On Spanning Trees in Catacondensed Molecules

Abstract

Among all catacondensed benzenoid molecules the unbranched systems possess the largest number of spanning trees. Some other relations for the spanning-tree count of catacondensed benzenoids are also reported. The results obtained continue to hold for catacondensed molecules composed of rings the size of which differ from six. The propositions reported do, however, depend for their validity on there being rings of only one size in the catacondensed system under study.

Complexity of Molecules

“…A lot of work on the enumeration of spanning trees for various classes of (molecular) graphs has been carried out in the past 25 years. [37][38][39][40][41][42][43][44][45] The interest in spanning trees is related to several chemical problems, such as the use of spanning trees in chemical kinetics, [33][34][35][36] in calculating the magnetic properties of conjugated systems by means of the ring-current model [46][47][48][49] in the framework of π-electron molecular orbital theory, 50 and in coding and enumerating polyhex hydrocarbons. 51 The number of spanning trees can be obtained in several ways.…”

Computation of spanning tree generators of fullerenes

“…Both Gutman and Mallion [281 and Brown et al have shown that ring current computations are dependent on the spanning trees of the structure under consideration [28][29][30]. Recently, Brown et al [29] have enumerated the number of spanning trees in buckminsterfullerene as 375291866372898816000 using a theorem of Gutman and Mallion [28] and modulo arithmetic.…”

An Algorithm for Counting Spanning Trees in Labeled Molecular Graphs Homeomorphic to Cata-Condensed Systems

“…In recent years, we and others [1][2][3][4][5][6][7][8][9][10][11][12][13][14] have extensively considered the counting of spanning trees in labeled molecular graphs, with particular emphasis on the novel family of carbon clusters now known as the fullerenes. 2,5,7,9,10,[12][13][14] The initial motivation for this was the relevance of spanning trees [15][16][17] in calculating the magnetic properties of conjugated systems Via the "ring-current" model 18,19 in the context of classical π-electron molecular orbital theory.…”

“…In recent years, we and others − have extensively considered the counting of spanning trees in labeled molecular graphs, with particular emphasis on the novel family of carbon clusters now known as the fullerenes. ,,,,,− The initial motivation for this was the relevance of spanning trees − in calculating the magnetic properties of conjugated systems via the “ring-current” model , in the context of classical π-electron molecular orbital theory. , In a previous work, two of the present authors devised a method that was suitable for calculating the complexities of ( i.e., the number of spanning trees in) the labeled molecular graphs of cata-condensed systems containing rings of only one size. In a later publication, another combination of two of us generalized this to such systems containing rings of more than one size.…”

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The algorithmic method of Gutman and Mallion (1993), for calculating the number of spanning trees in the (labeled) molecular graphs of cata-condensed systems containing rings of only one size, was subsequently generalized by John and Mallion (1996) to make it applicable to such systems comprising rings of more than one size; this latter algorithm is thus generally valid for enumerating the spanning trees in the molecular graphs of any cata-condensed system. This algorithmic philosophy is extended here in order to devise a procedure that is suitable for an even more general class of molecular graphssnamely, those homeomorphic to the molecular graphs of cata-condensed systems.

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