21,597 research outputs found
Planar k-cycle resonant graphs with k=1,2
AbstractA connected graph is said to be k-cycle resonant if, for 1⩽t⩽k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M-alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and sufficient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and sufficient conditions for a graph to be k-cycle resonant, and further investigate planar k-cycle resonant graphs with k=1,2. Some new necessary and sufficient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established
K-CYCLE RESONANT GRAPHS
A connected graph G is said to be k-cycle resonant if, for 1 less than or equal to t less than or equal to k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M-alternating cycle. In this paper, we at the first time introduce the concept of k-cycle resonant graphs, and investigate some properties of k-cycle resonant graphs. Some simple necessary and sufficient conditions for a graph to be k-cycle resonant are given. The construction of k-cycle resonant hexagonal systems are also characterized
Fullerene graphs have exponentially many perfect matchings
A fullerene graph is a planar cubic 3-connected graph with only pentagonal
and hexagonal faces. We show that fullerene graphs have exponentially many
perfect matchings.Comment: 7 pages, 3 figure
Stationary scattering from a nonlinear network
Transmission through a complex network of nonlinear one-dimensional leads is
discussed by extending the stationary scattering theory on quantum graphs to
the nonlinear regime. We show that the existence of cycles inside the graph
leads to a large number of sharp resonances that dominate scattering. The
latter resonances are then shown to be extremely sensitive to the nonlinearity
and display multi-stability and hysteresis. This work provides a framework for
the study of light propagation in complex optical networks.Comment: 4 pages, 4 figure
2-Resonant fullerenes
A fullerene graph is a planar cubic graph with exactly 12 pentagonal
faces and other hexagonal faces. A set of disjoint hexagons of
is called a resonant pattern (or sextet pattern) if has a perfect
matching such that every hexagon in is -alternating.
is said to be -resonant if any () disjoint hexagons of
form a resonant pattern. It was known that each fullerene graph is
1-resonant and all 3-resonant fullerenes are only the nine graphs. In this
paper, we show that the fullerene graphs which do not contain the subgraph
or as illustrated in Fig. 1 are 2-resonant except for the specific eleven
graphs. This result implies that each IPR fullerene is 2-resonant.Comment: 34 pages, 25 figure
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