Geometric Structure of Chemistry-Relevant Graphs

Deza, Michel-Marie; Sikirić, Mathieu Dutour; Штогрин, Михаил Иванович

doi:10.1007/978-81-322-2449-5

Cited by 14 publications

(36 citation statements)

References 20 publications

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“…By [9, Theorem 2], we have the following possibilities for the z-monodromy M F : 3 , −e 2 , e 1 ), where (e 1 , e 2 , e 3 ) is one of the cycles in the permutation D F , (M4) M F = (e 1 , −e 2 )(e 2 , −e 1 ), where (e 1 , e 2 , e 3 ) is one of the cycles in D F (e 3 and −e 3 are fixed points), (M5) M F = (D F ) −1 , (M6) M F = (−e 1 , e 2 , e 3 )(−e 3 , −e 2 , e 1 ), where (e 1 , e 2 , e 3 ) is one of the cycles in the permutation (D F ) −1 , (M7) M F = (e 1 , e 2 )(−e 1 , −e 2 ), where (e 1 , e 2 , e 3 ) is one of the cycles in D F (e 3 and −e 3 are fixed points). The triangulation Γ is locally z-knotted in the face F only in the cases (M1)-(M4).…”

Section: Z-monodromymentioning

confidence: 99%

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On two types of Z-monodromy in triangulations of surfaces

Pankov

Tyc

2019

Discrete Mathematics

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Let Γ be a triangulation of a connected closed 2-dimensional (not necessarily orientable) surface. Using zigzags (closed left-right paths), for every face of Γ we define the z-monodromy which acts on the oriented edges of this face. There are precisely 7 types of z-monodromies. We consider the following two cases: (M1) the z-monodromy is identity, (M2) the z-monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph Γ * formed by edges whose z-monodromies are of types (M1) and (M2), respectively, both are forests. We apply this statement to the connected sum of z-knotted triangulations.

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Section: Z-monodromymentioning

confidence: 99%

“…A zigzag of a graph embedded in a surface is a closed path, where any two consecutive edges, but not three, belong to a face [3,6]. Zigzags are known also as Petrie paths [2] or closed left-right paths [5,10].…”

Section: Introductionmentioning

confidence: 99%

On two types of Z-monodromy in triangulations of surfaces

Pankov

Tyc

2019

Discrete Mathematics

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“…All c 6 chordless 6-cycles of Icosahedron are exactly their 30 edge-containing ones and 10 face-containing ones, which are exactly the 10 equators and the weak zigzags ( [22]). All c 10 chordless 10-cycles of Dodecahedron are 30 edge-containing ones and 6 face-containing ones, which are exactly all 6 equators and the zigzags.…”

Section: 3mentioning

confidence: 99%

Generalized cut and metric polytopes of graphs and simplicial complexes

Deza¹,

Sikirić

2018

Optim Lett

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Given a graph G one can define the cut polytope CUTP(G) and the metric polytope METP(G) of this graph and those polytopes encode in a nice way the metric on the graph. According to Seymour's theorem, CUTP(G) = METP(G) if and only if K 5 is not a minor of G.We consider possibly extensions of this framework: (1) We compute the CUTP(G) and METP(G) for many graphs.(2) We define the oriented cut polytope WOMCUTP(G) and oriented multicut polytope OMCUTP(G) as well as their oriented metric version QMETP(G) and WQMETP(G). (3) We define an n-dimensional generalization of metric on simplicial complexes.

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“…Propositions 5 and 6 imply the following. Note that all values from Corollary 1 were given in [5,6], but the connection with Coxeter numbers is new. Now, we consider the half-cube 1 2 γ n and the E-polytopes 2 21 , 3 21 , 4 21 associated to the Coxeter systems D n and E i , i = 6, 7, 8 (respectively).…”

Section: 2mentioning

confidence: 99%

Zigzag Structure of Thin Chamber Complexes

Deza

Pankov

2017

Discrete Comput Geom

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Zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all zigzags in the simplex α n , the cross-polytope β n , the 24-cell, the icosahedron and the 600-cell are equal to the Coxeter numbers of A n , B n = C n , F 4 and H i , i = 3, 4, respectively. We also discuss in which cases two faces in a thin chamber complex can be connected by a zigzag.

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Geometric Structure of Chemistry-Relevant Graphs

Cited by 14 publications

References 20 publications

On two types of Z-monodromy in triangulations of surfaces

On two types of Z-monodromy in triangulations of surfaces

Generalized cut and metric polytopes of graphs and simplicial complexes

Zigzag Structure of Thin Chamber Complexes

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