JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III GRAPH THEORY Previous Question Paper (January/February - 2023 - R18)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define a complete graph.
Answer
A complete graph, denoted by Kn, is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. In a complete graph with n vertices, the total number of edges is given by the formula:
Draw at least three non-isomorphic graphs on four vertices.
Answer
Three non-isomorphic graphs on four vertices (V=4) can be distinguished by their number of edges and the connectivity of their vertices:
- Graph 1 (Empty Graph): A graph with 4 vertices and 0 edges.
- Graph 2 (Path Graph P4): A graph with 4 vertices connected in a single line, having 3 edges.
- Graph 3 (Star Graph K1,3): A central vertex connected to three leaf vertices, having 3 edges but a different vertex degree sequence (3, 1, 1, 1) compared to the path graph (2, 2, 1, 1).
These graphs are non-isomorphic because they have different structural properties, such as the number of edges or the degree sequence of their vertices.
What is block of graph?
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Define connected graph and draw an example.
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Define spanning tree.
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What is Eulerian graph? Give an example.
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Define a bipartite graph.
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Write the necessary conditions for complete matching.
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What is Chromatic number?
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What is the use of Brook’s Theorem?
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Part B
Explain in detail about Isomorphism with a neat diagram and steps for finding isomorphism.
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Explain matrix representation of graphs and state its advantages.
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What is Hamilton graph? State and define the Dirac’s theorem.
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Define a spanning forest and show that if T is any spanning forest of a graph G, then (i) each cut set of G has an edge in common with T; (ii) each cycle has an edge in common with the complement of T.
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Describe all pairs shortest path problem with an example.
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State every connected graph G with n>2, contains at least two vertices which are not cut-vertices.
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Explain the cut-vertices and cut-edges with a neat diagram.
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Explain the connected, disconnected graphs, and maximal connected subgraphs with their components.
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State and explain Kruskal’s algorithm in detail.
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Illustrate the Chinese postman problem with an example.
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Explain in detail about the Cayley’s theorem.
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Demonstrate the steps to show a graph is an Hamilton graph.
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What is independent set and explain max independent set problem.
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State and explain greedy algorithm for finding perfect match in graphs.
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Give detailed note on minimal vertex covering problem.
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Explain about König’s theorem.
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Show that every simple planar graph is 5-colourable.
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Explain about class-1 and class-2 graphs with suitable examples.
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Explain in detail about the Vizing’s theorem.
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Illustrate the graph coloring using greedy algorithm with suitable example.
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