By this definition, we can draw a conclusion that every connected and undirected Graph G has at least one spanning tree. A disconnected graph does not have any spanning tree, as it cannot be spanned to all its vertices.
We found three spanning trees off one complete graph. A complete undirected graph can have maximum n^{n-2} number of spanning trees, where n is the number of nodes. In the above addressed example, n is 3 , hence 3^{3-2} = 3 spanning trees are possible.
We now understand that one graph can have more than one spanning tree. Following are a few properties of the spanning tree connected to graph G:
A connected graph G can have more than one spanning tree.
All possible spanning trees of graph G, have the same number of edges and vertices.
The spanning tree does not have any cycle (loops).
Removing one edge from the spanning tree will make the graph disconnected, i.e. the spanning tree is minimally connected.
Adding one edge to the spanning tree will create a circuit or loop, i.e. the spanning tree is maximally acyclic.
Spanning tree has n-1 edges, where n is the number of nodes (vertices).
From a complete graph, by removing maximum e - n + 1 edges, we can construct a spanning tree.
A complete graph can have maximum n^{n-2} number of spanning trees.
Thus, we can conclude that spanning trees are a subset of connected Graph G and disconnected graphs do not have spanning tree.
Spanning tree is basically used to find a minimum path to connect all nodes in a graph. Common application of spanning trees are:
Civil Network Planning
Computer Network Routing Protocol
Cluster Analysis
Let us understand this through a small example. Consider, city network as a huge graph and now plans to deploy telephone lines in such a way that in minimum lines we can connect to all city nodes. This is where the spanning tree comes into picture.
In a weighted graph, a minimum spanning tree is a spanning tree that has minimum weight than all other spanning trees of the same graph. In real-world situations, this weight can be measured as distance, congestion, traffic load or any arbitrary value denoted to the edges.
We shall learn about two most important spanning tree algorithms here
Kruskal's Algorithm
Prim's Algorithm
Both are greedy algorithms.