Skip to main content
SpringerLink
Log in

Distance in Graphs

  • Chapter
  • First Online:
Structural Analysis of Complex Networks

Abstract

The distance between two vertices is the basis of the definition of several graph parameters including diameter, radius, average distance and metric dimension. These invariants are examined, especially how they relate to one another and to other graph invariants and their behaviour in certain graph classes. We also discuss characterizations of graph classes described in terms of distance or shortest paths. Finally, generalizations are considered.

MSC2000: Primary 05C12; Secondary 05C20, 05C38

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Althöfer I (1990) Average distances in undirected graphs and the removal of vertices. J Combin Theory B 48(1):140–142

    Article  MATH  Google Scholar 

  2. Bandelt H-J, Mulder HM (1986) Distance–hereditary graphs. J Combin Theory B 41:182–208

    Article  MATH  MathSciNet  Google Scholar 

  3. Beezer RA, Riegsecker JE, Smith BA (2001) Using minimum degree to bound average distance. Discrete Math 226:365–371

    Article  MATH  MathSciNet  Google Scholar 

  4. Beineke LW, Oellermann OR, Pippert RE (1996) On the Steiner median of a tree. Discrete Appl Math 68:249–258

    Article  MATH  MathSciNet  Google Scholar 

  5. Bienstock D, Györi E (1988) Average distance in graphs with removed elements. J Graph Theory 12(3):375–390

    Article  MATH  MathSciNet  Google Scholar 

  6. Bonato A (2005) A survey of models of the web graph. In: Combinatorial and algorithmic aspects of networking, vol 3405, Lecture notes in computer science. Springer, Berlin, pp 159–172

    Google Scholar 

  7. Buckley F, Harary F (1990) Distance in graphs. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA

    MATH  Google Scholar 

  8. Buckley F, Lewinter M (1988) A note on graphs with diameter-preserving spanning trees. J Graph Theory 12(4):525–528

    Article  MATH  MathSciNet  Google Scholar 

  9. Buckley F, Miller Z, Slater PJ (1981) On graphs containing a given graph as center. J Graph Theory 5:427–432

    Article  MATH  MathSciNet  Google Scholar 

  10. Cáceres J, Hernando C, Mora M, Pelayo IM, Puertas ML, Seara C, Wood DR (2007) On the metric dimension of the cartesian product of graphs. SIAM J Discrete Math 21(2):423–441

    Article  MATH  MathSciNet  Google Scholar 

  11. Cáceres J, Oellermann OR (2009) On 3-Steiner simplicial orderings. Discrete Math 309: 5828–5833

    Article  MATH  MathSciNet  Google Scholar 

  12. Chartrand G, Eroh L, Johnson MA, Oellermann OR (2000) Resolvability in graphs and the metric dimension of a graph. Discrete Appl Math 105:99–113

    Article  MATH  MathSciNet  Google Scholar 

  13. Chartrand G, Oellermann OR, Tian S, Zou HB (1989) Steiner distance in graphs. Časopis Pro Pěstování Matematiky 114(4):399–410

    MATH  MathSciNet  Google Scholar 

  14. Chung FRK (1988) The average distance and the independence number of a graph. J Graph Theory 12:229–235

    Article  MATH  MathSciNet  Google Scholar 

  15. Chvátal V, Thomassen C (1978) Distances in orientations of graphs. J Combin Theory B 24:61–75

    Article  Google Scholar 

  16. Cockayne EJ, Hedetniemi SM, Hedetniemi ST (1981) Linear algorithms for finding the Jordan center and path center of a tree. Transport Sci 15:98–114

    Article  MathSciNet  Google Scholar 

  17. Dankelmann P Average distance in graphs – a survey, (submitted)

    Google Scholar 

  18. Dankelmann P, Oellermann OR, Swart HC (1996) The average Steiner distance of a graph. J Graph Theory 22(1):15–22

    Article  MATH  MathSciNet  Google Scholar 

  19. Dankelmann P, Oellermann OR, Wu J-L (2004) Minimum average distance of strong orientations of graphs. Discrete Appl Math 143:204–212

    Article  MATH  MathSciNet  Google Scholar 

  20. D’Atri A, Moscarini M (1988) Steiner trees and connected domination. SIAM J Discrete Math 17:521–538

    MathSciNet  Google Scholar 

  21. Dirac GA (1961) On rigid circuit graphs. Abh Math Sem Univ Hamburg 25:71–76

    Article  MATH  MathSciNet  Google Scholar 

  22. Doyle JK, Graver JE (1977) Mean distance in a graph. Discrete Math 7(2):147–154

    Article  MathSciNet  Google Scholar 

  23. Dragan FF, Nicolai F, Brandstädt A (1999) Convexity and HHD-free graphs. SIAM J Discrete Math 12:119–135

    Article  MATH  MathSciNet  Google Scholar 

  24. Entringer RC, Jackson DE, Snyder DA (1976) Distance in graphs. Czech Math J 26:283–296

    MathSciNet  Google Scholar 

  25. Entringer RC, Kleitman DJ, Székely LA (1996) A note on spanning trees with minimum average distance. Bull Inst Combinator Appl 17:71–78

