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Cayleys formula states that for a complete graph on nvertices, the number of spanning trees is n^(n-2). For a complete bipartite graph we can use the formula p^q-1 q^p-1. for the number of spanning trees. A generalization of this for any graph is Kirchhoff's theorem or Kirchhoff's matrix tree theorem. This theorem looks at the Laplacian matrix of a graph. ( you may need to look up what that is with some examples). For graphs with a small number of edges and vertices, you can find all the spanning trees and this is often quicker. There are also algorithms such as depth-first and breadth-first for finding spanning trees.
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Prove that a graph G is connected and only if it has a spanning tree?
Proving this is simple. First, you prove that G has a spanning tree, it is connected, which is pretty obvious - a spanning tree itself is already a connected graph on the vertex set V(G), thus G which contains it as a spanning sub graph is obviously also connected. Second, you prove that if G is connected, it has a spanning tree. If G is a tree itself, then it must "contain" a spanning tree. If G is connected and not a tree, then it must have at least one cycle. I don't know if you know this or not, but there is a theorem stating that an edge is a cut-edge if and only if it is on no cycle (a cut-edge is an edge such that if you take it out, the graph becomes disconnected). Thus, you can just keep taking out edges from cycles in G until all that is left are cut-gees. Since you did not take out any cut-edges, the graph is still connected; since all that is left are cut-edges, there are no cycles. A connected graph with no cycles is a tree. Thus, G contains a spanning tree. Therefore, a graph G is connected if and only if it has a spanning tree!
How can a graph be used to determine how many solutions an equation has?
Count the number of many times the graph intersects the x-axis. Each crossing point is a root of the equation.
How do you find the perimeter of a rectangle with coordinates on graph paper?
Count the lengths of each side and sum them up (add them all together.)
When creating graph does the x and y axis have to go b the same count?
Yes it does! It has to be equal to connect with each other at the ''o''.
How can you graph and locate points that contain negative numbers in a coordinate plane?
To,plota point, start at the origin and count along the x axis until you reach the x coordinate, count right for positive numbers, left for negative.
How many spanning trees can be drawn with 5 labeled vertices?
No of spanning trees in a complete graph Kn is given by n^(n-2) so for 5 labelled vertices no of spanning trees 125
What is Difference between tree and spanning tree?
A tree is a connected graph in which only 1 path exist between any two vertices of the graph i.e. if the graph has no cycles. A spanning tree of a connected graph G is a tree which includes all the vertices of the graph G.There can be more than one spanning tree for a connected graph G.
What is the total number of spanning tree that can be drawn using five labeled vertices?
125 according to Cayley's formula for counting spanning trees. For a complete graph Kn, t(kn) = nn-2 where n is the number of vertices.
What is spanning tree in data structure?
A spanning tree is a tree associated with a network. All the nodes of the graph appear on the tree once. A minimum spanning tree is a spanning tree organized so that the total edge weight between nodes is minimized.
Prove that a graph G is connected if and only if it has a spanning tree?
Proving this is simple. First, you prove that G has a spanning tree, it is connected, which is pretty obvious - a spanning tree itself is already a connected graph on the vertex set V(G), thus G which contains it as a spanning sub graph is obviously also connected. Second, you prove that if G is connected, it has a spanning tree. If G is a tree itself, then it must "contain" a spanning tree. If G is connected and not a tree, then it must have at least one cycle. I don't know if you know this or not, but there is a theorem stating that an edge is a cut-edge if and only if it is on no cycle (a cut-edge is an edge such that if you take it out, the graph becomes disconnected). Thus, you can just keep taking out edges from cycles in G until all that is left are cut-gees. Since you did not take out any cut-edges, the graph is still connected; since all that is left are cut-edges, there are no cycles. A connected graph with no cycles is a tree. Thus, G contains a spanning tree. Therefore, a graph G is connected if and only if it has a spanning tree!
Prove that a graph G is connected and only if it has a spanning tree?
Proving this is simple. First, you prove that G has a spanning tree, it is connected, which is pretty obvious - a spanning tree itself is already a connected graph on the vertex set V(G), thus G which contains it as a spanning sub graph is obviously also connected. Second, you prove that if G is connected, it has a spanning tree. If G is a tree itself, then it must "contain" a spanning tree. If G is connected and not a tree, then it must have at least one cycle. I don't know if you know this or not, but there is a theorem stating that an edge is a cut-edge if and only if it is on no cycle (a cut-edge is an edge such that if you take it out, the graph becomes disconnected). Thus, you can just keep taking out edges from cycles in G until all that is left are cut-gees. Since you did not take out any cut-edges, the graph is still connected; since all that is left are cut-edges, there are no cycles. A connected graph with no cycles is a tree. Thus, G contains a spanning tree. Therefore, a graph G is connected if and only if it has a spanning tree!
What is krushkal algorithm?
Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. If the graph is not connected, then it finds a minimum spanning forest (a minimum spanning tree for each connected component). Kruskal's algorithm is an example of a greedy algorithm.
What are the Prim and Kruskal algorithms?
we use them to find minimum spanning trees.
How can you find minimum spanning trees?
Minimum spanning trees can be found using algorithms like Prim's algorithm or Kruskal's algorithm. These algorithms work by starting with an empty spanning tree and iteratively adding edges with the smallest weights until all vertices are connected. The resulting tree will have the minimum total weight possible.
What is the complexity of kruskal's minimum spanning tree algorithm on a graph with n nodes and e edges?
o(eloge)
What is the difference between a bar graph and a column graph?
A bar graph is used to compare and contrast while the column graph is used to count things
What is forest in graph theory?
a graph that contains at least one null vertex is called forestanswer from :abdul rasheed rind: "the collection of trees is called forest"SS:A forest is an undirected graph, all of whose connected components are trees; in other words, the graph consists of a disjoint union of trees.
