Graph Theory [Lecture Notes] by Christopher Griffin

By Christopher Griffin

Penn country Math 485

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Bridges (and small edge cuts) are a very important part of social network analysis [KY08, CSW05] because they represent connections between different communities. 7. We illustrate a vertex cut and a cut vertex (a singleton vertex cut) and an edge cut and a cut edge (a singleton edge cut). Cuts are sets of vertices or edges whose removal from a graph creates a new graph with more components than the original graph. 7(b) represents the communications connections between individuals in two terrorist cells.

A graph G = (V, E) is connected if and only if it has a spanning tree. Exercise 27. 73. 74. Let T = (V, E) be a graph with |V | = n. Then the following are equivalent: (1) T is a tree. (2) T is acyclic and has exactly n − 1 edges. (3) T is connected and has exactly n − 1 edges. (4) T is connected and every edge is a cut-edge. (5) Any two vertices of T are connected by exactly one path. (6) T is acyclic and the addition of any new edge creates exactly one cycle in the resulting graph. Proof. (1 =⇒ 2) Assume T is a tree.

8. If e lies on a cycle, then we can repair path w by going the long way around the cycle to reach vn+1 from v1 . (⇒) Suppose G = G − {e} is connected. Now let e = {v1 , vn+1 }. Since G is connected, there is a walk from v1 to vn+1 . 24, we can reduce this walk to a path p with: p = (v1 , e1 , . . , vn , en , vn+1 ) Since p is a path, there are no repeated vertices in p. We can construct a cycle c containing e in G as: p = (v1 , e1 , . . , vn , en , vn+1 , e, v1 ) since e = {v1 , vn+1 } = {vn+1 , v1 }.

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