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

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.

Figures (4)

• Figure 1: (a) The Harary-Palmer tree with pseudosimilar (hence non-similar, strongly walk-similar) vertices $x$ and $y$. (b) The same tree as an instance of the general construction of minimal trees with pseudosimilar vertices (obtained by removal of the vertex $v$ from a unicyclic graph with automorphism $\alpha$ of degree 3).
• Figure 2: The graphs $G=P_n$ and $H=Y_n$ for $n=5$.
• Figure 3: (a) The tail block $T_{6,s}$ for $s=4$. (b) The head block of $G_{4,t}$. (c) The head block of $H_{4,t}$.
• Figure 4: The graphs $G_{s,t}$ and $H_{s,t}$ for $s=t=3$.

Theorems & Definitions (36)

• Definition 1.1
• Definition 1.2
• Theorem 2.1
• Corollary 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof