Consider an arbitrary connected simple graph $G$. I'm having trouble with the definition of the least eccentricity path in $G$, because the definition looks like it could be "cheated" with.
If $G$ has an hamiltonian path, then what is the eccentricity of that path? It looks like it is undefined because there is no node in $G$ that is not in that path, and the definition considers the distance between a node that is not in the path and that path.
Also, it feels like we could simply consider a path that goes through every node in the graph (if possible) but one and that's the least eccentricity path, because the distance between the remaining node and the path is necessarily 1.
If anyone knows of a detailed book about these concepts I'd appreciate it, because I have been unable to find a good reference about concepts of distance in graphs. Thanks in advance!
I think your definition of Least eccentricity path is equivalent to the definition of Central Path given by Slater [Peter J. Slater, (1982) Locating Central Paths in a Graph. Transportation Science 16(1):1-18. [http://dx.doi.org/10.1287/trsc.16.1.1]],
A central path in a graph $G$ is a path $P$ with minimum value of $e(P)=\max\{v \in G: d(v,P)\}$, where $d(v,P)$ is the distance to $v$ to $P$, that is, the length of a shortest path from $v$ to a vertex in $P$. In words, it is a path that compared to other paths, it is closer to any vertex.
So answer your question, if $G$ has a hamiltonian path $P$, then $e(P)=0$ and $P$ will be a central path.
An observation about distances. When consider distances between two vertices $u$ and $v$, $d(u,v)$ is always the length of a shortest path from $u$ to $v$. You do not care about longer paths. Also, it is common to talk about distances between sets, for instance $d(X,Y)$ is the minimum distance between a vertex in $X$ and a vertex in $Y$.