A cycle of a graph , also called a circuit if the first vertex is not specified, is a subset of the edge set of that forms a path such that the first node of the path corresponds to the last. A maximal set of edge-disjoint cycles of a given graph can be obtained using ExtractCycles[g] in the Wolfram Language package Combinatorica` .
A cycle that uses each graph vertex of a graph exactly once is called a Hamiltonian cycle.
A graph containing no cycles of length three is called a triangle-free graph, and a graph containing no cycles of length four is called a square-free graph.
A graph containing no cycles of any length is known as an acyclic graph, whereas a graph containing at least one cycle is called a cyclic graph. A graph possessing exactly one (undirected, simple) cycle is called a unicyclic graph. A connected acyclic graph is known as a tree, and a possibly disconnected acyclic graph is known as a forest.
The length of the shortest graph cycle (if any) in a given graph is known as the girth, and the length of a longest cycle is known as the graph circumference.
Any cycle of a graph can be expressed as a sum modulo 2 over a fundamental cycle set of the graph (Gould 1959, Paton 1969).
The minimum number of swaps between vertices in a random circular embedding of a cycle to put in its standard configuration is considered by Björner and Wachs (1982) and (Stanley 1999).
An acyclic graph is bipartite, and a cyclic graph is bipartite iff all its cycles are of even length (Skiena 1990, p. 213).
The number of (undirected) closed -walks in a graph with adjacency matrix is given by , where denotes the matrix trace. In order to compute the number of -cycles, all closed -walks that are not cycles must be subtracted. One would think that by analogy with the matching-generating polynomial, independence polynomial, etc., a cycle polynomial whose coefficients are the numbers of cycles of length would be defined. While no such polynomial seems not to have been defined in the literature (instead, "cycle polynomials" commonly instead refers to a polynomial corresponding to cycle indices of permutation groups), they are defined in this work.
The number of -cycles is related to the matrix of path counts by
(1)
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where denotes the matrix trace and is the adjacency matrix (Perepechko and Voropaev).
Graphs corresponding to closed walks of length are known as k-cyclic graphs, or "-graphs" for short. The numbers of -graphs for , 4, ... are 1, 3, 3, 10, 12, 35, 58, 160, 341, 958, 2444, 7242, 21190, 67217, 217335, ... (OEIS A081809; FlowProblems).
Harary and Manvel (1972) gave the following closed forms for small :
(2)
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(4)
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(5)
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(6)
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(7)
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(8)
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(with variants from Perepechko and Voropaev), where is the number of edges of the graph, denotes the element of , is the matrix formed from the diagonal elements of , and is the th vertex degree. Alon et al. (1997) extended these results up to , although with explicit formulas given only up to . Exact formulas for and are given by
(9)
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where is the th vertex degree (Perepechko and Voropaev; S. Perepechko, pers. comm., Jan. 4, 2014).
Khomenko and Golovko (1972) gave a formula giving the number of cycles of any length, but its computation requires computing and performing matrix operations involving all subsets up to size , making it computationally expensive. A simplified and improved version of the Khomenko and Golovko formula is given by
(10)
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for , 4, ..., , where is the th matrix power of the submatrix of the adjacency matrix with the subset of rows and columns deleted (Perepechko and Voropaev). The case therefore gives the number of Hamiltonian cycles.
Giscard et al. (2016) gave the formula for the number of undirected -cycles in a graph as
(11)
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where the sum is over connected induced subgraphs of , denotes the number of neighbors of in (namely vertices of which are not in and such that there exists at least one edge from to a vertex of ), denotes the matrix trace, and is the th matrix power of the adjacency matrix of the graph .
Let denote the total number of undirected cycles in a graph and the circuit rank. Then
(12)
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(Kirchhoff 1847, Ahrens 1897, König 1936, Volkmann 1996). The total numbers of undirected cycles for all simple graphs of orders , 2, ... are 0, 0, 1, 13, 143, 1994, 39688, ... (OEIS A234601).
(13)
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iff any two cycles have no edge in common (Volkmann 1996). Among connected graphs, the equality therefore holds for (and only for) cactus graphs. Mateti and Deo (1976) proved that there are "essentially" only four graphs with : the complete graphs and , the complete bipartite graph , and (Volkmann 1996).
The total number of undirected cycles also satisfies
(14)
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and
(15)
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where is the number of vertices and is the minimum vertex degree (Volkmann 1996).
The following table gives the number of undirected graph cycles for various classes of graphs.
graph | OEIS | sequence |
Andrásfai graph | A234602 | 0, 1, 29, 1014, 72273, 9842527, ... |
antiprism graph | A077263 | X, X, 63, 179, 523, 1619, 5239, 17379, ... |
bishop graph | A234636 | X, 0, 3, 106, 17367, 24601058, 638520866656, ... |
-black bishop graph | A234603 | X, X, X, 53, 12424, 12300529, ... |
cocktail party graph | A167987 | 0, 1, 63, 2766, 194650, 21086055, 3257119761, ... |
complete bipartite graph | A070968 | 0, 1, 15, 204, 3940, 113865, 4662231, ... |
complete tripartite graph | A234616 | 1, 63, 6705, 1960804, 1271288295, 1541975757831, ... |
complete graph | A002807 | 1, 7, 37, 197, 1172, 8018, ... |
-crossed prism graph | A234617 | 28, 107, 380, 1345, 4878, 18219, ... |
crown graph | A234618 | 1, 28, 586, 16676, 674171, 36729512, ... |
cube-connected cycle graph | A000000 | X, X, 2664, ... |
cycle graph | A000012 | 1, 1, 1, 1, 1, 1, 1, 1, ... |
folded cube graph | A234619 | 0, 0, 7, 204, 322248, ... |
grid graph | A140517 | X, 1, 13, 213, 9349, 122236, 487150371, ... |
grid graph | A234620 | X, 28, 3426491, ... |
halved cube graph | A234621 | 0, 0, 7, 2766, 4678134804, ... |
Hanoi graph | A000000 | 1, 11, 1761, ... |
hypercube graph | A085408 | 0, 1, 28, 14704, 51109385408, ... |
-king graph | A234622 | X, 7, 348, 136597, 545217435, 21964731190911, ... |
-knight graph | A234623 | X, 0, 1, 222, 128769, 959427728, ... |
-ladder graph | A000217 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ... |
Möbius ladder | A020873 | X, X, 15, 29, 53, 95, 171, 313, 585, ... |
Mycielski graph | A234625 | 0, 0, 1, 337, 445228418, ... |
odd graph | A301558 | 0, 1, 57, 872137842, ... |
permutation star graph | A000000 | 0, 0, 1, 5442, ... |
prism graph | A077265 | 14, 28, 52, 94, 170, ... |
-queen graph | A234626 | 0, 7, 8215, 2080941496, 269529670654115055, ... |
rook graph | A234624 | 0, 1, 312, 3228524, 6198979538330, ... |
Sierpiński gasket graph | A234634 | 1, 11, 1033, ... |
sun graph | A234627 | X, X, 11, 44, 198, 1036, 6346, 45019, 364039, ... |
sunlet graph | A000000 | X, X, 1, 1, 1, 1, 1, 1, 1, 1, ... |
triangular graph | A234629 | 0, 1, 63, 15703, 58520309, ... |
web graph | A077265 | 14, 28, 52, 94, 170, 312, 584, 1114, ... |
wheel graph | A002061 | 7, 13, 21, 31, 43, 57, ... |
-white bishop graph | A234630 | X, X, X, 53, 4943, 12300529, ... |
Closed forms for some of these classes of graphs are summarized in the following table.