circulant graph

{{short description|Undirected graph acted on by a vertex-transitive cyclic group of symmetries}}

{{For|the square matrices|Circulant matrix}}

File:Paley13.svg of order 13, an example of a circulant graph.]]

In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings.

Equivalent definitions

Circulant graphs can be described in several equivalent ways:{{citation|first=V.|last=Vilfred|contribution=On circulant graphs|title=Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001)|publisher=Alpha Science|editor1-first=R.|editor1-last=Balakrishnan|editor2-first=G.|editor2-last=Sethuraman|editor3-first=Robin J.|editor3-last=Wilson|year=2004|url=https://books.google.com/books?id=wG-08Lv8E-0C&pg=PA34|pages=34–36}}.

  • The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has an automorphism which is a cyclic permutation of its vertices.
  • The graph has an adjacency matrix that is a circulant matrix.
  • The {{mvar|n}} vertices of the graph can be numbered from 0 to {{math|n − 1}} in such a way that, if some two vertices numbered {{mvar|x}} and {{math|(x + d) mod n}} are adjacent, then every two vertices numbered {{mvar|z}} and {{math|(z + d) mod n}} are adjacent.
  • The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing.
  • The graph is a Cayley graph of a cyclic group.{{citation

| last = Alspach | first = Brian | authorlink = Brian Alspach

| contribution = Isomorphism and Cayley graphs on abelian groups

| location = Dordrecht

| mr = 1468786

| pages = 1–22

| publisher = Kluwer Acad. Publ.

| series = NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.

| title = Graph symmetry (Montreal, PQ, 1996)

| url = https://books.google.com/books?id=-tIaXdII8egC&pg=PA1

| volume = 497

| year = 1997}}.

Examples

Every cycle graph is a circulant graph, as is every crown graph with number of vertices congruent to 2 modulo 4.

The Paley graphs of order {{mvar|n}} (where {{mvar|n}} is a prime number congruent to {{nowrap|1 modulo 4}}) is a graph in which the vertices are the numbers from 0 to {{math|n − 1}} and two vertices are adjacent if their difference is a quadratic residue modulo {{mvar|n}}. Since the presence or absence of an edge depends only on the difference modulo {{mvar|n}} of two vertex numbers, any Paley graph is a circulant graph.

Every Möbius ladder is a circulant graph, as is every complete graph. A complete bipartite graph is a circulant graph if it has the same number of vertices on both sides of its bipartition.

If two numbers {{mvar|m}} and {{mvar|n}} are relatively prime, then the {{math|m × n}} rook's graph (a graph that has a vertex for each square of an {{math|m × n}} chessboard and an edge for each two squares that a rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group Cmn \simeq Cm×Cn. More generally, in this case, the tensor product of graphs between any {{mvar|m}}- and {{mvar|n}}-vertex circulants is itself a circulant.

Many of the known lower bounds on Ramsey numbers come from examples of circulant graphs that have small maximum cliques and small maximum independent sets.[http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1 Small Ramsey Numbers], Stanisław P. Radziszowski, Electronic J. Combinatorics, dynamic survey 1, updated 2014.

A specific example

The circulant graph C_n^{s_1,\ldots,s_k} with jumps s_1, \ldots, s_k is defined as the graph with n nodes labeled 0, 1, \ldots, n-1 where each node i is adjacent to 2k nodes i \pm s_1, \ldots, i \pm s_k \mod n.

  • The graph C_n^{s_1, \ldots, s_k} is connected if and only if \gcd(n, s_1, \ldots, s_k) = 1.
  • If 1 \leq s_1 < \cdots < s_k are fixed integers then the number of spanning trees t(C_n^{s_1,\ldots,s_k}) = na_n^2 where a_n satisfies a recurrence relation of order 2^{s_k-1}.
  • In particular, t(C_n^{1,2}) = nF_n^2 where F_n is the n-th Fibonacci number.

Self-complementary circulants

A self-complementary graph is a graph in which replacing every edge by a non-edge and vice versa produces an isomorphic graph.

