Consider a complete undirected graph with vertex set {0,1,2,3,4}. Entry W_ij in the matrix W below is the weight of the edge {i, j} $$$W = \left( {\matrix{ 0 & 1 & 8 & 1 & 4 \cr 1 & 0 & {12} & 4 & 9 \cr 8 & {12} & 0 & 7 & 3 \cr 1 & 4 & 7 & 0 & 2 \cr 4 & 9 & 3 & 2 & 0 \cr } } \right)$$$ What is the minimum possible weight of a path P from vertex 1 to vertex 2 in this graph such that P contains at most 3 edges?

Consider the following sequence of nodes for the undirected graph given below.
1.a b e f d g c
2.a b e f c g d
3.a d g e b c f
4.a d b c g e f
A Depth First Search (DFS) is started at node a. The nodes are listed in the order they are first visited. Which all of the above is (are) possible output(s)?

What is the largest integer m such that every simple connected graph with n vertices and n edges contains at least m different spanning trees?

Consider the depth-first-search of an undirected graph with 3 vertices P, Q, and R. Let discovery time d(u) represent the time instant when the vertex u is first visited, and finish time f(u) represent the time instant when the vertex u is last visited. Given that
d(P) = 5 units      f(P) = 12 units
d(Q) = 6 units      f(Q) = 10 units
d(R) = 14 unit      f(R) = 18 units
Which one of the following statements is TRUE about the graph