# Various Graph Theory Questions

Through the use of appropriate algorithms minimal spanning tree solutions and optimal networking routes are discovered and explained. The document contains detailed solutions and drawings to help understand how to use the included algorithms.

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#### Solution Preview

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1. In creating minimal spanning trees its best to use either Prim's Algorithm or Kruskal's Algorithm.

Prim's is as follows:

Step 1: Pick any vertex as a starting vertex. (Call it S). Mark it with any given color, say red.

Step 2: Find the nearest neighbor of S (call it P1). Mark both P1 and the edge SP1 red. cheapest unmarked (uncolored) edge in the graph that doesn't close a colored circuit. Mark this edge with same color of Step 1.

Step 3: Find the nearest uncolored neighbor to the red subgraph (i.e., the closest vertex to any red vertex). Mark it and the edge connecting the vertex to the red subgraph in red.

Step 4: Repeat Step 3 until all vertices are marked red. The red subgraph is a minimum spanning tree.

And Kruskal's:

Step 1: Find the cheapest edge in the graph (if there is more than one, pick one at random). Mark it with any given color, say red.

Step 2: Find the cheapest unmarked (uncolored) edge in the graph that doesn't close a colored or red circuit. Mark this edge ...

#### Solution Summary

This solution includes the detailed steps to solving various discrete math graph theory questions. It includes minimal spanning trees, optimal paths, and optimal flow.