Do all minimum spanning trees have the same number of edges?

Since all the choices of F have the same size, the number of edges from Et required to complete F to a spanning tree is independent of the choice of F and we are done.

Is it possible for a minimum spanning tree to have the same total weight as its original graph?

There may be several minimum spanning trees of the same weight; in particular, if all the edge weights of a given graph are the same, then every spanning tree of that graph is minimum.

How many edges does a minimum spanning tree have?

How many edges does a minimum spanning tree has? A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.

Is the minimum spanning tree of a given weighted graph unique?

Any undirected, connected graph has a spanning tree. If the graph has more than one connected component, each component will have a spanning tree (and the union of these trees will form a spanning forest for the graph). The spanning tree of G is not unique. This is called the minimum spanning tree (MST) of G.

What is the total weight of the minimum spanning tree?

This is the minimal spanning tree and its total weight is (1 + 2 + 3 + 5 + 9) = 20.

How many different minimum spanning trees does it have?

There is only one minimum spanning tree in the graph where the weights of vertices are different.

Is minimum spanning tree of a graph unique?

When a graph has a unique minimum spanning tree?

If all edge weights in a connected graph G are distinct, then G has a unique minimum spanning tree. Proof: Let G be an arbitrary connected graph with two minimum spanning trees T and T0; we need to prove that some pair of edges in G have the same weight. The proof is essentially a greedy exchange argument.

How do you find the minimal spanning tree of the weighted graph?

Step 1 − Arrange all the edges of the given graph G(V,E) in ascending order as per their edge weight. Step 2 − Choose the smallest weighted edge from the graph and check if it forms a cycle with the spanning tree formed so far. Step 3 − If there is no cycle, include this edge to the spanning tree else discard it.

How many minimum spanning trees does a graph have?

one spanning tree
A spanning tree is a subset of Graph G, which has all the vertices covered with minimum possible number of edges. Hence, a spanning tree does not have cycles and it cannot be disconnected.. By this definition, we can draw a conclusion that every connected and undirected Graph G has at least one spanning tree.