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

14.3 Classification of Elementary ({2, 3, 4, 5}, 3)-Polycycles

For 2 ≤ m ≤ ∞, denote by Barrelm the ({5}, 3)-polycycle with two m-gonal holes separated by two m-rings of 5-gonal proper faces. This polycycle is nonextensible if and only if m ≥ 6. Its symmetry group, Dmd, coincideswith the symmetry group of the underlying 3-valent plane graph if and only if m ≠ 5. See below for pictures of Barrelm for m = 2, 3, 4, 5, and ∞:

images

Theorem 14.3 The list of elementary ({2, 3, 4, 5},3)-polycycles consists of:

(i) 204 sporadic ({2, 3, 4, 5},3)simp-polycycles, given in Appendix 1.

(ii) Six ({3, 4, 5},3)simp-polycycles, infinite in one direction:

images

(iii) 21 = images infinite series obtained by taking two ends of the infinite polycycles from (ii) above and concatenating them.

For example, merging α with itself produces the infinite series of elementary ({5},3)simp-polycycles, denoted by En in [12]. See Figure 14.1 for the first 3 members (starting with 6 faces) of two such series: αα and βε.

(iv) The infinite series of Barrelm, 2 ≤ m ≤ ∞, and its nonorientable quotient for m odd.

Proof. Take an elementary ({2, 3, 4, 5},3)simp-polycycle P that, by Theorem 14.2, is kernel-elementary. If its kernel is empty, ...

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