For instance, a five-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every Paley graph of prime order is a self-complementary circulant graph.{{Cite journal

| last = Sachs | first = Horst | authorlink = Horst Sachs

| mr = 0151953

| journal = Publicationes Mathematicae Debrecen

| pages = 270–288

| title = Über selbstkomplementäre Graphen

| volume = 9

| year = 1962}}. Horst Sachs showed that, if a number {{mvar|n}} has the property that every prime factor of {{mvar|n}} is congruent to {{nowrap|1 modulo 4}}, then there exists a self-complementary circulant with {{mvar|n}} vertices. He conjectured that this condition is also necessary: that no other values of {{mvar|n}} allow a self-complementary circulant to exist. The conjecture was proven some 40 years later, by Vilfred.

Ádám's conjecture

Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to {{math|n − 1}} in such a way that, if some two vertices numbered {{mvar|x}} and {{mvar|y}} are adjacent, then every two vertices numbered {{mvar|z}} and {{math|(zx + y) mod n}} are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix.

Let {{mvar|a}} be an integer that is relatively prime to {{mvar|n}}, and let {{mvar|b}} be any integer. Then the linear function that takes a number {{mvar|x}} to {{math|ax + b}} transforms a circulant numbering to another circulant numbering. András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if {{mvar|G}} and {{mvar|H}} are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for {{mvar|G}} into the numbering for {{mvar|H}}. However, Ádám's conjecture is now known to be false. A counterexample is given by graphs {{mvar|G}} and {{mvar|H}} with 16 vertices each; a vertex {{mvar|x}} in {{mvar|G}} is connected to the six neighbors {{math|x ± 1}}, {{math|x ± 2}}, and {{math|x ± 7}} modulo 16, while in {{mvar|H}} the six neighbors are {{math|x ± 2}}, {{math|x ± 3}}, and {{math|x ± 5}} modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.

Toida's conjecture refines Ádám's conjecture by considering only a special class of circulant graphs, in which all of the differences between adjacent graph vertices are relatively prime to the number of vertices. According to this refined conjecture, these special circulant graphs should have the property that all of their symmetries come from symmetries of the underlying additive group of numbers modulo {{math|n}}. It was proven by two groups in 2001 and 2002.{{citation

| last1 = Muzychuk | first1 = Mikhail

| last2 = Klin | first2 = Mikhail

| last3 = Pöschel | first3 = Reinhard

| contribution = The isomorphism problem for circulant graphs via Schur ring theory

| mr = 1816402

| pages = 241–264

| publisher = American Mathematical Society | location = Providence, Rhode Island

| series = DIMACS Ser. Discrete Math. Theoret. Comput. Sci.

| title = Codes and association schemes (Piscataway, NJ, 1999)

| volume = 56

| year = 2001}}{{citation

| last1 = Dobson | first1 = Edward

| last2 = Morris | first2 = Joy | author2-link = Joy Morris

| issue = 1

| journal = Electronic Journal of Combinatorics

| mr = 1928787

| page = R35:1–R35:14

| title = Toida's conjecture is true

| url = https://www.combinatorics.org/Volume_9/Abstracts/v9i1r35.html

| volume = 9

| year = 2002}}

Algorithmic questions

There is a polynomial-time recognition algorithm for circulant graphs, and the isomorphism problem for circulant graphs can be solved in polynomial time.{{Cite journal

| last1 = Muzychuk | first1 = Mikhail

| mr = 2018956

| journal = Proc. London Math. Soc.

| pages = 1–41

| title = A Solution of the Isomorphism Problem for Circulant Graphs

| volume = 88

| year = 2004

| doi=10.1112/s0024611503014412}}{{Cite journal

| last1 = Evdokimov | first1 = Sergei

| last2 = Ponomarenko | first2 = Ilia

| mr = 2044629

| journal = St. Petersburg Math. J.

| pages = 813–835

| title = Recognition and verification of an isomorphism of circulant graphs in polynomial time

| volume = 15

| year = 2004

| doi=10.1090/s1061-0022-04-00833-7| doi-access = free

}}

References

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