How many vertices have an even degree?
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- Question 13given the adjacency matrices construct the corresponding graphsWhich of the graph/s above contains an Euler Trail? Which of the graph/s above is/are Eulerian? Which of the graph/s above is/are Hamiltonian?Which of the graphs has a Hamiltonian circuit? a. graphs 1 and 2 onlyb. graph 2 onlyc. graph 1 onlyd. graphs 2 and 3 onlye. graph 3 onlyf. graphs 1, 2, and 3
- The following is an adjacency matrix for a graph: ? = [0 1 1 01 0 2 11 2 0 10 1 1 1]Answer the following questions by examining the matrix and its powers only, not bydrawing the graph:a) How many walks of length 2 are there from v2 to v3?b) How many walks of length 2 are there from v3 to v4?c) How many walks of length 3 are there from v1 to v4?d) How many walks of length 3 are there from v2 to v3?Theorem 3.5 states the following: Let G be a loopless graph with at least three vertices, and no isolated vertices. Then G is 2-connected if and only if, for every pair {e, f} of edges of G, there is a cycle of G that contains both e and f.Which of the following is false? A.) Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edge. B.) Every graph that contains a Hamiltonian cycle also contains a Hamiltonian path and vice versa is true. C.) There may exist more than one Hamiltonian paths and Hamiltonian cycle in a graph. D.) A connected graph has as Euler trail if and only if it has at most two vertices of odd degree
- Question 11 from Applied Combinatorics Section 1.3 (a) Show that if a circuit in a planar graph encloses exactly two regions, each of which has an even number of boundary edges, then the circuit has even length. (b) Show that if a circuit in a planar graph encloses a collection of regions, each of which has an even number of boundary edges, then the circuit has even length.Discrete Maths Oscar Levin 3rd eddition 4.1.15: Prove that any graph with at least two vertices must have two vertices of the same degree. ps: I'd be so glad if you include every detail of the solution.Consider the adjacency matrix for a graph that is shown below. Answer the following questions by examining the matrix and its powers only, not by drawing the graph. Show your work in a way that makes your reasoning clear. (a) How many walks of length 2 are there from v1 to v3? (b) How many walks of length 3 are there from v1 to v2?
- See figure: given the graph as above from the graph, determine: a. Adjacency Matrix b. Incidency Matrix c. Degrees at each vertex d. Subgraphs and subgraph componentsProblem 7. Determine if a graph with the property specified below exists. If it exists, draw the graph and give its adjacency matrix; otherwise, prove that the graph doesn't exist. (a) A simple digraph with in-degrees 0, 1, 2, 2 and out-degrees 0, 1, 1, 3. (b) A simple digraph with in-degrees 0, 1, 1, 2 and out-degrees 0, 1, 1, 1.Consider the complete bipartite graph K4,3. a) Does it have a Hamiltonian path? b) Does it have a Hamiltonian cycle? c) What is maximum size of a matching in this graph? d) What is minimum size of a vertex cover in this graph?
