Properties of stepwise irregular graphs

TL;DR

This paper explores properties and existence conditions of stepwise irregular graphs, including new constructions of connected bicyclic SI graphs and bounds on their irregularity, expanding understanding of their structural characteristics.

Contribution

The paper demonstrates the existence of certain connected bicyclic SI graphs previously thought impossible and characterizes SI graphs with specific cyclomatic numbers.

Findings

Existence of connected bicyclic SI graphs of order 5 and 9

Characterization of SI graphs with specific cyclomatic numbers

Bounds on the irregularity of SI graphs

Abstract

Stepwise irregular (SI) graphs were introduced by Ivan Gutman recently in 2018 and in these graphs the difference between the degrees of any two adjacent vertices is exactly one. In this work, we show the existence of connected bicyclic SI graphs of order 5 and 9 which has been claimed to be non-existent by Gutman. We also show the existence of SI graphs of different order and cyclomatic numbers when there exist a SI graph with a vertex of degree 1 or 2. We give a characterization on the order of the connected tricyclic SI graphs. At the end, we investigate some properties of the SI graphs under elementary graph operations along with the lower and upper bound of the irregularity of SI graphs.