Star Graph

The star graph  of order , sometimes simply known as an "-star" (Harary 1994, pp. 17-18; Pemmaraju and Skiena 2003, p. 248; Tutte 2005, p. 23), is a tree on  nodes with one node having vertex degree  and the other  having vertex degree 1. The star graph  is therefore isomorphic to the complete bipartite graph  (Skiena 1990, p. 146).

Note that there are two conventions for the indexing for star graphs, with some authors (e.g., Gallian 2007), adopting the convention that  denotes the star graph on  nodes.

 is isomorphic to "the" claw graph. A star graph is sometimes termed a "claw" (Hoffman 1960) or a "cherry" (Erdős and Rényi 1963; Harary 1994, p. 17).

Star graphs  are always graceful. Star graphs can be constructed in the Wolfram Language using StarGraph[n]. Precomputed properties of star graphs are available via GraphData["Star", n].

The chromatic polynomial of  is given by

and the chromatic number is 1 for , and  otherwise.

The line graph of the star graph  is the complete graph .

Note that -stars should not be confused with the "permutation" -star graph (Akers et al. 1987) and their generalizations known as -star graphs (Chiang and Chen 1995) encountered in computer science and information processing.

REFERENCES

Akers, S.; Harel, D.; and Krishnamurthy, B. "The Star Graph: An Attractive Alternative to the -Cube." In Proc. International Conference of Parallel Processing, pp. 393-400, 1987.

Chiang, W.-K. and Chen, R.-J. "The -Star Graph: A Generalized Star Graph." Information Proc. Lett. 56, 259-264, 1995.

Erdős, P. and Rényi, A. "Asymmetric Graphs." Acta Math. Acad. Sci. Hungar. 14, 295-315, 1963.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018.

 https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.

Hoffman, A. J. "On the Uniqueness of the Triangular Association Scheme." Ann. Math. Stat. 31, 492-497, 1960.

Pemmaraju, S. and Skiena, S. "Cycles, Stars, and Wheels." §6.2.4 in Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, pp. 248-249, 2003.

Skiena, S. "Cycles, Stars, and Wheels." §4.2.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 83 and 144-147, 1990.

Tutte, W. T. Graph Theory. Cambridge, England: Cambridge University Press, 2005.