Sharp bounds on some distance-based graph invariants of n-vertex k-trees are established in a unified approach, which may be viewed as the weighted Wiener index or weighted Harary index. The main techniques used in this paper are graph transformations and mathematical induction. Our results demonstrate that among k-trees with n vertices the extremal graphs with the maximal and the second maximal reciprocal sum-degree distance are coincident with graphs having the maximal and the second maximal reciprocal product-degree distance (and similarly, the extremal graphs with the minimal and the second minimal degree distance are coincident with graphs having the minimal and the second minimal eccentricity distance sum).
A graph G is a k-tree if either G is the complete graph on k + 1 vertices, or G has a vertex v whose neighborhood is a clique of order k and the graph obtained by removing v from G is also a k-tree. Clearly, a k-tree has at least k + 1 vertices, and G is a 1-tree (usual tree) if and only if it is a 1-connected graph and has no K_{3} -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of k-trees as follows: if G is a graph with at least k + 1 vertices, then G is a k-tree if and only if G has no K_{k+2} -minor, G does not contain any chordless cycle of length at least 4 and G is k-connected., De-Yan Zeng, Jian-Hua Yin., and Obsahuje seznam literatury