K4 Spanning Trees Calculation
Cayley's formula provides the number of spanning trees for a complete graph $K_n$. The formula is $n^{n-2}$.
For the given complete graph $K_4$, we have $n=4$. Applying Cayley's formula:
$ \text{Number of spanning trees} = 4^{4-2} = 4^2 = 16 $
Thus, $K_4$ has 16 spanning trees.
K2,2 Bipartite Graph Spanning Trees
The number of spanning trees in a complete bipartite graph $K_{m,n}$ can be calculated using the formula $m^{n-1} n^{m-1}$.
For the given complete bipartite graph $K_{2,2}$, we have $m=2$ and $n=2$. Applying the formula:
$ \text{Number of spanning trees} = 2^{2-1} \times 2^{2-1} = 2^1 \times 2^1 = 2 \times 2 = 4 $
Alternatively, note that $K_{2,2}$ is the same as the cycle graph $C_4$. A cycle graph $C_n$ has $n$ spanning trees. For $C_4$, this means 4 spanning trees.
Graph Spanning Trees Summary
The calculation shows that $K_4$ has 16 spanning trees and $K_{2,2}$ has 4 spanning trees.
Therefore, the counts are 16 and 4, respectively.
Important Questions from Graph Theory
Let $G(V, E)$ be an undirected and unweighted graph with 100 vertices. Let $d(u, v)$ denote the number of edges in a shortest path between vertices $u$ and $v$ in $V$. Let the maximum value of $d(u, v)$, $u, v \in V$ such that $u \neq v$, be 30. Let T be any breadth-first-search tree of G. Which ONE of the given options is CORRECT for every such graph G?
- Maintaining a graph in memory by means of its adjacency matrix is known as
Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on plane is equal to_________.
- The number of possible rooted trees in a phylogeny of three species is ________.
Consider the following statements regarding graph theory :
1. Chord is that branch of the graph that does not belong to the particular tree.
2. Cut-set is a unique set with respect to a given tree of a connected graph containing one chord and all of the free branches contained in the free path formed between two vertices of the chord.
3. If M represents the number of branches and N the number of nodes, the minimum number of variables involved in analyzing a network is equal to $(M + N − 1)$.
Which of the above statements are not correct?
