Determination of Zhang-Zhang Polynomials for Various Classes of Benzenoid Systems: Non-Heuristic Approach

Abstract

We employ a graphical proof-oriented tool, ZZDecomposer, to discover formal derivations of Zhang-Zhang (ZZ) polynomials for various families and subfamilies of benzenoid structures including tripods, zigzag-edge coronoids fused with a starphene, oblate rectangles Or(m, 2), hexagons 0 (2,2, n), 0 (2,3, n), and 0(33, n), and multiple zigzag chains Z(4, n), Z(5, n), Z(6, n), Z (7, n), Z (8, n), and Z(9, n). Current derivations are based on formal graph decompositions of the analyzed structures. The decompositions provide appropriate recurrence formulas, which are subsequently solved, yielding closed-form expressions for the ZZ polynomials. We hope that in addition to many new basic facts about ZZ polynomials of some important classes of benzenoids, the current study will provide the researchers who are interested in mathematical graph theory with a practical guide to the ZZDecomposer functionality and will enable and facilitate their research.