Mastering graph theory requires more than just reading theorems; it demands hands-on problem-solving. Narsingh Deo’s classic textbook, Graph Theory with Applications to Engineering and Computer Science , is a staple for students due to its emphasis on algorithms and real-world engineering.
Finding a comprehensive exercise solution guide is a common goal for those self-studying or preparing for competitive exams like GATE. Below is a guide on how to approach the exercises and where to find support. 1. Key Topics in Narsingh Deo’s Graph Theory
The book is structured into 15 chapters, with the first nine serving as a foundational introduction. Major topics covered in the exercises include:
Paths and Circuits: Understanding Eulerian and Hamiltonian paths.
Trees: Exploring properties of spanning trees and fundamental circuits.
Planarity: Determining if a graph can be drawn in a plane without edges crossing.
Matrix Representation: Using adjacency and incidence matrices to solve graph problems.
Algorithms: Implementing Kruskal’s, Prim’s, and Dijkstra’s algorithms. 2. Where to Find Exercise Solutions
While an official solutions manual was never widely published for the general public, several student-led and academic resources provide detailed answers:
Scribd: This platform hosts various student-uploaded documents, including a Graph Theory by Narsingh Deo Exercise Solution guide that covers many of the textbook’s core problems.
Academic Repositories: Some universities provide lecture notes that include solved examples directly from Narsingh Deo's text, such as these Graph Theory Lecture Notes from UO Anbar.
Study Groups: Platforms like Quora often have threads where CS undergraduates share tips and specific solutions for the book's trickier application-based questions. 3. Tips for Solving the Exercises
Focus on Algorithms: Narsingh Deo prioritizes constructive proofs over non-constructive ones. When solving, try to develop an algorithm rather than just a mathematical proof.
Use Visual Aids: Graph theory is inherently visual. Always sketch the graph mentioned in the exercise to identify paths, cycles, or cut-sets.
Leverage Coding: For larger graphs mentioned in the later chapters (10–15), try implementing the solutions in Python or C++ to verify your results, as the book emphasizes computer-aided analysis.
Graph Theory by Narsingh Deo is a foundational textbook for computer science and mathematics students. Its exercises are designed to test deep conceptual understanding of algorithms, trees, and connectivity. Overview of Narsingh Deo’s Graph Theory
The book covers everything from basic definitions to complex applications. It is widely used for competitive exams and university courses. Solving the exercises is essential for mastering the subject. Chapter 1: Introduction to Graphs
Chapter 1 introduces basic terminology like vertices, edges, and degrees. The exercises often focus on the Handshaking Lemma.
Key Concept: The sum of degrees of all vertices is twice the number of edges.
Problem Type: Proving the number of odd-degree vertices is always even.
Solution Strategy: Use the sum of degrees formula to show parity. Chapter 2: Paths and Circuits
This chapter delves into Euler paths and Hamiltonian circuits. These are the building blocks of network routing.
Eulerian Graphs: A connected graph has an Euler circuit if every vertex has an even degree. Graph Theory By Narsingh Deo Exercise Solution
Hamiltonian Graphs: Finding a cycle that visits every vertex once.
Exercise Tip: Use Dirac’s Theorem to check for Hamiltonian cycles in dense graphs. Chapter 3: Trees and Fundamental Circuits
Trees are acyclic connected graphs. The exercises here focus on properties and counting. Property: A tree with vertices has exactly
Distance and Center: Exercises often ask to find the center or radius of a tree. Spanning Trees: Using Cayley’s formula ( nn−2n raised to the n minus 2 power ) for labeled trees. Chapter 4: Cut-Sets and Cut-Vertices
Connectivity is the focus here. You will learn how to identify weak points in a graph.
Cut-Set: A set of edges whose removal increases the number of components.
Edge Connectivity vs. Vertex Connectivity: Understanding why
Solution Approach: Use Menger’s Theorem for flow-based connectivity problems. Tips for Solving Advanced Exercises 1. Master Matrix Representations
Many solutions in the later chapters require using Adjacency and Incidence matrices. Practice matrix multiplication to find the number of paths between vertices. 2. Focus on Planarity
Chapter 5 deals with planar graphs. Remember Euler’s Formula: . This is the "magic key" for most planarity proofs. 3. Algorithm Implementation
For algorithms like Kruskal’s or Prim’s, don't just solve them on paper. Try tracing them step-by-step to see how the "greedy" approach works.
💡 Pro-Tip: When stuck on a proof, try drawing a small counter-example first to see why a statement might be false.
While having a complete solution set for Narsingh Deo’s Graph Theory would be convenient, the real learning happens in the struggle. Use available partial solutions as checkpoints, not crutches. By working through the proofs, algorithms, and counterexamples yourself, you’ll gain a mastery of graph theory that serves you long after the final exam.
If you’re an instructor, consider publishing your own curated solution set for your students. If you’re a student, start a solutions wiki for your class—future learners will thank you.
