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Extremal fullerene graphs with the maximum Clar number
, 801
"... A fullerene graph is a cubic 3connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The ma ..."
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A fullerene graph is a cubic 3connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar number is no more than ⌊n−12 6 ⌋. Many fullerenes with experimental evidence attain the upper bound, for instance, C60 and C70. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−12 6. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.
On the role of hypercubes in . . .
 DISCRETE MATHEMATICS
, 2006
"... The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent 13 if their symmetric difference forms the edge set of a hexagon of B. A family P of pairwise disjoint hexagons of a benzenoid graph B is resonant in B if B.P contains a ..."
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The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent 13 if their symmetric difference forms the edge set of a hexagon of B. A family P of pairwise disjoint hexagons of a benzenoid graph B is resonant in B if B.P contains at least one perfect matching, or if B.P is empty. It is proven that there exists a surjective map f 15 from the set of hypercubes of R(B) onto the resonant sets of B such that a kdimensional hypercube is mapped into a resonant set of cardinality k.