Circulant graph

The Paley graph of order 13, an example of a circulant graph.

In graph theory, a circulant graph is an undirected graph that has a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph,^[1] but this term also is given other meanings.

Equivalent definitions

Circulant graphs can be described in several equivalent ways:^[2]

The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices.
The graph has an adjacency matrix that is a circulant matrix.
The $n$ vertices of the graph can be numbered from 0 to $n - 1$ in such a way that, if some two vertices numbered $x$ and $(x + d) mod n$ are adjacent, then every two vertices numbered $z$ and $(z + d) mod n$ 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.^[3]

Examples

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

The Paley graphs of order $n$ (where $n$ is a prime number congruent to 1 modulo 4) is a graph in which the vertices are the numbers from 0 to $n - 1$ and two vertices are adjacent if their difference is a quadratic residue modulo $n$ . Since the presence or absence of an edge depends only on the difference modulo $n$ 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 $m$ and $n$ are relatively prime, then the $m \times n$ rook's graph (a graph that has a vertex for each square of an $m \times n$ chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group C_mn $\simeq$ C_m×C_n. More generally, in this case, the tensor product of graphs between any $m$ - and $n$ -vertex circulants is itself a circulant.^[2]

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.^[1]

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 is a self-complementary circulant graph.^[4] Horst Sachs showed that, if a number $n$ has the property that every prime factor of $n$ is congruent to 1 modulo 4, then there exists a self-complementary circulant with $n$ vertices. He conjectured that this condition is also necessary: that no other values of $n$ allow a self-complementary circulant to exist.^[2]^[4] The conjecture was proven some 40 years later, by Vilfred.^[2]

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 $n - 1$ in such a way that, if some two vertices numbered $x$ and $y$ are adjacent, then every two vertices numbered $z$ and $(z - x + y) mod n$ 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 $a$ be an integer that is relatively prime to $n$ , and let $b$ be any integer. Then the linear function that takes a number $x$ to $ax + b$ 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 $G$ and $H$ are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for $G$ into the numbering for $H$ . However, Ádám's conjecture is now known to be false. A counterexample is given by graphs $G$ and $H$ with 16 vertices each; a vertex $x$ in $G$ is connected to the six neighbors $x \pm 1$ , $x \pm 2$ , and $x \pm 7$ modulo 16, while in $H$ the six neighbors are $x \pm 2$ , $x \pm 3$ , and $x \pm 5$ modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.^[2]

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.^[5]^[6]

References

1 2 Small Ramsey Numbers, Stanisław P. Radziszowski, Electronic J. Combinatorics, dynamic survey 1, updated 2014.
1 2 3 4 5 Vilfred, V. (2004), "On circulant graphs", in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J., Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001), Alpha Science, pp. 34–36 .
↑ Alspach, Brian (1997), "Isomorphism and Cayley graphs on abelian groups", Graph symmetry (Montreal, PQ, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 497, Dordrecht: Kluwer Acad. Publ., pp. 1–22, MR 1468786 .
1 2 Sachs, Horst (1962). "Über selbstkomplementäre Graphen". Publicationes Mathematicae Debrecen. 9: 270–288. MR 0151953. .
↑ Muzychuk, Mikhail (2004). "A Solution of the Isomorphism Problem for Circulant Graphs". Proc. London Math. Soc. 88: 1–41. doi:10.1112/s0024611503014412. MR 2018956.
↑ Evdokimov, Sergei; Ponomarenko, Ilia (2004). "Recognition and verification of an isomorphism of circulant graphs in polynomial time". St. Petersburg Math. J. 15: 813–835. doi:10.1090/s1061-0022-04-00833-7. MR 2044629.

External links

Weisstein, Eric W. "Circulant Graph". MathWorld.

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[ds1-1] 1 2 Small Ramsey Numbers, Stanisław P. Radziszowski, Electronic J. Combinatorics, dynamic survey 1, updated 2014.

[v04-2] 1 2 3 4 5 Vilfred, V. (2004), "On circulant graphs", in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J., Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001), Alpha Science, pp. 34–36 .

[3] Alspach, Brian (1997), "Isomorphism and Cayley graphs on abelian groups", Graph symmetry (Montreal, PQ, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 497, Dordrecht: Kluwer Acad. Publ., pp. 1–22, MR 1468786 .

[s62-4] 1 2 Sachs, Horst (1962). "Über selbstkomplementäre Graphen". Publicationes Mathematicae Debrecen. 9: 270–288. MR 0151953. .

[muz04-5] Muzychuk, Mikhail (2004). "A Solution of the Isomorphism Problem for Circulant Graphs". Proc. London Math. Soc. 88: 1–41. doi:10.1112/s0024611503014412. MR 2018956.

[ep04-6] Evdokimov, Sergei; Ponomarenko, Ilia (2004). "Recognition and verification of an isomorphism of circulant graphs in polynomial time". St. Petersburg Math. J. 15: 813–835. doi:10.1090/s1061-0022-04-00833-7. MR 2044629.