Shortest Path

Let G=(V,E) be an undirected unweighted connected graph. The diameter of G is defined as:
diam(G)=\displaystyle \max_{u,v \in V} \{ \text{the length of shortest path between }u \text{ and }v \}

Let M be the adjacency matrix of G.
Define graph G_2 on the same set of vertices with adjacency matrix N, where

N_{ij}=\left\{\begin{array} {lcl} 1 &\text{if}\; M_{ij}>0 \text{ or } P_{ij}>0, \text{ where }P=M^2\\0 &\text{otherwise} \end{array}\right.

Which one of the following statements is true?

Let G=(V,E) be a directed, weighed graph with weight function w:E\rightarrow \mathbb{R}. For some function f:V\rightarrow \mathbb{R}, for each edge (u,v) \in E, define w'(u,v) as w(u,v)+f(u)-f(v).

Which one of the options completes the following sentence so that it is TRUE?

"The shortest paths in G under w are shortest paths under w' too,_________".

Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive and distinct. Consider the following statements:

(I) Minimum spanning tree of G is always unique.
(II) Shortest path between any two vertices of G is always unique.

Which of the above statements is/are necessarily true?

Consider the weighted undirected graph with 4 vertices,where the weigh to edge {i, j} is given by the entry Wij in the matrix W.
W=\begin{bmatrix} 0 & 2&8 & 5\\ 2& 0& 5 &8 \\ 8 & 5 & 0& x\\ 5&8 & x&0 \end{bmatrix}
The largest possible integer value of x, for which at least one shortest path between some pair of vertices will contain the edge with weight x is ____.

Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increasedby the same value ,then which of the following statements is/are TRUE?
P: Minimum spanning tree of G does not change
Q: Shortest path between any pair of vertices does not change

Let G = (V, E) be a simple undirected graph, and s be a particular vertex in it called the source. For x \in V, let d(x) denote the shortest distance in G from s to x. A breadth first search (BFS) is performed starting at s. Let T be the resultant BFS tree. If (u,v) is an edge of G that is not in T, then which one of the following CANNOT be the value of d(u)-d(v)?

Consider the tree arcs of a BFS traversal from a source node W in an unweighted, connected, undirected graph. The tree T formed by the tree arcs is a data structure for computing

What is the time complexity of Bellman-Ford single-source shortest path algorithm on a complete graph of n vertices?

Consider the directed graph shown in the figure below. There are multiple shortest paths between vertices S and T. Which one will be reported by Dijkstra's shortest path algorithm? Assume that, in any iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is discovered

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}.

\begin{pmatrix} 0&1 & 8 & 1 &4 \\ 1& 0 & 12 & 4 & 9\\ 8 & 12 & 0 & 7 & 3\\ 1& 4& 7 & 0 &2 \\ 4& 9 & 3& 2 &0 \end{pmatrix}

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?