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Graph irregularity and its measures

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  • Abdo, Hosam
  • Dimitrov, Darko
  • Gutman, Ivan

Abstract

Let G be a simple graph. If all vertices of G have equal degrees, then G is said to be regular. Otherwise, G is irregular. There were various attempts to quantify the irregularity of a graph, of which the Collatz–Sinogowitz index, Bell index, Albertson index, and total irregularity are the best known. We now show that no two of these irregularity measures are mutually consistent, namely that for any two such measures, irrX and irrY there exist pairs of graphs G1, G2, such that irrX(G1) > irrX(G2) but irrY(G1) < irrY(G2). This implies that the concept of graph irregularity is not free of ambiguities.

Suggested Citation

  • Abdo, Hosam & Dimitrov, Darko & Gutman, Ivan, 2019. "Graph irregularity and its measures," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 317-324.
    • Handle: RePEc:eee:apmaco:v:357:y:2019:i:c:p:317-324
    DOI: 10.1016/j.amc.2019.04.013
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    References listed on IDEAS

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    1. Chen, Xiaodan & Hou, Yaoping & Lin, Fenggen, 2018. "Some new spectral bounds for graph irregularity," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 331-340.
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    Cited by:

    1. 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).
    2. Dimitrov, Darko & Stevanović, Dragan, 2023. "On the σt-irregularity and the inverse irregularity problem," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    3. Sakander Hayat & Amina Arif & Laiq Zada & Asad Khan & Yubin Zhong, 2022. "Mathematical Properties of a Novel Graph-Theoretic Irregularity Index with Potential Applicability in QSPR Modeling," Mathematics, MDPI, vol. 10(22), pages 1-24, November.

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