$($a$)$ Consider the following weighted undirected graph with $7$ vertices and $11$ edges. Apply Dijkstra's Algorithm on the graph above, to determine the shortest distance from vertex $G$ to each of the other six vertices $($A$,$ B$,$ C$,$ D$,$ E$,$ F$).$ Clearly show all of your steps.

$($b$)$ Now suppose we change the weight of edge EF from $+8$ to $-8.$ What happens?

Using this example, explain why Dijkstra$$s Algorithm can produce incorrect outputs when one or more edges are negative.

$($c $)$ Let$$s convert the above graph to a directed graph, where each edge is pointed in one direction $$ i$.$e$.,$ xy means the edge goes from x to y$.$ We will also make some of our edge weights negative.

Suppose our edges $($and weights$)$ are listed in this order: GE$=+9,$ FG$=+11,$ EF$=+8,$ ED$=15,$EC$=+5,$ DF$=+6,$ DB$=+9,$ CB$=8,$ BE$=+7,$ BA$=7,$ AD$=+5.$

Using the above order, apply the Bellman$-$Ford Algorithm to determine the shortest distance from the source vertex A to each of the other six vertices in the graph. Clearly show your steps.