Difference between Eulerian Path vs Minimum Spanning Trees

Minimum spanning tree (MST) and Eulerian graph are two different concepts in graph theory:

Minimum Spanning Tree:

• A minimum spanning tree is a tree that connects all the vertices of a connected, undirected graph with the minimum possible total edge weight.
• It is a subgraph of the original graph that is acyclic (no cycles) and spans all the vertices.
• The weight of an MST is the sum of the weights of its edges, which is minimized among all possible spanning trees.
• MST algorithms, such as Kruskal’s algorithm or Prim’s algorithm, are used to find the minimum spanning tree of a graph.

Eulerian Graph:

• An Eulerian graph is a graph that contains a closed trail (a path that visits each edge exactly once) called an Eulerian circuit.
• In an Eulerian circuit, the starting and ending vertices are the same.
• An Eulerian graph may have multiple Eulerian circuits, or it may have just one.
• For an undirected graph to be Eulerian, all vertices must have even degrees (the number of edges incident to a vertex).
• For a directed graph to be Eulerian, it must satisfy the condition that the indegree (incoming edges) is equal to the outdegree (outgoing edges) for every vertex.

In summary, a minimum spanning tree focuses on finding the minimum total weight tree that connects all vertices in a graph, while an Eulerian graph is concerned with the existence of a closed trail (Eulerian circuit) that visits each edge exactly once.