k-Wiener index of a k-plex

Abstract

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.