The path sequence of a graph

Let P ( G ) = ( P 0 ( G ) , P 1 ( G ) , … , P ρ ( G ) ) be the path sequence of a graph G , where P i ( G ) is the number of paths with length i and ρ is the length of a longest path in G . In this paper, we first give the path sequences of some graphs and show that the number of paths with length h in a starlike tree is completely determined by its branches of length not more than h − 2 . And then we consider whether the path sequence characterizes a graph from a different point of view and find that any two graphs in some graph families are isomorphic if and only if they have the same path sequence.