Average eccentricity, minimum degree and maximum degree in graphs

Abstract

Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as \(\frac{1}{|V(G)|}\sum _{v \in V(G)}e(v)\). We show that the average eccentricity of a connected graph of order n, minimum degree \(\delta \) and maximum degree \(\Delta \) does not exceed \(\frac{9}{4} \frac{n-\Delta -1}{\delta +1} \big ( 1 + \frac{\Delta -\delta }{3n} \big ) + 7\), and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.

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References

  1. Ali P, Dankelmann P, Morgan MJ, Mukwembi S, Vetrík T (2018) The average eccentricity, spanning trees of plane graphs, size and order. Util Math 107:37–49

    MathSciNet  MATH  Google Scholar 

  2. Buckley F, Harary F (1990) Distance in graphs. Addisson-Wesley, Redwood City

    Google Scholar 

  3. Dankelmann P, Entringer R (2000) Average distance, minimum degree and spanning trees. J Graph Theory 33(1):1–13

    MathSciNet  Article  Google Scholar 

  4. Dankelmann P, Mukwembi S (2014) Upper bounds on the average eccentricity. Discrete Appl Math 167:72–79

    MathSciNet  Article  Google Scholar 

  5. Dankelmann P, Osaye FJ (2019) Average eccentricity, \(k\)-packing and \(k\)-domination in graphs. Discrete Math 342:1261–1274

    MathSciNet  Article  Google Scholar 

  6. Dankelmann P, Goddard W, Swart CS (2004) The average eccentricity of a graph and its subgraphs. Util Math 41:41–51

    MathSciNet  MATH  Google Scholar 

  7. Dankelmann P, Osaye FJ, Mukwembi S, Rodrigues B (2019) Upper bounds on the average eccentricity of \(K_3\)-free and \(C_4\)-free graphs. Discrete Appl Math 270:106–114

    MathSciNet  Article  Google Scholar 

  8. Darabi H, Alizadeh Y, Klavzar S, Das KC (2018) On the relation between Wiener index and eccentricity of a graph

  9. Du Z, Ilic̆ A (2013) On AGX conjectures regarding average eccentricity. MATCH Commun Math Comput Chem 69:597–609

    MathSciNet  MATH  Google Scholar 

  10. Du Z, Ilic̆ A (2016) A proof of the conjecture regarding the sum of the domination number and average eccentricity. Discrete Appl Math 201:105–113

    MathSciNet  Article  Google Scholar 

  11. Erdös P, Pach J, Pollack R, Tuza Z (1989) Radius, diameter, and minimum degree. J Comb Theory B 47:73–79

    MathSciNet  Article  Google Scholar 

  12. Ilic̆ A (2012) On the extremal properties of the average eccentricity. Comput Math Appl 64(9):2877–2885

    MathSciNet  Article  Google Scholar 

  13. Smith H, Székely LA, Wang H (2016) Eccentricity sum in trees. Discrete Appl Math 207:120–131

    MathSciNet  Article  Google Scholar 

  14. Tang Y, Zhou B (2012) On average eccentricity. MATCH Commun Math Comput Chem 67:405–423

    MathSciNet  MATH  Google Scholar 

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Correspondence to P. Dankelmann.

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P. Dankelmann: Financial Support by the South African National Research Foundation, Grant Number 118521, is gratefully acknowledged. F.J. Osaye: The results presented in this paper form part of the second author’s PhD thesis.

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Dankelmann, P., Osaye, F.J. Average eccentricity, minimum degree and maximum degree in graphs. J Comb Optim (2020). https://doi.org/10.1007/s10878-020-00616-x

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Keywords

  • Average eccentricity
  • Distance
  • Minimum degree
  • Maximum degree
  • Eccentricity
  • Graph