## Crown 2 Lesson 4 Homework

## Crown Graph

The -crown graph for an integer is the graph with vertex set

(1) |

and edge set

(2) |

It is therefore equivalent to the complete bipartite graph with horizontal edges removed.

The -crown graph is isomorphic to the rook complement graph (e.g., Brouwer *et al. *1989, p. 222), where denotes the graph Cartesian product.

The crown graphs are distance-transitive (Brouwer *et al. *1989, p. 222), and therefore also distance-regular. They are also Taylor graphs.

The -crown graph is isomorphic to the Haar graph. Other special cases are summarized in the following table.

The numbers of vertices in for , 4, ... are therefore 6, 8, 10, 12, ... (the even numbers), while the numbers of edges are 6, 12, 20, 30, 42, ... (OEIS A002378).

The numbers of directed Hamiltonian cycles for , 4, 5, ... are given by 2, 12, 312, 9600, 416880, ... (OEIS A094047), which has the beautiful closed form

(3) |

(M. Alekseyev, pers. comm., Feb. 10, 2008).

Closed formulas for the numbers of -graph cycles of are given by for odd and

(E. Weisstein, Nov. 16, 2014).

The independence polynomial of the -crown graph is

(7) |

which satisfies the recurrence equation

(8) |

Another class of graph referred to as "crown" graphs by Gallian (2007) are cycle graphs with pendant cycles.

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