Clar structures vs Fries structures in hexagonal systems

Clar structures vs Fries structures in hexagonal systems A hexagonal system H is a 2-connected bipartite plane graph such that all inner faces are hexagons, which is often used to model the structure of a benzenoid hydrocarbon or graphen. A perfect matching of H is a set of disjoint edges which covers all vertices of H. A resonant set S of H is a set of hexagons in which every hexagon is M-alternating for some perfect matching M. The Fries number of H is the size of a maximum resonant set and the Clar number of H is the size of a maximum independent resonant set (i.e. all hexagons are disjoint). A pair of hexagonal systems with the same number of vertices is called a contra-pair if one has a larger Clar number but the other has a larger Fries number. In this paper, we investigates the Fries number and Clar number for hexagonal systems, and show that a catacondensed hexagonal system has a maximum resonant set containing a maximum independent resonant set, which is conjectured for all hexagonal systems. Further, our computation results demonstrate that there exist many contra-pairs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Computation Elsevier

Clar structures vs Fries structures in hexagonal systems

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Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier Inc.
ISSN
0096-3003
eISSN
1873-5649
D.O.I.
10.1016/j.amc.2018.02.014
Publisher site
See Article on Publisher Site

Abstract

A hexagonal system H is a 2-connected bipartite plane graph such that all inner faces are hexagons, which is often used to model the structure of a benzenoid hydrocarbon or graphen. A perfect matching of H is a set of disjoint edges which covers all vertices of H. A resonant set S of H is a set of hexagons in which every hexagon is M-alternating for some perfect matching M. The Fries number of H is the size of a maximum resonant set and the Clar number of H is the size of a maximum independent resonant set (i.e. all hexagons are disjoint). A pair of hexagonal systems with the same number of vertices is called a contra-pair if one has a larger Clar number but the other has a larger Fries number. In this paper, we investigates the Fries number and Clar number for hexagonal systems, and show that a catacondensed hexagonal system has a maximum resonant set containing a maximum independent resonant set, which is conjectured for all hexagonal systems. Further, our computation results demonstrate that there exist many contra-pairs.

Journal

Applied Mathematics and ComputationElsevier

Published: Jul 15, 2018

References

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