Graph Theory

$1.$ For this graph $($see the picture for diagram$),$ find the shortest paths from the source node $(0)$ to node

$4,$ using Dijkstra's Algorithm. Present your work in the format and style

similar to what you see the class lectures.

$2.$

a$.$ For the graph in Question $1,$ draw the shortest$-$path tree, with the

vertex $0$ as the root.

b$.$ Given any vertex in a graph, does a shortest$-$path tree exist from that

vertex to all other nodes within the graph? Justify your answer by

providing reasoning, proof, or an example.