Have you found a reliable source for Deo’s exercise solutions? Share your experience in the comments below (but please, no piracy).
The fluorescent lights of the engineering library hummed at a frequency that felt like a drill to Leo’s brain. Spread out before him was the "green bible"—Narsingh Deo’s Graph Theory with Applications to Engineering and Computer Science.
For most, it was a textbook. For Leo, it was a mountain. Specifically, Exercise 4-21.
He had been staring at the problem for three hours. It asked for a proof regarding the Hamiltonian circuits in a specific planar graph. The margins of his notebook were littered with failed sketches—webs of vertices and edges that looked more like crushed spiders than mathematical structures. "Still on the same page?" a voice whispered.
Leo looked up to see Sarah, a doctoral student who seemed to live in the stacks. She glanced at the book. "Ah, Deo. Chapter 4. That one’s a classic trap."
"I’ve tried everything," Leo admitted, his voice cracking. "Inductive steps, contradiction, even checking the Handshaking Lemma just to feel like I knew something. There’s no solution manual for this in the back."
Sarah pulled a chair over. "That’s because Deo doesn't want to give you an answer; he wants to change how you see the world. You’re looking at the edges as lines. Look at them as relationships. If every vertex has a degree of at least
"Dirac's Theorem," Leo finished. "But this graph is sparse. Dirac doesn't apply here." Mastering graph theory requires more than just reading
"Exactly," Sarah smiled. "So, look at the dual graph. What happens to the faces when you traverse the circuit?"
Leo blinked. He hadn't considered the dual. He grabbed his pen, his movements sudden and frantic. He began to draw—not the graph itself, but the spaces between the lines. As he mapped the dual vertices, the logic began to click like tumblers in a lock. The "impossible" Hamiltonian path revealed itself not through the points, but through the voids they created.
The proof flowed. Three pages of dense notation collapsed into a single, elegant conclusion.
Leo leaned back, his hands shaking slightly. He hadn't just found the solution to a textbook problem; he felt, for a fleeting second, like he’d mapped the hidden architecture of the universe. "Got it?" Sarah asked, already standing up to leave. "Got it," Leo said.
He closed the book. The cover was worn, the gold lettering fading, but as he walked out of the library, the city outside looked different. The streetlights, the intersections, the subway lines—they weren't just infrastructure anymore. They were vertices. They were edges. And now, he knew how to navigate them.
Creating a complete solution manual for Narsingh Deo’s Graph Theory with Applications to Engineering and Computer Science
is a massive undertaking, as the book contains hundreds of problems ranging from basic proofs to complex algorithms. However, I can provide structured solutions for representative problems
from the early chapters to help you get started with the logic and formatting. Chapter 1: Introduction
Exercise 1-1: Prove that the maximum number of edges in a simple graph with vertices is
In a simple graph, there are no self-loops or parallel edges. To maximize edges, every vertex must be connected to every other vertex (a Complete Graph, cap K sub n Each of the vertices can be connected to other vertices. Summing these gives Since each edge is the same as , we have counted every edge exactly twice. Therefore, the maximum number of edges is
the fraction with numerator n open paren n minus 1 close paren and denominator 2 end-fraction Chapter 2: Paths and Circuits
Exercise 2-1: Show that if a graph has exactly two vertices of odd degree, there must be a path between them.
This relies on the Handshaking Lemma and the properties of connected components. Let the two odd-degree vertices be
In any graph, the number of odd-degree vertices must be even (Handshaking Lemma).
were in different connected components, each component would have exactly one odd-degree vertex.
This is impossible, as each component is a graph itself and must have an even number of odd-degree vertices. Therefore,
must belong to the same connected component, meaning a path exists between them. Chapter 3: Trees and Fundamental Circuits Exercise 3-2: Prove that a tree with vertices has exactly Proof (by Induction): Base Case: , edges = 0 ( ). Correct. Inductive Step: Assume a tree with vertices has Consider a tree
vertices. Every tree has at least two pendant vertices (degree 1).
Remove one pendant vertex and its incident edge. The remaining graph is still a tree (it remains connected and circuit-less) with By our assumption, this smaller tree has Adding back the pendant vertex and its edge gives , the number of edges is Commonly Requested Topics for Solutions
If you are building a study guide, you should focus on these high-yield areas from the book: Dijkstra’s Algorithm (Chapter 11) – Finding the shortest path. Kruskal’s vs. Prim’s (Chapter 3) – Minimum spanning tree construction. Matrix Representation (Chapter 7) – Adjacency vs. Incidence matrices. (Chapter 5) – Using Euler’s formula ( or a particular from the book?
