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Extremal results on k-stepwise irregular graphs

Author

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  • Alizadeh, Yaser
  • Klavžar, Sandi
  • Langari, Javaher

Abstract

For a positive integer k, 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≥1 and any d≥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(Δ(G),k)=1. Along the way the degree complexity of a graph is introduced and used.

Suggested Citation

  • Alizadeh, Yaser & Klavžar, Sandi & Langari, Javaher, 2026. "Extremal results on k-stepwise irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 514(C).
    • Handle: RePEc:eee:apmaco:v:514:y:2026:i:c:s0096300325005430
    DOI: 10.1016/j.amc.2025.129818
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    References listed on IDEAS

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    1. Damnjanović, Ivan & Stevanović, Dragan & Al-Yakoob, Salem, 2024. "On transmission-irregular graphs and long pendent paths," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    2. Buyantogtokh, Lkhagva & Azjargal, Enkhbayar & Horoldagva, Batmend & Dorjsembe, Shiikhar & Adiyanyam, Damchaa, 2021. "On the maximum size of stepwise irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    3. Abdo, Hosam & Dimitrov, Darko & Gutman, Ivan, 2019. "Graph irregularity and its measures," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 317-324.
    4. Al-Yakoob, Salem & Stevanović, Dragan, 2022. "On stepwise transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    5. Gutman, Ivan, 2018. "Stepwise irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 234-238.
    6. Dobrynin, Andrey A. & Sharafdini, Reza, 2020. "Stepwise transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    7. Nahid Akhter & Al-Nashri Al-Hossain Ahmad & Muhammad Faisal Nadeem, 2022. "Stepwise Irregular Graphs and Their Metric-Based Resolvability Parameters," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-8, May.
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