Please make a table for Dijkstra's Shortest Path. Please provide steps with explanation for the proofs. Thanks!

Problem One: Algorithm Tracing In Blackboard, show the result of each iteration of applying Dijkstra's Shortest-Path Algorithm to the following graph, starting at node #1 : At each iteration, if a node has not yet been seen, then put an ? in the box for that node. After a node has been chosen, put an x in the box for the node. If there is a tie at a particular iteration, then pick the node having the lowest node number. (14 points.) Problem Two: Minimum Spanning Trees Let e=(u,v) be a minimum-weight edge in a graph G. Prove that (u,v) belongs to every minimum spanning tree of G. It's OK to assume that the edge costs of G are distinct. (Four points) Graduate Students Suppose we are given an undirected graph G=(V,E), with nonnegative edge weights ce. If the edge weights are not distinct, then there can be many distinct MSTs. Suppose we are then given a spanning tree TE such that each edge eT belongs to some minimum spanning tree of G. Is it then the case that T itself is a minimum spanning tree? Prove or show a counterexample. Problem One: Algorithm Tracing In Blackboard, show the result of each iteration of applying Dijkstra's Shortest-Path Algorithm to the following graph, starting at node #1 : At each iteration, if a node has not yet been seen, then put an ? in the box for that node. After a node has been chosen, put an x in the box for the node. If there is a tie at a particular iteration, then pick the node having the lowest node number. (14 points.) Problem Two: Minimum Spanning Trees Let e=(u,v) be a minimum-weight edge in a graph G. Prove that (u,v) belongs to every minimum spanning tree of G. It's OK to assume that the edge costs of G are distinct. (Four points) Graduate Students Suppose we are given an undirected graph G=(V,E), with nonnegative edge weights ce. If the edge weights are not distinct, then there can be many distinct MSTs. Suppose we are then given a spanning tree TE such that each edge eT belongs to some minimum spanning tree of G. Is it then the case that T itself is a minimum spanning tree? Prove or show a counterexample