    MATH  Google Scholar 

  26. Erdös P, Rényi A (1963) On two problems of information theory. Magyar Tud Akad Mat Kutató Int Közl 8:229–243

    MATH  Google Scholar 

  27. Fajtlowicz S (1986) On two conjectures of GRAFFITI. Congr Numer 55:51–56

    MathSciNet  Google Scholar 

  28. Fajtlowicz S (1987) On conjectures of GRAFFITI II. Congr Numer 60:187–197

    MathSciNet  Google Scholar 

  29. Farber M, Jamison RE (1986) Convexity in graphs and hypergraphs. SIAM J Algebra Discrete Methods 7:433–444

    Article  MATH  MathSciNet  Google Scholar 

  30. Favaron O, Kouider M, Mahéo M (1989) Edge-vulnerability and mean distance. Networks 19(5):493–504

    Article  MATH  MathSciNet  Google Scholar 

  31. Floyd RW (1962) Algorithm 97: shortest path. Commun ACM 5:345

    Article  Google Scholar 

  32. Freeman LC (1978/1979) Centrality in social networks: conceptual clarification. Soc Networks 1:215–239

    Google Scholar 

  33. Füredi Z, Horak P, Pareek CM, Zhu X (1998) Minimal oriented graphs of diameter 2. Graph Combinator 14:345–350

    Article  MATH  Google Scholar 

  34. Garey MR, Johnson DS (1979) Computers and intractibility: a guide to the theory of NP-completeness. W.H. Freeman and Company, New York

    MATH  Google Scholar 

  35. Goddard W, Swart CS, Swart HC (2005) On the graphs with maximum distance or k-diameter. Math Slovaca 55(2):131–139

    MATH  MathSciNet  Google Scholar 

  36. Györi E (1988) On Winkler’s four thirds conjecture on mean distance in graphs. Congr Numer 61:259–262

    MathSciNet  Google Scholar 

  37. Hammer PL, Maffray F (1990) Completely separable graphs. J Discrete Appl Math 27:85–99

    Article  MATH  MathSciNet  Google Scholar 

  38. Harary F (1962) The maximum connectivity of a graph. Proc Natl Acad Sci USA 48: 1142–1146

    Article  MATH  Google Scholar 

  39. Harary F, Melter RA (1976) On the metric dimension of a graph. Ars Combinatoria 2:191–195

    MathSciNet  Google Scholar 

  40. Harary F, Norman RZ (1953) The dissimilarity characteristic of Husimi trees. Ann Math (2) 58:134–141

    Google Scholar 

  41. Howorka E (1977) A characterization of distance-hereditary graphs. Quart J Math Oxford 28:417–420

    Article  MATH  MathSciNet  Google Scholar 

  42. Jeong H, Mason SP, Barabási A-L, Oltvai ZN (2001) Lethality and centrality in protein networks. Nature 411:41–42

    Article  Google Scholar 

  43. Johnson DS, Lenstra JK, Rinnooy-Kan AHG (1978) The complexity of the network design problem. Networks 8:279–285

    Article  MATH  MathSciNet  Google Scholar 

  44. Jordan C (1869) Sur les assembalges des lignes. J Reine Angew Math 70:185–190

    Google Scholar 

  45. Khuller S, Raghavachari B, Rosenfeld A (1996) Landmarks in graphs. Discrete Appl Math 70(3):217–229

    Article  MATH  MathSciNet  Google Scholar 

  46. Kouider M, Winkler P (1997) Mean distance and minimum degree. J Graph Theory 25:95–99

    Article  MATH  MathSciNet  Google Scholar 

  47. Landau HG (1953) On dominance relations and the structure of animal societies. III. The condition for a score structure. Bull Math Biophys 15:143–148

    Google Scholar 

  48. Laskar R, Shier D (1983) On powers and centers of chordal graphs. Discrete Appl Math 6(2):139–147

    Article  MATH  MathSciNet  Google Scholar 

  49. Lekkerkerker CG, Boland J (1962) Representation of a finite graph by a set of intervals on the real line. Fund Math 51:45–64

    MATH  MathSciNet  Google Scholar 

  50. Lindström B (1964) On a combinatory detection problem. I. Magyar Tud Akad Mat Kutató Int Közl 9:195–207

    MATH  Google Scholar 

  51. Lovász L (1979) Combinatorial problems and exercises. Akadémiai Kiadó, Budapset

    MATH  Google Scholar 

  52. March L, Steadman P (1974) The geometry of environment: an introduction to spatial organization in design. MIT Press, Cambridge, MA

