I am trying to understand what that sentence means. Intuitively, its obvious a radius ball means in a $ \mathbb{R}^{n}$ with respect to some norm. Its just the following set:$$ B(v, r) = \{ x \in \mathbb{R}^n : \| v - x\| \leq r\}$$i.e. its just all the points that are distance $ \| v - x\|^2 $ away from the center with respect to some norm $ \| \cdot\|$. However, in graph theory, if we have a graph $G = (V,E)$ I seem to have a much harder time to rigorously understand what this set means. Can someone help me define this set rigorously so that there is no doubt in my mind what it means to have a radius ball with at most radius $r$ in a fixed known graph $G = (V,E)$? In particular this is the sentence I am having a hard time understanding rigorously:a ball with center v and radius r is the set of vertices at most r  hops away from vFor example, if I have a path from v1 to v2 to v3, then it has 3 hops. But if v1 also has v4 as a neighbor and we want a ball radius 3, does it mean that v4 should also be part of the set of radius 3 with center v1?In fact, is the radius ball defined as follow:$$ B(v,r) = \{ x \in V : \exists path(v,x) s.t.| path(v,x)| \leq r \}$$or maybe it should be defined with shortest paths $\delta(v, \cdot)$:$$ B(v,r) = \{ x \in V : \delta(v,x) \leq r \}$$i.e. if you can get to a node from v in less that $r$ hops then its part of the radius ball on the graph.In particular, consider the example. If we have a bounded degree say $d$ and the graph is undirected, if we have the radius be some finite number, do we necessarily have to include everyone's neighbor i.e. consider all the nodes in the degree? 
What does a ball of center v and radius r with at most r hops away mean?
graph theory;graphs