Problem Statement

Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree.
This algorithm is often used in routing and as a subroutine in other graph algorithms.

For a given source vertex (node) in the graph, the algorithm finds the path with lowest cost (i.e. the shortest path) between that vertex and every other vertex.

If the vertices of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road,   Dijkstra's algorithm can be used to find the shortest route between one city and all other cities.
As a result, the shortest path first is widely used in network routing protocols, most notably:

The inputs to Dijkstra's algorithm are a directed and weighted graph consisting of 2 or more nodes, generally represented by:

A destination node is not specified. The output is a set of edges depicting the shortest path to each destination node.

You can use numbers or names to identify vertices in your program.

Let's start with the solution:

restart:
with(GraphTheory):
G:=Digraph([a,b,c,d,e,f],{[[a,b],7],[[a,c],9],[[a,f],14],[[b,c],10],[[b,d],15],[[c,d],11],[[c,f],2],[[d,e],6],[[e,f],9]}):
DijkstrasAlgorithm(G,a);
# [[[a], 0], [[a, b], 7], [[a, c], 9], [[a, c, d], 20], [[a, c, d, e], 26], [[a, c, f], 11]]