The -crown graph for an integer is the graph with vertex set
and edge set
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
(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
which satisfies the recurrence equation
Another class of graph referred to as "crown" graphs by Gallian (2007) are cycle graphs with pendant cycles.
Wolfram Web Resources
The #1 tool for creating Demonstrations and anything technical.
Explore anything with the first computational knowledge engine.
|Wolfram Demonstrations Project »|
Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
Join the initiative for modernizing math education.
|Online Integral Calculator »|
Solve integrals with Wolfram|Alpha.
|Step-by-step Solutions »|
Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.
|Wolfram Problem Generator »|
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
|Wolfram Education Portal »|
Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.
|Wolfram Language »|
Knowledge-based programming for everyone.
Are you a student or parent who dreads homework time at your house? You will find a great variety of resources on the internet that can make homework more manageable for everyone.
If you need information on a specific topic, go to www.askanexpert.com, which links students with the web sites of experts on such topics as science, animals, arts, health, trades, industry, and many others.
At Homework Helper you will find tips on writing, over 100 general reference links, and help with math problems - you can input a math problem and it will help you solve it!
Specific help in all the basic subjects is available at www.bigchalk.com. Click on the subject where help is needed.
Find help with science fair ideas, solve math problems, and create your own crossword puzzles and word searches at http://school.discovery.com. Also at this site, click on BJ Pinchbeck's Homework Helper, where you'll find over 700 links to homework help websites.
Homework Spot is full of helpful ideas and tips for homework and studying, as well as subject area assistance.
JPL Live Homework Help is an online classroom service designed to assist those from grades 4 to 12, Live Homework Help also provides assistance for those approaching college. Subjects include Math, Social Studies, Science, and English. Tutors are available every day, 2pm to midnight. Lessons and assistance in Spanish are also provided Sunday thru Thursday, 2pm to 10pm. Services are not available on Thanksgiving, Christmas, or New Year's Day.
The following links provide helpful information on specific subjects: