Algorithm and Formalism for Most Remote Vertices
One way to formulate your problem is well known as finding the diameter of a graph as the following description in Wikipedia.

The diameter $d$ d of a graph is the maximum eccentricity of any vertex in the graph. That is, $d$ is the greatest distance between any pair of vertices or, alternatively, $d=\max_{v\in V}\epsilon (v)$. To find the diameter of a graph, first find the shortest path between each pair of vertices. The greatest length of any of these paths is the diameter of the graph.
A peripheral vertex in a graph of diameter $d$ is one that is distance $d$ from some other vertex—that is, a vertex that achieves the diameter. Formally, $v$ is peripheral if $\epsilon (v)=d$.

Similarly to the way we talk about the diameter of a circle, the term &quot;diameter of a graph&quot; is overloaded. It can also mean, although less frequently, any shortest path between any two vertices whose length is equal to the maximum eccentricity.
If the graph is a tree, there is an algorithm that finds its diameter using only two BFS scans. Choose an arbitrary vertex $v$. Do a breadth first search from $v$ that  finds a vertex that is farthest away from $v$. Call the vertex $u$. Do another BFS from $u$ that finds a vertex that is farthest away from $u$. Call the vertex $u'$. The distance between $u$ and $u'$ is the diameter of the graph.
In a general graph, many algorithms can be used to to find the diameter of a graph. Please check this CS StackExchange question and answer, where most algorithms are in fact designed to solve the all-pairs-shortest-path problem.
In the example graph given in the question, the distance between $N$ and $M$, 9, is the diameter. In fact, $N$ and $M$ are the only two peripheral vertices in that graph. (OP calls peripheral vertices as most remote). However, $H$ is not a peripheral vertex, since the furthest distance from any vertex to $H$ is only 8. ($H$ is not a pseudo-peripheral vertex, either.)
We could define an extremum of a graph as a vertex whose eccentricity is not smaller than the eccentricity of any of its neighbors. We could also define a strong extremum of a graph as a vertex whose distance to any vertex $v$ is not smaller than the distance from any of its neighbors to $v$. Every peripheral vertices is an extremum and every strong extremum is an extremum. We can recall a leaf vertex of a graph means a vertex of degree 1. Every leaf is a strong extremum. Note that $H$ is a leaf. All those algorithms that compute all-pairs-shortest-paths can be adapted easily to find all (strong) extremum vertices as well. A simple degree-counting algorithm will find all leaves of a graph. (Please note this definition of &quot;extremum&quot; might not be standard at all. As far as I have tried, I have not found a single article that defines that concept. Well, I can only check probably a negligible number of articles among a myriad of them. There is probably some existing term for it in papers about social network.)