, Volume 33, Issue 5, pp 531–548 | Cite as

On the density of triangles and squares in regular finite and unimodular random graphs

Original paper


We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these densities. As a corollary one gets an explicit description of the possible pairs of the third and fourth moments of the spectral measure of r-regular unimodular random graphs.

Mathematics Subject Classification (2010)

Primary 05C38 Secondary 05C80, 05C81 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Abért, Y. Glasner and B. Virág: The measurable Kesten theorem, Preprint, 2011. arXiv:1111.2080.Google Scholar
  2. [2]
    D. Aldous and R. Lyons: Processes on unimodular random networks, Electron. J. Probab. 12 (2007), 1454–1508.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    B. Bollobás: On complete subgraphs of different orders, Math. Proc. Camb. Phil. Soc., 79 (1976), 19–24.CrossRefMATHGoogle Scholar
  4. [4]
    B. Bollobás: Extremal graph theory Dover Books on Mathematics, Dover Publications, 2004.Google Scholar
  5. [5]
    P. Erdős and H. Sachs: Regul are Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Univ. Halle, Math.-Nat. 12 (1963), 251–258.Google Scholar
  6. [6]
    E. Győri: On the number of C 5’s in a triangle-free graph, Combinatorica 9 (1989), 101–102.MathSciNetCrossRefGoogle Scholar
  7. [7]
    H. Hatami, J. Hladký, D. Král’, S. Norine and A. Razborov: On the number of pentagons in triangle-free graphs, Preprint, 2011. arXiv:1102.1634v2.Google Scholar
  8. [8]
    L. Lovász and M. Simonovits: On the number of complete subgraphs of a graph, ii, in: Studies in pure mathematics, 459–495. Birkhäuser, 1983.CrossRefGoogle Scholar
  9. [9]
    B. Mohar and W. Woess: A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), 209–234.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    A. Razborov: On the minimal density of triangles in graphs, Combin. Probab. Comput. 17 (2008), 603–618.MathSciNetMATHGoogle Scholar
  11. [11]
    H. Sachs: Regular graphs with given girth and restricted circuits, J. London Math. Soc. 38 (1963), 423–429.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    N. Sauer: On the existence of regular n-graphs with given girth, J. Combin. Theory 9 (1970), 144–147.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.A. Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

Personalised recommendations