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Tr. Mat. Inst. Steklova, 2002, Volume 239, Pages 127–145 (Mi tm364)  

This article is cited in 6 scientific papers (total in 6 papers)

Extremal and Nonextendible Polycycles

M. Dezaa, M. I. Shtogrinb

a Ècole Normale Supérieure, Département de mathématiques et applications
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We continue the study of $(r,q)$-polycycles, i.e. planar graphs $G$ that admit a realization on the plane such that all internal vertices have degree $q$, all boundary vertices have degree at most $q$, and all internal faces are combinatorial $r$-gons; moreover, the vertices, edges, and internal faces form a cell complex. Two extremal problems related to chemistry are solved: the description of $(r,q)$-polycycles with the maximal number of internal vertices for a given number of faces, and the description of nonextendible $(r,q)$-polycycles. Numerous examples of isohedral polycycles (whose symmetry groups are transitive on faces) are presented. The main proofs involve an abstract cell complex $\mathbf P(G)$ obtained from a planar realization of the graph $G$ by replacing all its internal faces by regular Euclidean $r$-gons.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 239, 117–135

Bibliographic databases:
UDC: 514.17+519.17
Received in May 2002

Citation: M. Deza, M. I. Shtogrin, “Extremal and Nonextendible Polycycles”, Discrete geometry and geometry of numbers, Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov, Tr. Mat. Inst. Steklova, 239, Nauka, MAIK Nauka/Inteperiodika, M., 2002, 127–145; Proc. Steklov Inst. Math., 239 (2002), 117–135

Citation in format AMSBIB
\Bibitem{DezSht02}
\by M.~Deza, M.~I.~Shtogrin
\paper Extremal and Nonextendible Polycycles
\inbook Discrete geometry and geometry of numbers
\bookinfo Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov
\serial Tr. Mat. Inst. Steklova
\yr 2002
\vol 239
\pages 127--145
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm364}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1975140}
\zmath{https://zbmath.org/?q=an:1067.05021}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 239
\pages 117--135


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. Deza, M. I. Shtogrin, “Archimedean polycycles”, Russian Math. Surveys, 59:3 (2004), 564–566  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. M. Deza, M. I. Shtogrin, “A metric of constant curvature on polycycles”, Math. Notes, 78:2 (2005), 204–212  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. M. Deza, M. Dutour, M. I. Shtogrin, “Elliptic polycycles with holes”, Russian Math. Surveys, 60:2 (2005), 349–351  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. M. Deza, S. V. Shpektorov, M. I. Shtogrin, “Non-extendible finite polycycles”, Izv. Math., 70:1 (2006), 1–18  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Sikiric M.D., Deza M., Shtogrin M., “Filling of a given boundary by p–gons and related problems”, Discrete Applied Mathematics, 156:9 (2008), 1518–1535  crossref  mathscinet  zmath  isi  elib  scopus
    6. M. Deza, M. Dutour Sikirić, M. I. Shtogrin, “Fullerenes and disk-fullerenes”, Russian Math. Surveys, 68:4 (2013), 665–720  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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