Here's an alternative proof that a connected graph with n vertices and $n-1$ edges must be a tree, modified from yours, but without having to rely on the first derivation:
Show that a connected graph on $n$ vertices is a tree if and only if it ...
13 Problem Find the cut vertices and cut edges for the following graphs My understanding of the definitions: A cut vertex is a vertex that when removed (with its boundary edges) from a graph creates more components than previously in the graph.
1 This result is immediate by induction once you have established (as lemma) that in every connected graph with at least two vertices there are at least two vertices that can be individually removed (with all adjacent edges) such that the remaining graph is still connected. (The inductive proof applies removal of such a vertex.)
discrete mathematics - prove that a connected graph with $n$ vertices ...
$\frac {n (n-1)} {2} = \binom {n} {2}$ is the number of ways to choose 2 unordered items from n distinct items. In your case, you actually want to count how many unordered pair of vertices you have, since every such pair can be exactly one edge (in a simple complete graph). suppose $ (v,u)$ is an edge, then v can be any of the vertices in the graph - you have n options for this. u can be any ...
In a few examples i noted that the existence of $k$-regular graph on n vertices is : True , for k or n even. False , for k and n odd . But we can find a graph with $n ...