# Star (graph theory)

__: Tree graph with one central node and leaves of length 1__

**Short description**Star | |
---|---|

The star S_{7}. (Some authors index this as S_{8}.) | |

Vertices | k + 1 |

Edges | k |

Diameter | 2 |

Girth | ∞ |

Chromatic number | 2 |

Chromatic index | k |

Properties | Edge-transitive Tree Unit distance Bipartite |

Notation | S_{k} |

Table of graphs and parameters |

In graph theory, a **star** *S*_{k} is the complete bipartite graph *K*_{1,k} : a tree with one internal node and k leaves (but no internal nodes and *k* + 1 leaves when *k* ≤ 1). Alternatively, some authors define *S*_{k} to be the tree of order k with maximum diameter 2; in which case a star of *k* > 2 has *k* − 1 leaves.

A star with 3 edges is called a **claw**.

The star *S*_{k} is edge-graceful when k is even and not when k is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when *l* > 1), girth ∞ (it has no cycles), chromatic index k, and chromatic number 2 (when *k* > 0). Additionally, the star has large automorphism group, namely, the symmetric group on k letters.

Stars may also be described as the only connected graphs in which at most one vertex has degree greater than one.

## Relation to other graph families

Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph.^{[1]}^{[2]} They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomorphic line graphs are themselves isomorphic, with the exception of the claw and the triangle *K*_{3}.^{[3]}

A star is a special kind of tree. As with any tree, stars may be encoded by a Prüfer sequence; the Prüfer sequence for a star *K*_{1,k} consists of *k* − 1 copies of the center vertex.^{[4]}

Several graph invariants are defined in terms of stars. Star arboricity is the minimum number of forests that a graph can be partitioned into such that each tree in each forest is a star,^{[5]} and the star chromatic number of a graph is the minimum number of colors needed to color its vertices in such a way that every two color classes together form a subgraph in which all connected components are stars.^{[6]} The graphs of branchwidth 1 are exactly the graphs in which each connected component is a star.^{[7]}

## Other applications

The set of distances between the vertices of a claw provides an example of a finite metric space that cannot be embedded isometrically into a Euclidean space of any dimension.^{[8]}

The star network, a computer network modeled after the star graph, is important in distributed computing.

A geometric realization of the star graph, formed by identifying the edges with intervals of some fixed length, is used as a local model of curves in tropical geometry. A tropical curve is defined to be a metric space that is locally isomorphic to a star-shaped metric graph.

## See also

- Star (simplicial complex) - a generalization of the concept of a star from a graph to an arbitrary simplicial complex.

## References

- ↑ "Claw-free graphs — A survey",
*Discrete Mathematics***164**(1–3): 87–147, 1997, doi:10.1016/S0012-365X(96)00045-3. - ↑ "The structure of claw-free graphs",
*Surveys in combinatorics 2005*, London Math. Soc. Lecture Note Ser.,**327**, Cambridge: Cambridge Univ. Press, 2005, pp. 153–171, http://www.columbia.edu/~mc2775/claws_survey.pdf. - ↑ "Congruent Graphs and the Connectivity of Graphs",
*American Journal of Mathematics***54**(1): 150–168, January 1932, doi:10.2307/2371086. - ↑ Gottlieb, J.; Julstrom, B. A.; Rothlauf, F.; Raidl, G. R. (2001). "Prüfer numbers: A poor representation of spanning trees for evolutionary search".
*GECCO-2001: Proceedings of the Genetic and Evolutionary Computation Conference*. Morgan Kaufmann. pp. 343–350. ISBN 1558607749. https://web.archive.org/web/20060926171652/http://www.ads.tuwien.ac.at/publications/bib/pdf/gottlieb-01.pdf. - ↑ Hakimi, S. L.; Mitchem, J.; Schmeichel, E. E. (1996), "Star arboricity of graphs",
*Discrete Math.***149**: 93–98, doi:10.1016/0012-365X(94)00313-8 - ↑ Fertin, Guillaume; Raspaud, André (2004), "Star coloring of graphs",
*Journal of Graph Theory***47**(3): 163–182, doi:10.1002/jgt.20029, https://hal.archives-ouvertes.fr/hal-00307788. - ↑ "Graph minors. X. Obstructions to tree-decomposition",
*Journal of Combinatorial Theory***52**(2): 153–190, 1991, doi:10.1016/0095-8956(91)90061-N. - ↑ "Finite metric spaces–combinatorics, geometry and algorithms",
*Proc. International Congress of Mathematicians, Beijing*,**3**, 2002, pp. 573–586, Bibcode: 2003math......4466L

Original source: https://en.wikipedia.org/wiki/Star (graph theory).
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