Mastering Graphs: A Comprehensive Guide for Graduate Students

Question 1

What is a key characteristic of graphs?

Accepted Answer

Ability to represent complex relationships

Answer

Linearity

Answer

Presence of a single root node

Question 2

What is the time complexity of finding the minimum spanning tree of a graph using Kruskal's algorithm?

Accepted Answer

O(E log V)

Answer

O(V log V)

Answer

O(V^2)

Answer

O(E^2)

Question 3

What is the difference between a directed and an undirected graph?

Accepted Answer

In a directed graph, edges have a direction, while in an undirected graph, edges do not have a direction.

Answer

Directed graphs are used to represent relationships between objects, while undirected graphs are used to represent connections between objects.

Answer

Directed graphs can be used to represent hierarchies, while undirected graphs can be used to represent networks.

Answer

There is no difference between directed and undirected graphs.

Question 4

Which of the following is NOT a fundamental property of a graph?

Accepted Answer

Node color

Answer

Degree

Answer

Edge weight

Question 5

In a directed graph, the relationship between in-degree and out-degree of a node is that:

Accepted Answer

They may be different.

Answer

In-degree is always greater than or equal to out-degree.

Answer

They are always equal.

Question 6

What is the complexity of finding a minimum spanning tree in an undirected graph using Kruskal's algorithm?

Accepted Answer

E log V

Answer

E^2

Answer

V log V

Answer

V^2

Question 7

Which type of graph is appropriate for representing a social network where friends can have bidirectional connections?

Accepted Answer

Undirected graph

Answer

Directed graph

Answer

Directed acyclic graph

Answer

Weighted graph

Question 8

What is the time complexity of checking for a cycle in an undirected graph using the DFS algorithm?

Accepted Answer

O(V + E)

Answer

O(log V)

Answer

O(V)

Answer

O(E^2)

Question 9

Which data structure is commonly used to represent the adjacency list of a graph?

Accepted Answer

Array of linked lists

Answer

Hash table

Answer

Array of arrays

Answer

Adjacency matrix

Question 10

What is the purpose of edge weights in a weighted graph?

Accepted Answer

To represent the cost or distance associated with traversing an edge.

Answer

To identify the nodes that the edge connects.

Answer

To indicate the direction of the edge.

Question 11

What is the key difference between a simple graph and a multigraph?

Accepted Answer

A multigraph can have multiple edges between the same pair of nodes, while a simple graph cannot.

Answer

A simple graph has a unique identifier for each node, while a multigraph does not.

Answer

A multigraph can have directed edges, while a simple graph cannot.

Question 12

Which of the following is not a type of graph data structure?

Accepted Answer

Hash table

Answer

Undirected graph

Answer

Weighted graph

Answer

Directed graph

Question 13

What is the purpose of a minimum spanning tree in the context of graphs?

Accepted Answer

To find the subset of edges that connect all nodes with the minimum total weight

Answer

To detect cycles in a graph

Answer

To compute the shortest path between two nodes

Question 14

What is the primary distinction between directed and undirected graphs?

Accepted Answer

Directed graphs have edges with a specific direction, while undirected graphs do not

Answer

Directed graphs have cycles, while undirected graphs do not

Answer

Directed graphs have weighted edges, while undirected graphs do not

Question 15

Provide an example of a real-world application that utilizes graphs

Accepted Answer

Social networks

Answer

Web browsers

Answer

Spreadsheets

Answer

Databases

Question 16

For a simple graph with n vertices, what is the maximum number of edges it can have?

Accepted Answer

n(n-1)/2

Answer

n log n

Answer

2^n

Answer

n^2

Question 17

Which algorithm is commonly used to find the minimum spanning tree of a graph?

Accepted Answer

Kruskal's algorithm

Answer

Dijkstra's algorithm

Answer

Bellman-Ford algorithm

Answer

Floyd-Warshall algorithm

Question 18

Which graph search algorithm is designed to find the shortest path between two nodes?

Accepted Answer

Dijkstra's Algorithm

Answer

A* Search

Answer

Depth-First Search

Answer

Breadth-First Search

Question 19

What is the time complexity of a breadth-first search on a graph with 'V' vertices and 'E' edges?

