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On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs

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  • Alizadeh, Yaser
  • Klavžar, Sandi

Abstract

If u is a vertex of a graph G, then the transmission of u is the sum of distances from u to all the other vertices of G. The Wiener complexity CW(G) of G is the number of different complexities of its vertices. G is transmission irregular if CW(G)=n(G). It is proved that almost no graphs are transmission irregular. Let Tn1,n2,n3 be the tree obtained from paths of respective lengths n1, n2, and n3, by identifying an end-vertex of each of them. It is proved that T1,n2,n3 is transmission irregular if and only if n3=n2+1 and n2∉{(k2−1)/2,(k2−2)/2} for some k ≥ 3. It is also proved that if T is an asymmetric tree of order n, then the Wiener index of T is bounded by (n3−13n+48)/6 with equality if and only if T=T1,2,n−4. A parallel result is deduced for asymmetric uni-cyclic graphs.

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  • 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.
    • Handle: RePEc:eee:apmaco:v:328:y:2018:i:c:p:113-118
    DOI: 10.1016/j.amc.2018.01.039
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    References listed on IDEAS

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    1. Ghebleh, M. & Kanso, A. & Stevanović, D., 2016. "Open quipus with the same Wiener index as their quadratic line graph," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 130-136.
    2. Andova, Vesna & Orlić, Damir & Škrekovski, Riste, 2017. "Leapfrog fullerenes and Wiener index," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 281-288.
    3. Črepnjak, Matevž & Tratnik, Niko, 2017. "The Szeged index and the Wiener index of partial cubes with applications to chemical graphs," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 324-333.
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    Cited by:

    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. 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.
    4. Martin Knor & Riste Škrekovski, 2020. "Wiener Complexity versus the Eccentric Complexity," Mathematics, MDPI, vol. 9(1), pages 1-9, December.
    5. Al-Yakoob, Salem & Stevanović, Dragan, 2020. "On transmission irregular starlike trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    6. Dobrynin, Andrey A. & Sharafdini, Reza, 2020. "Stepwise transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    7. Hamid Darabi & Yaser Alizadeh & Sandi Klavžar & Kinkar Chandra Das, 2021. "On the relation between Wiener index and eccentricity of a graph," Journal of Combinatorial Optimization, Springer, vol. 41(4), pages 817-829, May.
    8. Sharon, Jane Olive & Rajalaxmi, T.M. & Klavžar, Sandi & Rajan, R. Sundara & Rajasingh, Indra, 2021. "Transmission in H-naphtalenic nanosheet," Applied Mathematics and Computation, Elsevier, vol. 406(C).
    9. Bezhaev, Anatoly Yu. & Dobrynin, Andrey A., 2021. "On quartic transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    10. Andrey A. Dobrynin & Andrei Yu Vesnin, 2019. "On the Wiener Complexity and the Wiener Index of Fullerene Graphs," Mathematics, MDPI, vol. 7(11), pages 1-17, November.

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