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 and star graphs on nodes are series-reduced trees.

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 . The simplex graph of is the book 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.

A different generalization of the star graph in which points are placed along each of the arms of the star (as opposed to 1 for the usual star graph) might be termed the -spoke graph.