Star (graph theory)
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
Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. 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.
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