ELA, Volume 10, pp. 155-162, June 2003, abstract.

The Path Polynomial of a Complete Graph

C.M. da Fonseca

Let P_k(x) denote the polynomial of the path on k vertices. 
A complete description of the matrix that is the obtained 
by evaluating P_k(x) at the adjacency matrix of the complete 
graph, along with computing the effect of evaluating P_k(x) 
with Laplacian matrices of a path and of a circuit.