graph_questions_answers

Question 1

What is a graph in computer science? How does it differ from other data structures like trees?

Answer 1

A graph is a collection of nodes (vertices) connected by edges. It differs from a tree because a tree is a special type of graph with no cycles, hierarchical structure, and a unique path between nodes.

Question 2

What are the different types of graphs? Explain each one briefly.

Answer 2

Undirected, Directed, Weighted, Unweighted.

Question 3

What is the difference between a dense and a sparse graph?

Answer 3

Dense graphs have many edges (close to V^2), sparse graphs have few edges relative to the number of nodes.

Question 4

What are the representations of graphs in computer science? Explain adjacency matrix and adjacency list.

Answer 4

Adjacency Matrix: 2D array representing connections. Adjacency List: Each vertex has a list of neighbors.

Question 5

What is a connected graph? What is a strongly connected and weakly connected graph in the case of directed graphs?

Answer 5

Connected graph: A path exists between any two vertices. Strongly connected (directed): A path exists between all pairs. Weakly connected: Connected only if edges were treated as undirected.

Question 6

What is a cyclic graph? How is it different from an acyclic graph?

Answer 6

Cyclic graph has at least one cycle, acyclic has none.

Question 7

What is a bipartite graph? How can you check if a given graph is bipartite?

Answer 7

A bipartite graph has vertices divided into two sets where no edges connect vertices in the same set. Use BFS/DFS for checking.

Question 8

What is a path in a graph? How does it differ from a simple path or a cycle?

Answer 8

A path is a sequence of edges connecting vertices, a simple path has no repeated vertices, and a cycle returns to the starting vertex.

Question 9

What are the applications of graphs in real-world problems? Provide at least three examples.

Answer 9

Social networks (e.g., friends), GPS navigation, data routing in networks.

Question 10

How do you perform a depth-first search (DFS) and breadth-first search (BFS) on a graph?

Answer 10

DFS uses a stack (or recursion), BFS uses a queue. DFS explores deeper first, BFS explores level by level.

Question 11

What is Dijkstra’s algorithm? Explain how it is used to find the shortest path in a graph.

Answer 11

Dijkstra’s finds the shortest path from a source to all vertices with non-negative weights using a priority queue.

Question 12

What is the Bellman-Ford algorithm? How does it compare with Dijkstra’s algorithm?

Answer 12

Bellman-Ford handles negative weights, has O(VE) time complexity, slower than Dijkstra but more flexible.

Question 13

Explain Prim’s and Kruskal’s algorithms for finding the minimum spanning tree in a graph.

Answer 13

Prim’s adds edges greedily to grow the MST, Kruskal’s sorts edges and adds them if they don’t form a cycle.

Question 14

What is topological sorting in a directed acyclic graph (DAG), and why is it important?

Answer 14

Linear ordering of vertices where each directed edge points from an earlier to a later vertex.

Question 15

How do you detect a cycle in a directed and undirected graph?

Answer 15

DFS/Recursion stack to detect cycles in directed graphs. In undirected graphs, a back edge during DFS indicates a cycle.

Question 16

What is the Floyd-Warshall algorithm, and how is it used to find shortest paths between all pairs of nodes?

Answer 16

Finds shortest paths between all pairs, has O(V^3) complexity.

Question 17

What is graph coloring? Explain the significance of the chromatic number of a graph.

Answer 17

Assign colors to vertices so no two adjacent vertices share the same color, chromatic number is the minimum needed.

Question 18

What are strongly connected components (SCCs) in a directed graph? How can Tarjan’s or Kosaraju’s algorithm be used to find them?

Answer 18

SCCs are maximal strongly connected subgraphs. Tarjan’s and Kosaraju’s algorithms use DFS to find SCCs.

Question 19

What is the concept of a ‘cut vertex’ or ‘cut edge’ in a graph? How do you find them?

Answer 19

Cut vertex increases the number of connected components if removed. Cut edge disconnects the graph if removed.

Question 20

What are Eulerian and Hamiltonian paths/cycles? How do you check if a graph contains one?

Answer 20

Eulerian path visits every edge once, Hamiltonian path visits every vertex once.

Question 21

What is the time complexity of common graph algorithms like DFS, BFS, Dijkstra’s, and Bellman-Ford?

Answer 21

DFS/BFS: O(V + E). Dijkstra: O((V + E) log V). Bellman-Ford: O(VE). Floyd-Warshall: O(V^3).

Question 22

How can graphs be used to represent and solve problems in social networks, like finding influencers or community detection?

Answer 22

Graphs represent social networks where nodes are people and edges are relationships. Algorithms can find influencers, communities.

Question 23

What is the max-flow/min-cut theorem? How does the Ford-Fulkerson algorithm solve the maximum flow problem in a flow network?

Answer 23

Max-flow/min-cut theorem states max flow equals min cut capacity in a flow network. Ford-Fulkerson finds max flow.

Question 24

Explain how graph partitioning can help in parallel computing. What are some common techniques for graph partitioning?

Answer 24

Graph partitioning divides a graph into subgraphs to minimize cut size, useful for parallelizing work.

Question 25

What are planar graphs, and how does Kuratowski’s Theorem relate to them?

Answer 25

Planar graphs can be drawn without edge crossings. Kuratowski’s Theorem defines conditions for planarity.

Question 26

How does the PageRank algorithm, used by search engines, apply graph theory?

Answer 26

PageRank measures node importance based on link structure, used in ranking web pages.

Question 27

What are random graphs? How are they modeled in network science (e.g., Erdős–Rényi model)?

Answer 27

Random graphs have edges with a given probability, Erdős–Rényi model is one such type.

Question 28

How can graphs be used in machine learning models? Explain graph neural networks (GNNs).

Answer 28

Graphs represent structured data. Graph neural networks (GNNs) are used for node classification, link prediction, etc.

Question 29

Explain the importance of spectral graph theory and how it is used to understand the properties of a graph.

Answer 29

Spectral graph theory studies graphs using eigenvalues/eigenvectors of their matrices (adjacency, Laplacian).

Question 30

What are dynamic graphs, and how can changes over time (e.g., edge/node insertion and deletion) be efficiently managed?

Answer 30

Dynamic graphs handle real-time changes in nodes and edges. Efficient algorithms track updates.