digplanet beta 1: Athena
Share digplanet:


Applied sciences






















Star network 7.svg
The star S7. (Some authors index this as S8.)
Vertices k+1
Edges k
Diameter minimum of (2, k)
Chromatic number minimum of (2, k + 1)
Chromatic index k
Properties Edge-transitive
Unit distance
Notation Sk

In graph theory, a star Sk is the complete bipartite graph K1,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 Sk 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 Sk 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 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[edit]

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 K3.[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 K1,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 star graphs S3, S4, S5 and S6.

Other applications[edit]

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.


  1. ^ Faudree, Ralph; Flandrin, Evelyne; Ryjáček, Zdeněk (1997), "Claw-free graphs — A survey", Discrete Mathematics 164 (1–3): 87–147, doi:10.1016/S0012-365X(96)00045-3, MR 1432221 .
  2. ^ Chudnovsky, Maria; Seymour, Paul (2005), "The structure of claw-free graphs", Surveys in combinatorics 2005 (PDF), London Math. Soc. Lecture Note Ser. 327, Cambridge: Cambridge Univ. Press, pp. 153–171, MR 2187738 .
  3. ^ 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 .
  4. ^ 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 .
  5. ^ 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 
  6. ^ Fertin, Guillaume; Raspaud, André; Reed, Bruce (2004), "Star coloring of graphs", Journal of Graph Theory 47 (3): 163–182, doi:10.1002/jgt.20029 .
  7. ^ Robertson, Neil; Seymour, Paul D. (1991), "Graph minors. X. Obstructions to tree-decomposition", Journal of Combinatorial Theory 52 (2): 153–190, doi:10.1016/0095-8956(91)90061-N .
  8. ^ Linial, Nathan (2002), "Finite metric spaces–combinatorics, geometry and algorithms", Proc. International Congress of Mathematicians, Beijing 3, pp. 573–586, arXiv:math/0304466 

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Star_(graph_theory) — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
21615 videos foundNext > 

Graph Theory: 64. Vertex Colouring

In this video we define a (proper) vertex colouring of a graph and the chromatic number of a graph. We discuss some basic facts about the chromatic number as ...

Graph Theory: 57. Planar Graphs

A planar graph is a graph that can be drawn in the plane without any edge crossings. Such a drawing (with no edge crossings) is called a plane graph. A given ...

[Discrete Math 2] Graph Coloring and Chromatic Polynomials

Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW Like us on Facebook: http://on.fb.me/1vWwDRc Submit your questions on ...

Lecture - 11 The Graph Theory Approach for Electrical Circuits(Part-I)

Lecture series on Dynamics of Physical System by Prof. Soumitro Banerjee, Department of Electrical Engineering, IIT Kharagpur.For more details on NPTEL visit ...

Graph Theory: Kruskal's Algorithm

This lesson explains how to apply Kruskal's algorithm to find the minimum cost spanning tree. Site: http://mathispower4u.com.

Graph Theory: 48. Complement of a Graph

In this video I define the complement of a graph and what makes a graph self-complementary. I show some examples, for orders 4 and 5 and discuss a ...

Graph Theory: 63. Petersen Graph is Non-Planar

In this video we give two proofs for why the Petersen graph is non-planar. -- Bits of Graph Theory by Dr. Sarada Herke. BIG NEWS - Check out my new course on ...

Basic Graph Theory Example in Hindi

Graph theory in hindi buy ipathshala, learn how to calculate points and edges in a graph.

Graph Theory [ Tree, Co-Tree, Branches, Links...]

A video that clearly explains what different terms in graph theory mean and how to form them. The formation of Tree, Co-Tree, identification of branches, links, ...

Network Theory: 6 Network Topology

For full courses, transcriptions & downloads please see: http://complexitylearning.io We call the overall structure to a network its topology where topology simply ...

21615 videos foundNext > 

We're sorry, but there's no news about "Star (graph theory)" right now.


Oops, we seem to be having trouble contacting Twitter

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!

Searchlight Group

Digplanet also receives support from Searchlight Group. Visit Searchlight