228 research outputs found
Fullerenes with the maximum Clar number
The Clar number of a fullerene is the maximum number of independent resonant
hexagons in the fullerene. It is known that the Clar number of a fullerene with
n vertices is bounded above by [n/6]-2. We find that there are no fullerenes
whose order n is congruent to 2 modulo 6 attaining this bound. In other words,
the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is
bounded above by [n/6]-3. Moreover, we show that two experimentally produced
fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a
graph-theoretical characterization for fullerenes, whose order n is congruent
to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3
(respectively, [n/6]-2)
Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs
AbstractRobert E. Jamison characterized chordal graphs by the edge set of every k-cycle being the symmetric difference of k−2 triangles. Strongly chordal (and chordal bipartite) graphs can be similarly characterized in terms of the distribution of triangles (respectively, quadrilaterals). These results motivate a definition of ‘strongly chordal bipartite graphs’, forming a class intermediate between bipartite interval graphs and chordal bipartite graphs
Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
An integrable self-adjoint 7-point scheme on the triangular lattice and an
integrable self-adjoint scheme on the honeycomb lattice are studied using the
sublattice approach. The star-triangle relation between these systems is
introduced, and the Darboux transformations for both linear problems from the
Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A
geometric interpretation of the Laplace transformations of the self-adjoint
7-point scheme is given and the corresponding novel integrable discrete 3D
system is constructed.Comment: 15 pages, 6 figures; references added, some typos correcte
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