The path sequence of a graph

TL;DR

This paper investigates the path sequence of graphs, providing explicit sequences for certain graphs, analyzing how path sequences determine graph structure, and establishing conditions under which path sequences uniquely identify graphs within specific families.

Contribution

It introduces the concept of path sequences for graphs, characterizes their behavior in starlike trees, and proves that in some graph families, the path sequence uniquely determines the graph.

Findings

Path sequences of some graphs are explicitly determined.

Number of paths in starlike trees depends on branch lengths.

Path sequence can uniquely identify graphs within certain families.

Abstract

Let $P(G)=(P_{0}(G),P_{1}(G),\cdots, P_{\rho}(G))$ be the path sequence of a graph $G$, where $P_{i}(G)$ is the number of paths with length $i$ and $\rho$ 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.