Terminal polynomials and star-like graphs
A terminal polynomial is the characteristic polynomial of the distance matrix between all pairs of leaves (valence one vertices) of a given graph. Two graphs are isoterminal if they share the same terminal polynomial. In this paper we prove the Clarke-type theorem about terminal polynomial of a given graph and prove that there are countably many isoterminal pairs of graphs. We investigate isoterminal pairs of star-like graphs in more details and calculate the isoterminal pair of star-like graphs with three rays that has the smallest number of vertices (in both graphs together) compared to all isoterminal pairs of star-like graphs with three rays.