Genus polynomials and crosscap‐number polynomials for ring‐like graphs

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...

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Bibliographic Details
Published inMathematische Nachrichten Vol. 292; no. 4; pp. 760 - 776
Main Authors Chen, Yichao, Gross, Jonathan L.
Format Journal Article
LanguageEnglish
Published Weinheim Wiley Subscription Services, Inc 01.04.2019
Subjects
42C05
bar‐ring
Cayley–Hamilton theorem
Chains
Chebyshev approximation
crosscap‐number polynomial
genus polynomial
Graphs
Polynomials
Primary: 05C10; Secondary: 30B70
Rings (mathematics)
ring‐like graph sequence
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Summary: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.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201800132