Wiener indices of maximal $k$-degenerate graphs

TL;DR

This paper establishes precise bounds on the Wiener indices of maximal $k$-degenerate graphs, including special cases like chordal and $k$-trees, and characterizes extremal graphs for these bounds.

Contribution

It provides sharp bounds on Wiener indices for maximal $k$-degenerate graphs and characterizes extremal structures in specific subclasses.

Findings

Sharp lower and upper bounds on Wiener indices for maximal $k$-degenerate graphs

Characterization of extremal graphs for the upper bounds in $k$-trees

Identification of properties of chordal 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.