Use Dijkstra's algorithm to find the length of the shortest path from $a$ to $z$ in the weighted undirected graph below.

Show the steps of the algorithm on the graph by updating the labels of vertices, $L(v).$

Use Breadth$-$First$-$Search $($BFS$)$ algorithm to produce a spanning tree for the graph.

Use vertex a as the root of the spanning tree.

Show all the steps $($put numbers along the edges showing which edge was added first, which one was added next, and so forth$).$

$8.$ Use Prim's or Kruskal's algorithms to find the minimum spanning tree for the graph below. State which algorithm you are using.

Show all the steps $($put numbers along the edges showing which edge was added first, which one was added next, and so forth$).$

Relations $R$ and $S$ are represented by graphs. Find $RS.$

Feel free to give the answer as the list of ordered pairs, as a graph, or as a matrix.