On Construction of the Characteristic Polynomial

Calculating the characteristic polynomial for larger systems could be tedious, but with current computer software, such as MATLAB [7], this is no longer an issue. There are very useful recursion formulas for the characteristic polynomial of acyclic graphs due to Heilbronner [8], the simplest of which is 

Here Ch(G - C) is the characteristic polynomial of a graph G obtained from G by deleting the cycle C and the summation goes over all cycles of the graph G containing the edge mv. Below, we illustrate application of the above recursion on naphthalene (as reported in [11])  

Some may be skeptical, not about the accuracy of the results but about, in this time of computers, wasting time and energy on something that you can get easily 

FIGURE 4.1 Heilbronner s general recursion expression for the characteristic polynomial of a tree illustrated on a graph of 3-methylhexane. The recursion applies also to the bridge bond of polycyclic graphs. 

Characteristic Polynomials of Carbon Skeletons of n-Alkanes for n = 1 to n = 12 (Path Graphs of Lengths 1-12) 

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