Star (graph theory)
Star  

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


Vertices  k+1 
Edges  k 
Diameter  minimum of (2, k) 
Girth  ∞ 
Chromatic number  minimum of (2, k + 1) 
Chromatic index  k 
Properties  Edgetransitive Tree Unit distance Bipartite 
Notation  S_{k} 
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 edgegraceful when k is even and not when k is odd. It is an edgetransitive matchstick graph, and has diameter 2 (when k > 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 clawfree 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.
References
Wikimedia Commons has media related to Star graphs. 
 ↑ Faudree, Ralph; Flandrin, Evelyne; Ryjáček, Zdeněk (1997), "Clawfree graphs — A survey", Discrete Mathematics, 164 (1–3): 87–147, doi:10.1016/S0012365X(96)000453, MR 1432221<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Chudnovsky, Maria; Seymour, Paul (2005), "The structure of clawfree graphs", Surveys in combinatorics 2005 (PDF), London Math. Soc. Lecture Note Ser., 327, Cambridge: Cambridge Univ. Press, pp. 153–171, MR 2187738<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Whitney, Hassler (January 1932), "Congruent Graphs and the Connectivity of Graphs", American Journal of Mathematics, 54 (1): 150–168, doi:10.2307/2371086, JSTOR 2371086<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Gottlieb, J.; Julstrom, B. A.; Rothlauf, F.; Raidl, G. R. (2001), "Prüfer numbers: A poor representation of spanning trees for evolutionary search", Proc. Genetic and Evolutionary Computation Conference (PDF), Morgan Kaufmann, pp. 343–350<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Hakimi, S. L.; Mitchem, J.; Schmeichel, E. E. (1996), "Star arboricity of graphs", Discrete Math., 149: 93–98, doi:10.1016/0012365X(94)003138<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Fertin, Guillaume; Raspaud, André; Reed, Bruce (2004), "Star coloring of graphs", Journal of Graph Theory, 47 (3): 163–182, doi:10.1002/jgt.20029<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Robertson, Neil; Seymour, Paul D. (1991), "Graph minors. X. Obstructions to treedecomposition", Journal of Combinatorial Theory, 52 (2): 153–190, doi:10.1016/00958956(91)90061N<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Linial, Nathan (2002), "Finite metric spaces–combinatorics, geometry and algorithms", Proc. International Congress of Mathematicians, Beijing, 3, pp. 573–586, arXiv:math/0304466<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>