On Transmission Irregular Cubic Graphs of an Arbitrary Order

Anatoly Yu. Bezhaev and Andrey A. Dobrynin
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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

Mathematics, 2022, vol. 10, issue 15, 1-15

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.

Keywords: cubic graph; graph invariant; vertex transmission; transmission irregular graph; Wiener complexity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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