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ON CYCLIC EDGECONNECTIVITY OF FULLERENES
, 2007
"... A graph is said to be cyclic kedgeconnected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclickedge cutset and it is called a trivial cyclickedge cutset if at least one of the resulting two components ind ..."
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A graph is said to be cyclic kedgeconnected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclickedge cutset and it is called a trivial cyclickedge cutset if at least one of the resulting two components induces a single kcycle. It is known that fullerenes, that is, 3connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5edgeconnected. In this article it is shown that a fullerene F containing a nontrivial cyclic5edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is n ⌊ shown that F has a Hamilton cycle, and as a consequence at least 15 ·2 20 ⌋ perfect matchings, where n is the order of F.
On kresonant fullerene graphs
, 2009
"... A fullerene graph F is a 3connected plane cubic graph with exactly 12 pentagons and the remaining hexagons. Let M be a perfect matching of F. A cycle C of F is Malternating if the edges of C appear alternately in and off M. A set H of disjoint hexagons of F is called a resonant pattern (or sextet ..."
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A fullerene graph F is a 3connected plane cubic graph with exactly 12 pentagons and the remaining hexagons. Let M be a perfect matching of F. A cycle C of F is Malternating if the edges of C appear alternately in and off M. A set H of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M such that all hexagons in H are Malternating. A fullerene graph F is kresonant if any i (0 ≤ i ≤ k) disjoint hexagons of F form a resonant pattern. In this paper, we prove that every hexagon of a fullerene graph is resonant and all leapfrog fullerene graphs are 2resonant. Further, we show that a 3resonant fullerene graph has at most 60 vertices and construct all nine 3resonant fullerene graphs, which are also kresonant for every integer k> 3. Finally, sextet polynomials of the 3resonant fullerene graphs are computed.
Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems
"... It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum c ..."
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It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number.
ON CYCLIC EDGECONNECTIVITY OF FULLERENES
, 2007
"... A graph is said to be cyclic kedgeconnected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclickedge cutset and it is called a trivial cyclickedge cutset if at least one of the resulting two components ind ..."
Abstract
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A graph is said to be cyclic kedgeconnected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclickedge cutset and it is called a trivial cyclickedge cutset if at least one of the resulting two components induces a single kcycle. It is known that fullerenes, that is, 3connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5edgeconnected. In this article it is shown that a fullerene F containing a nontrivial cyclic5edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is n ⌊ shown that F has a Hamilton cycle, and as a consequence at least 15 ·2 20 ⌋ perfect matchings, where n is the order of F.
Some Distance and Degree Graph INVARIANTS AND FULLERENE STRUCTURES
, 2013
"... In the thesis we concentrate to the part of graph theory that can be applied in chemistry. One of the aims is applying graphtheoretical methods to predict the properties of a chemical compound based on its molecule structure. Molecular descriptors or topological indices is one way of predicting so ..."
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In the thesis we concentrate to the part of graph theory that can be applied in chemistry. One of the aims is applying graphtheoretical methods to predict the properties of a chemical compound based on its molecule structure. Molecular descriptors or topological indices is one way of predicting some properties. We dedicate our attention to Zagreb indices, a modification of Randic ́ index called R ′ index, and Gutman index. For a simple graphG with n vertices andm edges, the inequalityM1(G)/n ≤M2(G)/m, where M1(G) and M2(G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. We characterize the intervals of vertex degrees that satisfy this inequality, and find an infinite family of connected graphs dissatisfying this inequality. We also present an algorithm that decides if an arbitrary set of vertex degrees satisfies the inequality and consider variable Zagreb index inequality. We also determine the graphs with extremal values for Gutman and R ′ indices, and find a trianglefree graph on n vertices with minimum R′(G) index. Fullerene molecules are carbon cage molecules arranged only in pentagons and hexagons.