<p> M.Sc. Maths I (First) Semester Examination 2014-15</p><p>Course Code:MAT103 Paper ID: 0171102</p><p>Graph Theory </p><p>Time: 3 Hours Max. Marks: 70 </p><p>Max Marks: 75</p><p>Note: Attempt six questions in all. Q. No. 1 is compulsory.</p><p>1. Answer any five of the following (limit your answer to 50 words): (4x5=20) a) Describe a discrete structure based on a graph that can be used to model airlines routes and their flight times. b) State two applications of each of the i) State two applications of each of the ii) Hamiltonian’s path and circuits c) Find the rooted tree representing the proposition : . d) State the significance of the max-flow min-cut theorem. e) State the characterizations for the planar graph. f) If G is a graph without loops what can you say about the sum of entries in: i) any row of column of the adjacent matrix ii) any row/column of incidence matrix g) Illustrate with example how graph coloring can be used in modeling. h) Determine whether the simple graphs with the adjacency matrices: b) Find the geometric duals of a wheel Wn and a cube . graph. (5) </p><p>6. a) Write the incidence matrix and adjacency matrix for 2. a) raw the graph G whose vertex set is and Km, n. (5) such that for . b) Determine the rank of a circuit matrix of a connected Also state its type. (5) graph with e edges and n vertices. (5) b) Determine the number of edges in a simple graph with n vertices and k components. 7. a) Illustrate with example how Hamiltonian graph can be (5) used in modeling. (5) 3. a) Solve the TSP for the weighted graph. b) Construct a graph with minimum degree k for which (5) , where, is vertex-connectivity and the edge-connectivity. (5)</p><p>8. a) Discuss the importance of study of a vector space associated with a graph. (5) b) Find (5) i) the chromatic polynomial of K ,s b) State and discuss various characterizations for 2 ii) Eulerian graphs. (5)</p><p>4. a) Illustrate the applications of binary trees in search procedures or analysis of algorithms. (5) b) Use’s Prim’s algorithm (matrix version) to find MST for the weighted graph. (5)</p><p>5. a) The vertices of a certain graph G have degrees . Examine G for a planer graph. (5) </p>