ABSTRACT
A plane cubic graph is called a (5,6,7)-fullerene if its faces are only composed of pentagons, hexagons and heptagons. In this paper, we completely characterize the cyclic edge-connectivity of the (5,6,7)-fullerene. Furthermore, we obtain that the anti-Kekulé number of the (5,6,7)-fullerene is 4 when the cyclic edge-connectivity is larger then three. In particular, we obtain some properties with respect to the anti-Kekulé number of the (5,6,7)-fullerene with cyclic 3 edge-connectivity if it is 2-extendable.
Acknowledgments
The authors are very grateful to the referees for the constructive suggestions and comments.
Funding
This work is supported by NSFC (11761056), by the NSF of Qinghai (2016-ZJ-947Q), by the Education Department of Shaanxi Province (15JK1135) and Human Society Department of Gansu Province, and by Shaanxi University of Technology (SLGQD14-14).