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Wiener Complexity versus the Eccentric Complexity

Author

Listed:
  • Martin Knor

    (Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 81368 Bratislava, Slovakia
    These authors contributed equally to this work.)

  • Riste Škrekovski

    (Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
    Faculty of Information Studies, 8000 Novo Mesto, Slovenia
    These authors contributed equally to this work.)

Abstract

Let w G ( u ) be the sum of distances from u to all the other vertices of G . The Wiener complexity, C W ( G ) , is the number of different values of w G ( u ) in G , and the eccentric complexity, C ec ( G ) , is the number of different eccentricities in G . In this paper, we prove that for every integer c there are infinitely many graphs G such that C W ( G ) − C ec ( G ) = c . Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c ≥ 0 we prove this statement using trees, and if c < 0 we prove it using unicyclic graphs. Further, we prove that C ec ( G ) ≤ 2 C W ( G ) − 1 if G is a unicyclic graph. In our proofs we use that the function w G ( u ) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees C ec ( G ) ≤ C W ( G ) . Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that C ec ( G ) < C W ( G ) .

Suggested Citation

  • Martin Knor & Riste Škrekovski, 2020. "Wiener Complexity versus the Eccentric Complexity," Mathematics, MDPI, vol. 9(1), pages 1-9, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:79-:d:473120
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    References listed on IDEAS

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    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. 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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