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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!
What is the least count of graph paper?
The least count of graph paper is determined by the smallest measurable unit on the paper, typically represented by the smallest division or grid square. This value is essential for accurately measuring distances and plotting points on the graph. The least count can vary depending on the type of graph paper, with common values including 1 mm, 0.5 mm, or even 0.1 mm.
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 many students practiced the piano more than 3 hours a week With a plot graph?
To determine how many students practiced the piano for more than 3 hours a week, you would need to analyze the data presented in the plot graph. Look for the section of the graph that represents practice times exceeding 3 hours and count the number of students indicated in that range. The total from that segment will give you the answer. If you have the graph, you can visually identify and count those students directly.
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.)
