O'Reilly logo

Analysis of Complex Networks by Frank Emmert-Streib, Matthias Dehmer

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

14

Elementary Elliptic (R, q)-Polycycles

Michel Deza, Mathieu Dutour Sikirić, and Mikhail Shtogrin

A(R, q)-polycycle is a map whose faces, besides some disjoint holes, are i-gons, iR, and whose vertices have a degree between 2 and q with vertices outside of holes being q-valent. This notion arises in organic chemistry and crystallography as well as in purely mathematical contexts.

A(R, q)-polycycle is called elementary if it cannot be cut along an edge. Every (R, q)-polycycle can be uniquely decomposed into elementary ones. This decomposition is useful for computer enumeration and determination of classes of plane graphs and (R, q)-polycycles [6, 11–14, 17, 20]. A critical step for using the decomposition theorem is to be able to list all elementary polycycles occurring in a given problem.

A(R, q)-polycycle is called elliptic, parabolic, or hyperbolic if images (where r = maxiRi) is positive, zero, or negative, respectively). Here we determine all elementary elliptic (R, q)-polycycles. For parabolic and hyperbolic cases, there is a continuum of possibilities, so this method is less useful.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required