Cyclic edge-cuts in fullerene graphs

Kardoš, František; Škrekovski, Riste

doi:10.1007/s10910-007-9296-9

Cited by 47 publications

(56 citation statements)

References 12 publications

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“…In light of the information available for M 1 and M 2 of trees, unicyclic graphs, bicyclic graphs et al, it is natural to consider other classes of graphs, and the connected graphs with k cut edges are a reasonable starting point for such an investigation. The connected graphs with k cut edges have been considered in mathematical literature [26,36,41,53], whereas to our best knowledge, the Zagreb indices of connected graphs with k cut edges were, so far, not considered in the chemical literature. On the other hand, connected graphs with k cut edges represent an important class of molecules.…”

Section: Introductionmentioning

confidence: 99%

On the Extremal Zagreb Indices of Graphs with Cut Edges

Feng

2009

Acta Appl Math

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For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M 2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M 1 -and M 2 -value are obtained, respectively, and uniquely, at K k n (resp. P k n ), where K k n is a graph obtained by joining k independent vertices to one vertex of K n−k and P k n is a graph obtained by connecting a pendent path P k+1 to one vertex of C n−k .

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Section: Introductionmentioning

confidence: 99%

On the Extremal Zagreb Indices of Graphs with Cut Edges

Feng

2009

Acta Appl Math

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“…For the structure of k-belts on simple 3-polytopes with at most hexagonal facets see [20]. Related results on cyclic k-edge cuts see in [21,22,23,24,25].…”

Section: Definition 4 a Cyclic Sequence Of K Facetsmentioning

confidence: 99%

“…The significance of the family D 5k is described by the following theorem, which follows directly from [23] or [24].…”

Section: Corollary 1 a Polytope P Is Flag If And Only If Straighteninmentioning

confidence: 99%

Finite sets of operations sufficient to construct any fullerene from C 20

Buchstaber

Erokhovets

2016

Struct Chem

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We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene C 20 ). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge-and two edges-truncations.

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“…There were several partial results [17,19,21], until this special case of Barnette's conjecture was settled by Kardoš, who provided a computer-assisted proof [15]. .…”

Section: M-generalized Fullerene Graphsmentioning

confidence: 99%

Matchings in m-generalized fullerene graphs

2015

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A connected planar graph is called m-generalized fullerene if two of its faces are mgons and all other faces are pentagons and hexagons. In this paper we first determine some structural properties of m-generalized fullerenes and then use them to obtain new results on the enumerative aspects of perfect matchings in such graphs. We provide both upper and lower bounds on the number of perfect matchings in m-generalized fullerene graphs and state exact results in some special cases.

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Cyclic edge-cuts in fullerene graphs

Cited by 47 publications

References 12 publications

On the Extremal Zagreb Indices of Graphs with Cut Edges

On the Extremal Zagreb Indices of Graphs with Cut Edges

Finite sets of operations sufficient to construct any fullerene from C 20

Matchings in m-generalized fullerene graphs

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