The following is a solution to Exercise 2-18 from Narsingh Deo's
Graph Theory with Applications to Engineering and Computer Science Exercise 2-18: Union of Two Paths Problem Statement: Show that if the union of two paths P1cap P sub 1 P2cap P sub 2 with the same endpoints has no common edges, then is a circuit. 1. Identify the Structure of the Union P1cap P sub 1 consists of a sequence of vertices are the endpoints. If P2cap P sub 2 is another path between the same endpoints , and they share no common edges, the union forms a single closed loop. 2. Verify the Degree of Vertices Conclusion: The Value of Struggling (Ethically) While having
To prove the union is a circuit, we check the degree of each vertex in Endpoints ( ): In P1cap P sub 1 , the degree of an endpoint is 1. In P2cap P sub 2
, the degree of the same endpoint is also 1. Since there are no common edges, the degree of in the union is Intermediate Vertices: Any vertex that is internal to P1cap P sub 1 has a degree of 2. If it is also in P2cap P sub 2
, its degree increases, but since a circuit only requires all vertices to have a degree of at least 2 and for the graph to be connected, this condition is satisfied. 3. Conclusion P1cap P sub 1 P2cap P sub 2
are connected at both ends and share no edges, traversing from P1cap P sub 1 and returning to P2cap P sub 2
creates a continuous walk where no edge is repeated and the start and end vertices are the same. This is the definition of a circuit. ✅ Result
The union of two edge-disjoint paths with the same endpoints forms a circuit because every vertex in the union has an even degree (specifically degree 2 if they share no intermediate vertices) and the resulting subgraph is connected.
Do you have a specific chapter or exercise number you are working on that you would like the solution for? Graph Theory: Narsingh Deo , Chapter 2, problem 2-18
Graph theory is a cornerstone of computer science and discrete mathematics, serving as the language used to model relationships and networks. Among the various textbooks on the subject, Narsingh Deo’s
Graph Theory with Applications to Engineering and Computer Science
stands out as a classic. First published in the 1970s, it remains a heavily utilized resource for students and educators worldwide. However, the true mastery of this subject lies not just in reading the definitions of paths, trees, and matrices, but in actively engaging with the textbook's exercises. Solving the problems in Deo's book is a rigorous intellectual journey that bridges abstract mathematical theory with practical computational execution. The Pedagogical Bridge
At its core, Deo’s book is designed for application. While many pure mathematics texts focus on existence proofs and abstract topological properties, Deo forces the reader to think algorithmically. The exercises at the end of each chapter are not merely repetitive drills; they are carefully crafted extensions of the text.
For instance, after introducing the concept of trees and spanning trees, the exercises push the student to understand the bounds of tree enumeration and the efficiency of finding a shortest spanning tree. When a student sits down to work through these solutions, they are forced to transition from passive recognition to active construction. Solving a problem about finding the cut-sets of a graph requires a student to deeply internalize the physical meaning of disconnecting a network, a skill directly applicable to modern network reliability and circuit design. The Challenge of Rigor and Intuition
One of the defining features of working through Narsingh Deo’s exercises is the balance between visual intuition and algebraic rigor. Graph theory is inherently visual. We draw dots and lines to represent complex systems. Early exercises often allow students to rely on this visual intuition to find Eulerian paths or check for planarity.
However, as the chapters progress into vector spaces of graphs, matrix representation (such as incidence and adjacency matrices), and coloring problems, visual intuition fails. The exercises demand a shift toward matrix algebra and boolean operations. Developing solutions for these advanced problems teaches students how to translate a physical, visual network into a system of equations that a computer can process. This specific transition—from picture to matrix to algorithm—is the exact workflow of a modern software engineer or data scientist working on network routing, social media mapping, or logistics. Bridging Theory and Algorithmic Thinking
Perhaps the greatest value in solving Deo's exercises is the exposure to classical algorithms in their native environment. Problems revolving around the shortest path (Dijkstra’s or Warshall’s algorithms), flow problems, and traveling salesman approximations are heavily featured.
By deriving these solutions manually or proving their correctness through the exercises, students gain a profound respect for computational complexity. They learn why certain graph problems are easily solvable in polynomial time, while others remain NP-complete. In a world where pre-built software libraries can instantly find the shortest route between two points, manually working through Deo’s exercises ensures that the engineer understands
the algorithm works, its limitations, and how it can be optimized for specific hardware constraints.
The exercise solutions to Narsingh Deo’s graph theory text are far more than just answers to homework questions; they are the crucible in which a student's mathematical maturity is forged. Deo did not design his problems to be easily looked up or memorized. They require a synthesis of logic, visual spatial reasoning, and algorithmic strategy. To successfully solve them is to truly understand the skeletal framework upon which much of our modern digital infrastructure is built. For any aspiring computer scientist or engineer, the sweat equity put into solving these problems yields a lifetime of analytical dividends. from Narsingh Deo's book?
In many engineering circles, shared PDFs of “Deo solution manuals” circulate. Be careful: Many are incorrectly solved or contain typos in graph diagrams. Use them only for inspiration, not gospel.
Searching for "Graph Theory By Narsingh Deo Exercise Solution" yields a scattered landscape. You will find:
The risk: Many free PDF sites claiming to offer "complete solutions" are either incomplete or contain malicious ads. Proceed with caution.
Graph theory is visual. For problems involving isomorphism, traversability, or planarity: