a basic doubt about definition in graph theory

Question

Friends, I have a very basic doubt about neighborhood of a vertex. I was going through some pdf and their it was written about i-th neighbor of v, $v \in V(G)$. Can anybody explain me the term i-th neighbor. Thanks a lot.

Answer 1

Notice this statement in Definition $\bf1$: for each vertex $v\in G$ fix an ordering of $v$’s neighbors.

Here that means that if $v\in G$ and $N$ is the set of vertices adjacent to $v$, we number the vertices in $N$, so that $N=\{v_1,v_2,\dots,v_m\}$, where $m=|N|$. The $i$-th neighbor of $v$ is then the vertex $v_i\in N$. If $v$ has $3$ neighbors, say $a,b$, and $c$, we could order them in any of $3!=6$ different ways, and each of those ways would produce a slightly different zig-zag product. This blog post about the zig-zag graph product may be helpful: it has some diagrams as well as some discussion.

Answer 2

I think it is the set of vertices with the distance $\le i$ from $v$.