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arXiv · 2411.15765

Extremal results on $k$-stepwise irregular graphs

Yaser Alizadeh·Sandi Klavžar·Javaher Langari
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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.

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Yaser Alizadeh, Sandi Klavžar, Javaher Langari. 2024-11-24. Extremal results on $k$-stepwise irregular graphs. https://doi.org/10.1016/j.amc.2025.129818

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