On Characteristic Polynomial

We will start this section on mathematical chemistry by briefly outlining the construction and several properties of the characteristic polynomial, one of the basic attributes of graphs. There are several reasons that we decided to elaborate on the characteristic polynomial. The characteristic polynomial emerges in a number of topics that overlap with problems we discuss in this book  

Isospectral graphs that we have already mentioned have the same characteristic polynomial. 

The coefficients of the characteristic polynomial offer a set of useful graph invariants. 

The characteristic polynomial is related to another important polynomial of interest in physics and chemistry, the matching polynomial, the coefficients of which count independent sets of edges in a graph [1]. When added, these counts give the Hosoya index Z [2], one of the early molecular descriptors for structure-property regressions. 

A relationship between the characteristic polynomial and its derivatives lead to Clarke s theorem [3], which is of interest as a possible alternative route to solving graph reconstruction problems. 

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J. Brocas, Comments on characteristic polynomials of chemical graphs, Theor. Chim. Acta 68 (1985) 155-148. 

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