Algorithm

Let G(V, E) be a directed graph, where V = \{1, 2, 3, 4, 5 \} is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A.
A[i][j]= \left \{ \begin{matrix} 1 &1 \leq j \leq i \leq 5 \\ 0& otherwise \end{matrix} \right.
A[i][j]=1 indicates a directed edge from node i to node j . A directed spanning tree of G , rooted at r \in V , is defined as a subgraph T of G such that the undirected version of T is a tree, and T contains a directed path from r to every other vertex in V . The number of such directed spanning trees rooted at vertex 5 is ____

Consider a simple undirected weighted graph G, all of whose edge weights are distinct. Which of the following statements about the minimum spanning trees of G is/are TRUE?
MSQ

Which one of the following statements is TRUE for all positive functions f(n)?

For constants a\geq 1 and b \gt 1, consider the following recurrence defined on the non-negative integers:

T(n) = a \cdot T \left(\dfrac{n}{b} \right) + f(n)

Which one of the following options is correct about the recurrence T(n)?

Consider the string abbccddeee. Each letter in the string must be assigned a binary code satisfying the following properties:

For any two letters, the code assigned to one letter must not be a prefix of the code assigned to the other letter.
For any two letters of the same frequency, the letter which occurs earlier in the dictionary order is assigned a code whose length is at most the length of the code assigned to the other letter.

Among the set of all binary code assignments which satisfy the above two properties, what is the minimum length of the encoded string?

Let G be a connected undirected weighted graph. Consider the following two statements.

S1: There exists a minimum weight edge in G which is present in every minimum spanning tree of G.
S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.

Which one of the following options is correct?

An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components.
Let T be a DFS tree obtained by doing DFS in a connected undirected graph G.
Which of the following options is/are correct?

Define R_n to be the maximum amount earned by cutting a rod of length n meters into one or more pieces of integer length and selling them. For i > 0, let p[i] denote the selling price of a rod whose length is i meters. Consider the array of prices:

\text{p}[1]=1,\text{p}[2]=5,\text{p}[3]=8,\text{p}[4]=9,\text{p}[5]=10,\text{p}[6]=17,\text{p}[7]=18

which of the following statements is/are correct about R_7?
[MSQ]

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?

Consider the following recurrence relation.
T\left ( n \right )=\left\{\begin{array} {lcl} T(n/2)+T(2n/5)+7n & \text{if} \; n > 0\\1 & \text{if}\; n=0 \end{array}\right.
Which one of the following options is correct?