Gathering Information about a Graph by Counting Walks from a Single Vertex

TL;DR

This paper investigates how walk counts from a single vertex can uniquely identify a graph or distinguish vertices, revealing conditions under which vertices are decisive or ambivalent, with implications for graph isomorphism and canonical labeling.

Contribution

It establishes conditions for vertex decisiveness based on spectral properties and proves that walk counts for lengths up to 2n suffice for vertex identification, answering a question in chemical graph theory.

Findings

Ambivalent vertices exist in almost all trees.

Vertices in graphs with certain spectral properties are decisive.

Walk counts up to length 2n distinguish vertices in random graphs.

Abstract

We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the other hand, $v$ is called ambivalent if it has the same walk counts as a vertex in a non-isomorphic connected graph with the same number of vertices as $G$. Using the classical constructions of cospectral trees, we first observe that ambivalent vertices exist in almost all trees. If a graph $G$ is determined by spectrum and its characteristic polynomial is irreducible, then we prove that all vertices of $G$ are decisive. Note that both assumptions are conjectured to be true for almost all graphs. Without using any assumption, we are able to prove that the vertices of a random graph are with high probability distinguishable from each other by the numbers of closed walks of length at most 4. As a consequence, the closed walk counts for lengths 2, 3, and 4 provide a canonical labeling of a random graph. Answering a question posed in chemical graph theory, we finally show that all walk counts for a vertex in an $n$-vertex graph are determined by the counts for the $2n$ shortest lengths, and the bound $2n$ is here asymptotically tight.