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On Transmission Irregular Cubic Graphs of an Arbitrary Order

Author

Listed:
  • Anatoly Yu. Bezhaev

    (Institute of Computational Mathematics and Mathematical Geophysics, The Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia)

  • Andrey A. Dobrynin

    (Sobolev Institute of Mathematics, The Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia)

Abstract

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices of G . A transmission irregular graph (TI graph) has mutually distinct vertex transmissions. In 2018, Alizadeh and Klavžar posed the following question: do there exist infinite families of regular TI graphs? An infinite family of TI cubic graphs of order 118 + 72 k , k ≥ 0 , was constructed by Dobrynin in 2019. In this paper, we study the problem of finding TI cubic graphs for an arbitrary number of vertices. It is shown that there exists a TI cubic graph of an arbitrary even order n ≥ 22 . Almost all constructed graphs are contained in twelve infinite families.

Suggested Citation

  • Anatoly Yu. Bezhaev & Andrey A. Dobrynin, 2022. "On Transmission Irregular Cubic Graphs of an Arbitrary Order," Mathematics, MDPI, vol. 10(15), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2741-:d:879091
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    References listed on IDEAS

    as
    1. Klavžar, Sandi & Azubha Jemilet, D. & Rajasingh, Indra & Manuel, Paul & Parthiban, N., 2018. "General Transmission Lemma and Wiener complexity of triangular grids," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 115-122.
    2. Dobrynin, Andrey A., 2019. "Infinite family of 2-connected transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 1-4.
    3. Bezhaev, Anatoly Yu. & Dobrynin, Andrey A., 2021. "On quartic transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    4. Al-Yakoob, Salem & Stevanović, Dragan, 2020. "On transmission irregular starlike trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    5. Alizadeh, Yaser & Klavžar, Sandi, 2018. "On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 113-118.
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