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| cycle graph | |
|---|---|
| Name | Cycle graph |
| Chromatic number | 2 or 3 |
| Diameter | floor(n/2) |
cycle graph is a simple, undirected graph consisting of a single cycle through all vertices. It appears as a fundamental example in graph theory and combinatorics and serves as a building block for more complex structures studied in algebraic graph theory and topological graph theory. Cycle graphs feature prominently in classical texts and in problems related to symmetry, spectral analysis, and network design.
A cycle graph on n vertices is a 2-regular connected graph whose edges form a single closed chain, often denoted by C_n in standard graph-theoretic literature such as works by Paul Erdős, Branko Grünbaum, Béla Bollobás, Frank Harary. For n ≥ 3 the graph is simple and has n edges and n vertices; for n = 2 it reduces to a multigraph variant historically discussed in expositions by Dénes Kőnig and Claude Berge. Basic combinatorial properties—degree sequence, connectivity, Eulerian and Hamiltonian character—are treated in texts from Richard Stanley and in surveys by László Lovász.
The adjacency spectrum of the cycle graph is explicit: eigenvalues are 2 cos(2πk/n) for k = 0,...,n−1, a formula appearing in analyses by Issai Schur and in applications by Hermann Weyl and Eugene Wigner. The characteristic polynomial and resulting eigenspaces connect to circulant matrix theory discussed in works by Philippe Flajolet and Peter J. Cameron. Automorphism group of the cycle is the dihedral group D_n, an observation used in group-theoretic studies by Emmy Noether and Évariste Galois, and exploited in representations in papers by William Burnside and Ferdinand Georg Frobenius.
Chromatic number of a cycle equals 2 for even n and 3 for odd n, a fact used in coloring problems studied by Kurt Gödel and Alfred Tarski in related decision contexts. Independence number, matching number, domination number, and toughness of cycles are standard values recorded in compendia by Harold N. Shapiro and R. L. Brooks; for example, matching and independence extremes appear in extremal results by Turán and Paul Turán. Connectivity and edge-connectivity equal 2; circumference equals n; and treewidth equals 2, parameters referenced in algorithmic frameworks by Richard Karp and Stephen Cook.
Cycle graphs embed naturally on the plane as simple polygons and on surfaces of higher genus in classical treatments by Henri Poincaré and Bernhard Riemann. Geometric graph representations as unit-distance graphs relate to results by Paul Erdős and Leo Moser; circular embeddings feature in studies by John Conway and Michael Atiyah concerning knot projections and polygonal linkages. Planar straight-line drawings and convex polygon realizations are standard constructs in computational geometry literature by Herbert Edelsbrunner and Jörg-Rüdiger Sack.
Cycle graphs model simple closed chains in chemistry as in August Kekulé’s descriptions of aromatic rings, and appear in molecular graph theory used by Linus Pauling and Robert Robinson. In electrical engineering and network design, cycles describe loop topologies referenced in works from Claude Shannon and Norbert Wiener. They arise in scheduling and routing problems in algorithmic studies by Donald Knuth and Edsger Dijkstra, and in physics in lattice models and spin chains treated by Hans Bethe and Ludwig Boltzmann.
Generalizations include polygonal chains, wheel graphs and circulant graphs; wheel graphs W_n combine a cycle with a universal hub node as in constructions studied by G. H. Hardy and J. E. Littlewood. Cartesian products and tensor products of cycles yield toroidal grids and ladder-like structures explored by John H. Conway and H. S. M. Coxeter. Random cycle ensembles and cycle spaces appear in algebraic topology and matroid theory in works by Samuel Eilenberg and Saunders Mac Lane, while directed cycles and multicycles are treated in combinatorial optimization literature by Jack Edmonds and László Lovász.