 1 \ atopwithdelims（）2}界定\）及以上的\（k ^ 2 {n-k + 2 \ atopwithdelims（）3}-（nk）{k \ atopwithdelims（）2} \）。仅当k -tree是k -star和k时才达到边界路径的第n次方。我们的结果概括了众所周知的结果，即n阶树的Wiener索引在\（（n-1）^ 2 \）和\（{n + 1 \ atopwithdelims（）3} \）之间，仅当树是从一维树到k维树的星形（分别是路径）时，才能达到下限（分别是上限）。