Extremal results on $k$-stepwise irregular graphs

TL;DR

This paper investigates the properties of $k$-stepwise irregular graphs, establishing their existence for various diameters, providing bounds on degrees and size, and introducing the concept of degree complexity.

Contribution

It proves the existence of $k$-SI graphs with arbitrary diameters using graph products and establishes bounds on degrees and size, also introducing degree complexity.

Findings

Existence of $k$-SI graphs for any diameter and degree

Sharp upper bounds for maximum degree of $k$-SI graphs

Bounds on size of $k$-SI graphs, especially when $ ext{gcd}( riangle(G), k) = 1$

Abstract

For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated that for any $k\ge 1$ and any $d \ge 2$ there exists a $k$-SI graph of diameter $d$. A sharp upper bound for the maximum degree of a $k$-SI graph of a given order is proved. The size of $k$-SI graphs is bounded in general and in the special case when $\gcd(\Delta(G), k) = 1$. Along the way the degree complexity of a graph is introduced and used.