Distance-regular graph

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In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i = d(v, w).

Graph families defined by their automorphisms
distance-transitive distance-regular strongly regular
symmetric (arc-transitive) t-transitive, t ≥ 2 skew-symmetric
(if connected)
vertex- and edge-transitive
edge-transitive and regular edge-transitive
vertex-transitive regular (if bipartite)
biregular
Cayley graph zero-symmetric asymmetric

Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.

Intersection arraysEdit

It turns out that a graph   of diameter   is distance-regular if and only if there is an array of integers   such that for all  ,   gives the number of neighbours of   at distance   from   and   gives the number of neighbours of   at distance   from   for any pair of vertices   and   at distance   on  . The array of integers characterizing a distance-regular graph is known as its intersection array.

Cospectral distance-regular graphsEdit

A pair of connected distance-regular graphs are cospectral if and only if they have the same intersection array.

A distance-regular graph is disconnected if and only if it is a disjoint union of cospectral distance-regular graphs.

PropertiesEdit

Suppose   is a connected distance-regular graph of valency   with intersection array  . For all  : let   denote the  -regular graph with adjacency matrix   formed by relating pairs of vertices on   at distance  , and let   denote the number of neighbours of   at distance   from   for any pair of vertices   and   at distance   on  .

Graph-theoretic propertiesEdit

  •   for all  .
  •   and  .

Spectral propertiesEdit

  •   for any eigenvalue multiplicity   of  , unless   is a complete multipartite graph.
  •   for any eigenvalue multiplicity   of  , unless   is a cycle graph or a complete multipartite graph.
  •   if   is a simple eigenvalue of  .
  •   has   distinct eigenvalues.

If   is strongly regular, then   and  .

ExamplesEdit

Some first examples of distance-regular graphs include:

Classification of distance-regular graphsEdit

There are only finitely many distinct connected distance-regular graphs of any given valency  .[1]

Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity  [2] (with the exception of the complete multipartite graphs).

Cubic distance-regular graphsEdit

The cubic distance-regular graphs have been completely classified.

The 13 distinct cubic distance-regular graphs are K4 (or Tetrahedral graph), K3,3, the Petersen graph, the Cubical graph, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the Dodecahedral graph, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.

ReferencesEdit

  1. ^ Bang, S.; Dubickas, A.; Koolen, J. H.; Moulton, V. (2015-01-10). "There are only finitely many distance-regular graphs of fixed valency greater than two". Advances in Mathematics. 269 (Supplement C): 1–55. arXiv:0909.5253. doi:10.1016/j.aim.2014.09.025. S2CID 18869283.
  2. ^ Godsil, C. D. (1988-12-01). "Bounding the diameter of distance-regular graphs". Combinatorica. 8 (4): 333–343. doi:10.1007/BF02189090. ISSN 0209-9683. S2CID 206813795.

Further readingEdit