Accepted Answer

O(V + E)

Answer

O(V^2)

Answer

O(V*E)

Answer

O(E^2)

Question 20

Which of the following is a type of undirected graph with no cycles?

Accepted Answer

Tree

Answer

Directed Acyclic Graph

Answer

Complete Graph

Answer

Cyclic Graph

Question 21

What is the purpose of topological sorting in a directed acyclic graph (DAG)?

Accepted Answer

To order vertices in a way that respects their dependency relationships

Answer

To detect cycles in the graph

Answer

To find the shortest path between two vertices

Question 22

Which of the following best describes the degree of a node in a graph?

Accepted Answer

The number of edges connected to it

Answer

The number of nodes connected to it

Answer

The distance from the nearest node

Answer

The weight of the edges connected to it

Question 23

Which data structure is commonly used to represent a graph efficiently?

Accepted Answer

Adjacency list

Answer

Tree

Answer

Linked list

Answer

Array

Question 24

What is the time complexity of a breadth-first search (BFS) on a graph with 'n' nodes and 'm' edges?

Accepted Answer

O(n + m)

Answer

O(m^2)

Answer

O(n^2)

Answer

O(n log n)

Question 25

Which type of graph assigns values to its edges?

Accepted Answer

Weighted graph

Answer

Undirected graph

Answer

Cyclic graph

Answer

Directed graph

Question 26

Which of the following properties is characteristic of a bipartite graph?

Accepted Answer

Its nodes can be divided into two distinct sets with no edges between nodes in the same set

Answer

It has no cycles

Answer

It is a complete graph

Answer

It is always connected

Question 27

Which technique is commonly used to detect cycles in a directed graph?

Accepted Answer

Depth-First Search (DFS) with cycle detection

Answer

Bellman-Ford algorithm

Answer

Breadth-First Search (BFS)

Answer

Dijkstra's algorithm

Question 28

Which of the following is a characteristic of a graph?

Accepted Answer

Consists of nodes and edges

Answer

Can only be represented as an adjacency matrix

Answer

Nodes must be labeled with unique integers

Answer

Always contains cycles

Question 29

A graph with all possible edges between its nodes is called a:

Accepted Answer

Complete graph

Answer

Simple graph

Answer

Regular graph

Answer

Bipartite graph

Question 30

Which of the following algorithms is primarily used for finding the shortest path in a weighted graph?

Accepted Answer

Dijkstra's algorithm

Answer

Depth-First Search (DFS)

Answer

Kruskal's algorithm

Answer

Prim's algorithm

Question 31

What is the time complexity of Breadth-First Search (BFS) on a graph with n nodes and m edges?

Accepted Answer

O(n + m)

Answer

O(m^2)

Answer

O(n log n)

Answer

O(n^2)

Question 32

Which data structure is commonly used to represent a graph in memory where each node stores a list of its connected nodes?

Accepted Answer

Adjacency list

Answer

Hash table

Answer

Linked list

Answer

Array

Question 33

Which of the following representations is NOT valid for an undirected graph?

Accepted Answer

Adjacency Matrix with negative weights

Answer

Adjacency list with weighted edges

Answer

Incidence matrix with symmetric structure

Answer

Hash table of vertex pairs with edge weights

Question 34

What is the time complexity of Prim's algorithm for finding the Minimum Spanning Tree (MST) of a connected and weighted graph?

Accepted Answer

E log V

Answer

V + E

Answer

V log E

Answer

E^2

Question 35

Which data structure is most appropriate for storing the vertices of a graph during a Depth-First Search (DFS) traversal?

Accepted Answer

Stack

Answer

Queue

Answer

Priority Queue

Answer

Hash Table

Question 36

What is the purpose of a maximum flow algorithm in graph theory?

Accepted Answer

Finding the maximum amount of flow that can be sent from a source vertex to a sink vertex

Answer

Determining the shortest path between two vertices

Answer

Checking if a graph is bipartite

Answer

Identifying strongly connected components

Question 37

What is the defining characteristic of a Hamiltonian cycle?

