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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 kresonant fullerene graphs ∗
, 801
"... 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.