The characteristic polynomials of structures with pending bonds |
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Authors: | K Balasubramanian M Randić |
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Institution: | (1) Department of Chemistry and Lawrence Berkeley Laboratory, University of California, 94720 Berkeley, California, USA;(2) Department of Mathematics and Computer Science, Drake University, 50311 Des Moines, Iowa;(3) Ames Laboratory-DOE, Iowa State University, 50011 Ames, Iowa, USA |
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Abstract: | It is well known 1] that the calculation of characteristic polynomials of graphs of interest in Chemistry which are of any size is usually extremely tedious except for graphs having a vertex of degree 1. This is primarily because of numerous combinations of contributions whether they were arrived at by non-imaginative expansion of the secular determinant or by the use of some of the available graph theoretical schemes based on the enumeration of partial coverings of a graph, etc. An efficient and quite general technique is outlined here for compounds that have pending bonds (i.e., bonds which have a terminal vertex). We have extended here the step-wise pruning of pending bonds developed for acyclic structures by one of the authors 2] for elegant evaluation of the characteristic polynomials of trees by accelerating this process, treating pending group as a unit. Further, it is demonstrated that this generalized pruning technique can be applied not only to trees but to cyclic and polycyclic graphs of any size. This technique reduces the secular determinant to a considerable extent. The present technique cannot handle only polycyclic structures that have no pending bonds. However, frequently such structures can be reduced to a combination of polycyclic graphs with pending bonds 3] so that the present scheme is applicable to these structures too. Thus we have arrived at an efficient and quite a simple technique for the construction of the characteristic polynomials of graphs of any size. |
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Keywords: | Graph theory Characteristic polynomials |
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