A terminal polynomial is the characteristic polynomial of the distance matrix between all pairs of leaves of a given graph. Graphs are isoterminal if they share the same terminal polynomial. 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 and calculate the isoterminal pair of star-like graphs with three rays that has the smallest number of vertices (in both graphs together) among all isoterminal pairs of star-like graphs with three rays.
COBISS.SI-ID: 14904153
A graph may be the Kronecker cover in more than one way. In this note we explore this phenomenon and apply it to show that the minimal common cover of two graphs need not be unique. We provide examples of graphs of the same size with non-unique common double covers and of isovalent graphs of the same size with non-unique minimal regular common covers.
COBISS.SI-ID: 14763353