    Google Scholar 

  53. Merris R (1989) An edge version of the matrix-tree theorem and the Wiener index. Linear Multilinear Algebra 25(4):291–296

    Article  MATH  MathSciNet  Google Scholar 

  54. Mohar B (1991) Eigenvalues, diameter, and mean distance in graphs. Graph Combinator 7(1):53–64

    Article  MATH  MathSciNet  Google Scholar 

  55. Mulder HM (1980) The interval function of a graph. Mathematical Centre Tracts, Amsterdam

    MATH  Google Scholar 

  56. Mulder HM (1980) n-cubes and median graphs. J Graph Theory 4:107–110

    Article  MATH  MathSciNet  Google Scholar 

  57. Ng CP, Teh HH (1966/1967) On finite graphs of diameter 2. Nanta Math 1:72–75

    Google Scholar 

  58. Nielsen M, Oellermann OR (2009) Steiner trees and convex geometries. SIAM J Discrete Math 23:680–693

    Article  MATH  MathSciNet  Google Scholar 

  59. Oellermann OR (1995) From Steiner centers to Steiner medians. J Graph Theory 20(2): 113–122

    Article  MATH  MathSciNet  Google Scholar 

  60. Oellermann OR, Tian S (1990) Steiner centers in graphs. J Graph Theory 14(5):585–597

    Article  MATH  MathSciNet  Google Scholar 

  61. Plesník J (1975) Note on diametrically critical graphs. In: Recent advances in graph theory (Proceedings of the 2nd Czechoslovak symposium, Prague, 1974). Academia, Prague, pp 455–465

    Google Scholar 

  62. Plesník J (1984) On the sum of all distance in a graph or digraph. J Graph Theory 8:1–24

    Article  MATH  MathSciNet  Google Scholar 

  63. Robbins HE (1939) Questions, discussions, and notes: a theorem on graphs, with an application to a problem of traffic control. Am Math Mon 46(5):281–283

    Article  MathSciNet  Google Scholar 

  64. Rodriguez JA, Yebra JLA (1999) Bounding the diameter and the mean distance of a graph from its eigenvalues: Laplacian versus adjacency matrix methods. Discrete Math 196(1–3):267–275

    Article  MATH  MathSciNet  Google Scholar 

  65. Sebö A, Tannier E (2004) On metric generators of graphs. Math Oper Res 29(2):383–393

    Article  MATH  MathSciNet  Google Scholar 

  66. Slater PJ (1975) Leaves of trees. Congr Numer 14:549–559

    MathSciNet  Google Scholar 

  67. Slater PJ (1978) Centers to centroids in graphs. J Graph Theory 2:209–222

    Article  MATH  MathSciNet  Google Scholar 

  68. Slater PJ (1981) Centrality of paths and vertices in a graph: cores and pits. In: The theory and applications of graphs. Wiley, New York, pp 529–542

    Google Scholar 

  69. S̆oltés L (1991) Transmission in graphs: a bound and vertex removing. Math Slovaca 41(1): 11–16

    Google Scholar 

  70. Swart CS (1996) Distance measures in graphs and subgraphs. Master’s thesis, University of Natal, Durban

    Google Scholar 

  71. Vizing VG (1967) On the number of edges in a graph with a given radius. Dokl Akad Nauk SSSR 173:1245–1246

    MathSciNet  Google Scholar 

  72. Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69(1): 17–20

    Article  Google Scholar 

  73. Winkler P (1986) Mean distance and the four thirds conjecture. Congr Numer 54:53–61

    Google Scholar 

  74. Winkler P (1989) Graph theory in memory of G.A. Dirac. In: Proceedings of meeting in Sandbjerg, Denmark, 1985, Ann Discrete Math, North-Holland, Amsterdam

    Google Scholar 

  75. Winter P (1987) Steiner problem in networks: a survey. Networks 17:129–167

    Article  MATH  MathSciNet  Google Scholar 

  76. Wong R (1980) Worst-case analysis of network design problem heuristics. SIAM J Algebra Discrete Methods 1:51–63

    Article  MATH  Google Scholar 

  77. Zelinka B (1968) Medians and peripherians of trees. Arch Math (Brno) 4:87–95

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank Peter Dankelmann for sharing his thoughts on average distance with us.

Author information

Authors and Affiliations

  1. School of Computing and Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634-1906, USA

    Wayne Goddard

Authors
  1. Wayne Goddard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ortrud R. Oellermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wayne Goddard .

Editor information

Editors and Affiliations

  1. Medizinische Informatik und Technik, Institute for Bioinformatics and Transla, UMIT-Private Universität für Gesundheits, Eduard Wallnöfer-Zentrum 1, Hall in Tirol, A-6060, Austria

    Matthias Dehmer

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Goddard, W., Oellermann, O.R. (2011). Distance in Graphs. In: Dehmer, M. (eds) Structural Analysis of Complex Networks. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4789-6_3

Download citation

Publish with us

Policies and ethics

Search

Navigation