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Carbon nanotubes are chemical compounds made of carbon which possess a cylindrical structure. Open-ended single-walled carbon nanotubes are called tubulenes. The resonance graph of a tubulene reflects the interference among its Kekule structures. Our work is motivated by [15] where some basic properties of resonance graphs of benzenoid systems were...
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... If h 1 and h 4 are edge joint, we get a situation in Figure 9. Clearly, there is Assumption that T is a thick tubulene is needed since otherwise a part of a tubulene in Figure 9 could be glued together and hexagons h 3 and h 4 would not be disjoint. ...
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Citations
... Since we may tile the Klein bottle by hexagons [21], we may also speak of kleinbottlenes. There is a whole menagerie of proposed finite and infinite theoretical carbon nanostructures, such as Möbiusenes, tubulenes, hexagonal systems, hexagonal animals, toroidal benzenoids, Schwarzites, Haeckelites, etc. [38,70,71,78,77,76,74,75]. The theory of maps [34,57] offers a toolbox for a general treatment of these diverse structures. ...
Catacondensed benzenoids (those benzenoids having no carbon atom belonging to three hexagonal rings) form the simplest class of polycyclic aromatic hydrocarbons (PAH). They have a long history of study and are of wide chemical importance. In this paper, mathematical possibilities for natural extension of the notion of a catacondensed benzenoid are discussed, leading under plausible chemically and physically motivated restrictions to the notion of a catacondensed chemical hexagonal complex (CCHC). A general polygonal complex is a topological structure composed of polygons that are glued together along certain edges. A polygonal complex is flat if none of its edges belong to more than two polygons. A connected flat polygonal complex determines an orientable or nonorientable surface, possibly with boundary. A CCHC is then a connected flat polygonal complex all of whose polygons are hexagons and each of whose vertices belongs to at most two hexagonal faces. We prove that all CCHC are Kekulean and give formulas for counting the perfect matchings in a series of examples based on expansion of cubic graphs in which the edges are replaced by linear polyacenes of equal length. As a preliminary assessment of the likely stability of molecules with CCHC structure, all-electron quantum chemical calculations are applied to molecular structures based on several CCHC, using either linear or kinked unbranched catafused polyacenes as the expansion motif. The systems examined all have closed shells according to H\"uckel theory and all correspond to minima on the potential surface, thus passing the most basic test for plausibility as a chemical species.
... Nowadays the structure of resonance graphs is well investigated for different families of graphs. Recently, some properties of resonance graphs or closely related concepts were established for benzenoid graphs [20,21], fullerenes [5,12,15,19], nanotubes [13,14], and plane bipartite graphs [3]. ...
In this paper we generalize the binary coding procedure of perfect matchings from catacondensed benzenoid graphs to catacondensed even ring systems (also called cers). Next, we study cers with isomorphic resonance graphs. For this purpose, we define resonantly equivalent cers. Finally, we investigate cers whose resonance graphs are isomorphic to the resonance graphs of catacondensed benzenoid graphs. As a consequence we show that for each phenylene there exists a catacondensed benzenoid graph such that their resonance graphs are isomorphic.
... The resonance graph R(G) of G reflects the structure of the set of its perfect matchings. In this paper we show that if a connected component of the resonance graph of a fullerene is not a path, then this component without vertices of degree one (its 2-core) is 2-connected, extending thus analogous results already established for benzenoid systems [14] and later for open-ended carbon nanotubes [11]. ...
... Therefore, it is not surprising that it has been independently introduced in the chemical [3,4] as well as in the mathematical literature [14] (under the name Z-transformation graph) and then later rediscovered in [9,8]. Some basic properties of resonance graphs were shown for benzenoid systems in [14] and for open-ended carbon nanotubes (tubulenes) in [11]. For a survey on resonance graphs see also [13]. ...
A fullerene G is a 3-regular 3-connected plane graph consisting of only pentagonal and hexagonal faces. The resonance graph R(G) of G reflects the structure of the set of its perfect matchings. In this paper we show that if a connected component of the resonance graph of a fullerene is not a path, then this component without vertices of degree one (its 2-core) is 2-connected, extending thus analogous results already established for benzenoid systems [14] and later for open-ended carbon nanotubes [11].
... The resonance graph of a hydrocarbon reflects the structure of its perfect matchings. There are many results on resonance graphs of benzenoid systems [14] and open-ended carbon nanotubes (tubulenes) [11,19], but nothing has been done on resonance graphs of fullerenes. ...
... Further, the equivalence of Zhang-Zhang polynomial of a fullerene and the cube polynomial of its resonance graph is established. The corresponding results are already known for benzenoid systems and tubulenes (see [14,12,11,17,3]). Moreover, we prove that every subgraph of the resonance graph of a fullerene that is isomorphic to a hypercube is actually an induced subgraph. ...
... Proof. The proof is the same as the proof of Lemma 3.1 in [11]. ...
A fullerene G is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. The resonance graph R(G) of G reflects the structure of its perfect matchings. The Zhang-Zhang polynomial of a fullerene is a counting polynomial of resonant structures called Clar covers. The cube polynomial is a counting polynomial of induced hypercubes in a graph. In the present paper we show that the resonance graph of every fullerene is bipartite and each connected component has girth 4 or is a path. Also, the equivalence of the Zhang-Zhang polynomial of a fullerene and the cube polynomial of its resonance graph is established. Furthermore, it is shown that every subgraph of the resonance graph isomorphic to a hypercube is an induced subgraph in the resonance graph. For benzenoid systems and tubulenes each connected component of the resonance graph is the covering graph of a distributive lattice; for fullerenes this is not true, as we show with an example.
... The definition has a chemical meaning since perfect matchings of G are Kekulé structures of a corresponding hydrocarbon molecule. Some basic properties of the resonance graph of a tubulene T were shown in [3,13,14], such as bipartiteness, connectedness, 2-connectedness. Furthermore, the equality between the Clar covering polynomial (Zhang-Zhang polynomial) of T and the cube polynoimal of R(T ) was established and it was also shown that every connected component of the resonance graph of a tubulene is either a path or a graph of girth 4. ...
Carbon nanotubes are composed of carbon atoms linked in hexagonal shapes, with each carbon atom covalently bonded to three other carbon atoms. Carbon nanotubes have diameters as small as 1 nm and lengths up to several centimeters. Carbon nanotubes can be open-ended or closed-ended (fullerenes). Open-ended single-walled carbon nanotubes are also called tubulenes. The resonance graph R(T) of a tubulene T reflects interactions between Kekulé structures—i.e. perfect matchings of T. With the orientation of edges the resonance digraph \(\overrightarrow{R}(T)\) of a tubulene is obtained. As the main result we show that \(\overrightarrow{R}(T)\) is isomorphic to the Hasse diagram of the direct sum of some distributive lattices. Similar results were proved in [10, 16], but one can not directly apply them to tubulenes. As a consequence of the main result it is proved that every connected component of R(T) is a median graph. Further we show that the block graph of every connected component H of the resonance graph of a tubulene is a path and that H contains at most two vertices of degree one.
