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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.

Star | |
---|---|

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 |

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.

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]}

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.

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

Wikimedia Commons has media related to Star graphs.

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