Genus polynomials and crosscap‐number polynomials for ring‐like graphs. Issue 4 (12th November 2018)

• Record Type: Journal Article
• Authors: Chen, Yichao; Gross, Jonathan L.
• Is Part Of: Mathematische Nachrichten. Volume 292:Issue 4(2019)
• Page Start: 760
• Page End: 776
• Publication Date: 2018-11-12
• Subjects: bar‐ring -- Cayley–Hamilton theorem -- crosscap‐number polynomial -- genus polynomial -- ring‐like graph sequence -- Primary: 05C10; Secondary: 30B70 -- 42C05; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201800132
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: An H ‐ linear graph is obtained by transforming a collection of copies of a fixed graph H into a chain. An H ‐ ring‐like graph is formed by binding the two end‐copies of H in such a chain to each other. Genus polynomials have been calculated for bindings of several kinds. In this paper, we substantially generalize the rules for constructing sequences of H ‐ring‐like graphs from sequences of H ‐linear graphs, and we give a general method for obtaining a recursion for the genus polynomials of the graphs in a sequence of ring‐like graphs. We use Chebyshev polynomials to obtain explicit formulas for the genus polynomials of several such sequences. We also give methods for obtaining recursions for partial genus polynomials and for crosscap‐number polynomials of a bar‐ring of a sequence of disjoint graphs.