General solution to the spanning tree enumeration problem in arbitrary multigraph joins

Waller, Derek A.

doi:10.1109/tcs.1976.1084242

Cited by 7 publications

(4 citation statements)

References 11 publications

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“…The path with n vertices is denoted by P". For instance, ID(G X ) and ID(G 2 ) are isomorphic to P 7 .…”

Section: Some Inequalities For the Characteristic Polynomials Of Treesmentioning

confidence: 99%

“…Interest in the enumeration of spanning trees in graphs [1,2] -originally accomplished by Kirchhoff [3] in the context of electrical networks nearly a century and a half ago -has, in recent years, been renewed [4][5][6][7][8][9][10][11][12][13][14][15]. This revival has been both practical [8,9,14,15] (for example, at the microscopic level, in connection with the calculation of 7i-electron "ring-currents" in conjugated systems, in which the concept of spanning tree is of relevance [9]) and mathematical [4][5][6][7][10][11][12][13] (extending the classic "Matrix-Tree" theorem [2], thereby leading to devices [5][6][7][10][11][12][13] that give ever-more efficient ways of counting spanning trees).…”

Section: Introductionmentioning

confidence: 99%

“…This revival has been both practical [8,9,14,15] (for example, at the microscopic level, in connection with the calculation of 7i-electron "ring-currents" in conjugated systems, in which the concept of spanning tree is of relevance [9]) and mathematical [4][5][6][7][10][11][12][13] (extending the classic "Matrix-Tree" theorem [2], thereby leading to devices [5][6][7][10][11][12][13] that give ever-more efficient ways of counting spanning trees). These new methods [10][11][12][13] have already been applied [14,15] to establish the "complexity" of (i.e., the number of spanning trees in) systems such as the topical buckminsterfullerene [14,15], in which the spanning-tree count is of the order of IO 20 .…”

Section: Introductionmentioning

confidence: 99%

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On Spanning Trees in Catacondensed Molecules

Gutman

Mallion

1993

Zeitschrift Für Naturforschung A

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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.

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“…The path with n vertices is denoted by P". For instance, ID(G X ) and ID(G 2 ) are isomorphic to P 7 .…”

Section: Some Inequalities For the Characteristic Polynomials Of Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Spanning Trees in Catacondensed Molecules

Gutman

Mallion

1993

Zeitschrift Für Naturforschung A

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“…Trees of these subgraphs are combined with the identified edge to obtain spanning tree, appropriately including or excluding the identified edge. Waller [2] obtained number of spanning trees via a general formula involving eigenvalues of an associated matrix. Gabow and G. W. Mayers [3] gave an algorithm to generate weighted trees arranged in an increasing order.…”

mentioning

confidence: 99%

Efficient Spanning Tree Enumeration Using Loop Domain

Bhagat

2009

IEEE Systems Journal

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Abstract-Exhaustive enumeration of spanning trees in a graph requires substantial time and computational overheads. The proposed method, using the concept of LOOP DOMAIN, greatly reduces the complexity of enumeration. The initial graph is transformed into a graph in loop domain where the spanning tree enumeration is then carried out. Subsequently spanning trees of initial graph are generated. The proposed method eliminates all the two degree vertices in a graph. This finds application in power system distribution networks, which have substantial number of two degree vertices. The consequent reduction in size of graph and its corresponding trees is highly desirable. The proposed method is demonstrated on a 33 vertex system and reduction incomplexity is evaluated.

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The number of spanning trees in buckminsterfullerene

et al. 1991

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The theorem of Gutman et al. (1983) is applied to calculate the number of spanning trees in the carbon-carbon connectivity-network of the recently diagnosed C,,cluster buckminsterfullerene. This "complexity" turns out to be approximately 3.75 x lozo and it is found necessary to invoke the device of modulo arithmetic and the "Chinese Remainder Theorem" in order to evaluate it precisely on a small computer. The exact spanningtree count for buckminsterfullerene is 375 291 866 372 898 816 000, or, ZZ5 X 34 X 53 X 115 X 19'. A "ringcurrent" calculation by the method of McWeeny may be based on any desired one of this vast number of spanning trees.

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General solution to the spanning tree enumeration problem in arbitrary multigraph joins

Cited by 7 publications

References 11 publications

On Spanning Trees in Catacondensed Molecules

On Spanning Trees in Catacondensed Molecules

Efficient Spanning Tree Enumeration Using Loop Domain

The number of spanning trees in buckminsterfullerene

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