A note on some coefficients of the Chebyshev polynomial form of the characteristic polynomial

The characteristic polynomial of a graph, which traditionally is written down in descending powers ofX can also be expressed in the mathematically equivalent form of a linear combination of the characteristic polynomials of linear chains, and often this expression is a simpler one. Investigation of the first few coefficients reveals that in this form the even ones are of smaller magnitude because they are more closely related to the cyclomatic number of the graph. On the other hand, the early odd coefficients are the same or are more complicated in composition.