Accepted Answer

Traverses every vertex in the graph exactly once, returning to the starting vertex

Answer

Is always the shortest cycle in the graph

Answer

Contains no repeated edges

Answer

Divides the graph into two connected components

Question 38

What fundamental difference exists between weighted and unweighted graphs?

Accepted Answer

Weighted graphs assign values (weights) to their edges, while unweighted graphs do not

Answer

Unweighted graphs can have negative edge weights

Answer

Weighted graphs have exclusively positive edge weights

Question 39

In which real-world context is graph theory commonly applied?

Accepted Answer

Modeling complex relationships within social networks

Answer

Identifying prime numbers

Answer

Evaluating integrals

Answer

Solving systems of linear equations

Question 40

What is the maximum possible number of edges in a simple graph with 'n' vertices?

Accepted Answer

n(n-1)/2

Answer

2^n

Answer

n^2

Answer

n(n+1)

Question 41

Which of the following is NOT an intrinsic property of a graph?

Accepted Answer

Vertices must be ordered

Answer

Graphs consist of a finite set of vertices

Answer

Edges connect pairs of vertices

Answer

Vertices represent entities and edges represent relationships

Question 42

What is the maximum number of edges in an undirected graph with 'n' vertices?

Accepted Answer

n*(n-1)/2

Answer

n^2

Answer

n-1

Answer

n*(n+1)/2

Question 43

Which data structure is commonly used to represent a graph in computer science?

Accepted Answer

Adjacency List

Answer

Queue

Answer

Tree

Answer

Stack

Question 44

What is the time complexity of a breadth-first search (BFS) on a graph with 'V' vertices and 'E' edges?

Accepted Answer

O(V+E)

Answer

O(V^2)

Answer

O(E^2)

Question 45

In which domain of computer science are graphs frequently used?

Accepted Answer

Social network analysis

Answer

Hashing algorithms

Answer

Merge sort implementation

Answer

Binary search optimization

Question 46

What is the difference between a digraph and an undirected graph?

Accepted Answer

In a digraph, edges have a direction, while in an undirected graph, edges do not have a direction.

Answer

A digraph is always connected, while an undirected graph is not.

Answer

A digraph can have cycles, while an undirected graph cannot.

Answer

A digraph has a weight associated with each edge, while an undirected graph does not.

Question 47

Which of the following is a characteristic of a complete graph?

Accepted Answer

All pairs of vertices are connected by edges.

Answer

The graph has a Hamiltonian cycle.

Answer

The graph has no cycles.

Answer

The graph has a unique Eulerian path.

Question 48

Which data structure is typically used to represent graphs?

Accepted Answer

Adjacency list or adjacency matrix

Answer

Stack

Answer

Binary heap

Answer

Queue

Question 49

What is the output of a depth-first search on a graph?

Accepted Answer

A list of vertices in the order they were visited

Answer

A list of strongly connected components in the graph

Answer

A list of edges in the order they were traversed

Question 50

What is the purpose of a minimum spanning tree?

Accepted Answer

To find a subset of edges that connects all vertices with the minimum total weight

Answer

To identify the shortest path between two vertices

Answer

To determine the diameter of the graph

Answer

To find all the cycles in the graph

Question 51

Which graph traversal method guarantees to visit all vertices in a graph, regardless of its structure?

Accepted Answer

Depth-first search or breadth-first search

Answer

Kruskal's algorithm

Answer

Dijkstra's algorithm

Answer

Topological sort

Question 52

Which of the following is NOT a key characteristic of a tree?

Accepted Answer

Contains cycles

Answer

Acyclic

Answer

Connected

Question 53

What is the time complexity of finding the shortest path between two vertices in a weighted graph using Dijkstra's algorithm?

Accepted Answer

O((V + E) * log V)

Answer

O(E^2)

Answer

O(V^2)

Question 54

What is the difference between a directed and an undirected graph?

Accepted Answer

Directed graphs have edges with orientations, while undirected graphs do not

Answer

Directed graphs can contain cycles, while undirected graphs cannot

Answer

Undirected graphs are always connected, while directed graphs may not be

Question 55

What is the chromatic number of a graph?

Accepted Answer

The minimum number of colors needed to color the vertices of a graph so that no adjacent vertices have the same color

Answer

The maximum number of edges in a graph