Spanning Tree graphs...?
In each case below, either draw a graph with the required properties, or prove that it doesn't exist
(a) A tree on 8 vertices with degrees 1,1,1,1,1,2,3,4.
(b) A tree on 6 vertices with degrees 1,2,2,3,3,3.
(c) A graph on ve vertices with degrees 1,2,3,4,5.
(d) A graph on 7 vertices with degrees 1,2,3,4,5,6,7.
(e) A graph on 4 vertices without cycles and with degree sum equal to 4 (Hint: it can be done.)
(f) A graph with 6 vertices and 5 edges that is not a tree (Hint: it can be done.)
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A spanning tree is a graph in which all vertices are connected but no cycles exist.
The degrees of a vertex is the number of edges leaving it. (e.g. a vertex directly joined to two other vertices, with two edges, has the degree 2)
Another rule of spanning trees are that the number edges in a spanning tree graph is equal to "one less than the number of vertices", e.g. tree with 4 vertices has 3 edges.
So to answer your questions, start with plotting the vertex with the largest degree (For (a), dot with 4 edges coming off it).
Next pick one of the vertices at the end of one of the edges and draw more edges from this vertex so that this vertex now has the next largest degree stated in the question (in the case of (a) draw two more edges off a vertex so that its degree is 3).
Repeat the previous step with any of the remaining vertices... (so in the case of (a) draw an edge of any of the vertices (not the one with degree 3 or degree 4 as they can't be made into a degree 2 vertex by adding an edge) and then count to see whether you have 8 vertices on your graph, and following on from the rules mentioned above, 7 edges and no cycles.
Hope this helps feel free to contact me if it doesn't.
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reality: each and each area in a tree is a bridge (and upon removal of an area the tree breaks into 2 wood) A Bridge is an area whose removal disconnects the graph OR will advance the kind of factors reality:between any 2 vertices there's a undeniable course. information: If there are 2 paths (ie. 2 edges) between vertices a cycle is created. This contradicts the definition of a tree (a tree could desire to be acyclic). This proves that an edge of a suited graph is a bridge.