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.