TOPICS
Search

Polygon Diagonal Intersection Graph


PolygonDiagonalIntersectionGraph

Consider the plane figure obtained by drawing each diagonal in a regular polygon with n vertices. If each point of intersection is associated with a node and diagonals are split ar each intersection to form segments associated with edges, the resulting figure is a planar graph here termed the polygon diagonal intersection graph and denoted R_n.

For n=1, 2, ..., the vertex counts v_n of R_n are 1, 2, 3, 5, 10, 19, 42, 57, 135, 171, ... (OEIS A007569), which are given by a finite sum of

 delta_m(n)={1   if n=0 (mod m); 0   otherwise.
(1)

times polynomials in n with m=2, 4, 6, 12, 18, 24, 30, 42, 60, 84, 90, 120, and 210 (Poonen and Rubinstein 1998).

For n=1, 2, ..., the edge counts e_n of R_n are 0, 1, 3, 8, 20, 42, 91, 136, 288, ... (OEIS A135565), which are again given by a finite sum of polynomials times delta_m(n).

Similarly, for n=1, 2, ..., the numbers of regions f_n into which the polygon is divided are given by 1, 4, 11, 24, 50, 80, 154, 220, 375, ... (OEIS A007678), where the nth term is given in closed form by

 f_n=1/(24)(n^4-6n^3+23n^2-42n+24)+1/(48)(-5n^3+42n^2-40n-48)delta_2(n)-3/4ndelta_4(n)+1/(12)(-53n^2+310n)delta_6(n)+(49)/2ndelta_(12)(n)+32ndelta_(18)(n)+19ndelta_(24)(n)-36ndelta_(30)(n)-50ndelta_(42)(n)-190ndelta_(60)(n)-78ndelta_(84)(n)-48ndelta_(90)(n)-78ndelta_(120)(n)-48ndelta_(210)(n).
(2)

For n odd, all terms but the first drop out, so the numbers of regions are given by

 f_n=1/(24)(n^4-6n^3+23n^2-42n+24).
(3)

Precomputed properties of polygonal diagonal intersection graphs are implemented in the Wolfram Language as GraphData[{"DiagonalIntersection", n}].


See also

Complete Graph, Euler's Polygon Division Problem, Polygon Diagonal, Regular Polygon, Regular Polygon Division by Diagonals

Explore with Wolfram|Alpha

References

Dan. "Regular n-Gon With Diagonals: Bounds on Area of Largest Cell?" Dec. 17, 2022. https://mathoverflow.net/q/436753.Griffiths, M. "Counting the Regions in a Regular Drawing of K_(n,n)." J. Int. Seq. 13, No. 10.8.5, 2010. https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths2/griffiths.html.Meeus, J. Wiskunde Post (Belgium) 10, 62-63, 1972.Pickover, C. A. "The Beauty of Polygon Slicing." Ch. 58 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 132-134 and 314, 2002.Poonen, B. and Rubinstein, M. "Number of Intersection Points Made by the Diagonals of a Regular Polygon." SIAM J. Disc. Math. 11, 135-156, 1998.Sloane, N. J. A. Sequences A007678, A007569/M0724, and A135565 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Polygon Diagonal Intersection Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html

Subject classifications