Irregularities of maximal k-degenerate graphs

Abstract

A graph is maximal k-degenerate if every subgraph has a vertex of degree at most k, and the property does not hold if any new edge is added to the graph. A k-tree is a maximal k-degenerate graph that does not contain any induced cycle with more than three edges. We study Albertson irregularity and sigma irregularity for a maximal k-degenerate graph of order n≥k+2. Sharp upper bounds on both irregularity indices of maximal k-degenerate graphs are provided and their extremal graphs are characterized as k-stars Kk+K¯n−k. Sharp lower bounds on both irregularity indices of k-trees are obtained and their extremal graphs are characterized as kth powers of paths Pnk. Sharp lower bounds on irregularities of maximal 2-degenerate graphs are also proved.