Wiener indices of maximal $k$-degenerate graphs

Abstract

A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal $k$-degenerate graphs of order $n\ge k\ge 1$. A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal $k$-degenerate graphs of order $n\ge k$ are $k$-trees. For $k$-trees of order $n\ge 2k+2$, we characterize all extremal graphs for the upper bound.