数学代码代写 MT4512代写 自动机、语言和复杂性代写
708MT4512 Automata, Languages and Complexity 数学代码代写 EXAM DURATION: 2 hours EXAM INSTRUCTIONS: Attempt ALL questions. The number in square brackets shows the maximum marks obtainable for tha...
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图论代写 This assignment forms 5% of the assessment for this module. The assignment will be marked out of 20 marks. Please answer all questions,
This assignment forms 5% of the assessment for this module.
The assignment will be marked out of 20 marks.
Please answer all questions, fully justifying your answers. Think about your mathematical style – the aim is for clear, streamlined solutions. Good choice of notation can help clarify your arguments. You may cite results from the notes without proof.
• for each edge e of G, create a vertex ve of G’ ;
• two vertices ve1 , ve2 of V (G’ ) are joined by an edge in G’ if and only if the corresponding edges e1, e2 of E(G) share an endpoint in G.
In other words, G’ has vertices in one-to-one correspondence with the edges of G, and two vertices are adjacent in G’ precisely if the corresponding edges are adjacent in G. For example,
(i) Let Pn denote the path on n vertices (n ≥ 2). Prove that P’n is isomorphic to Pn−1.
(ii) There exists an infinite family of graphs Gn (n ≥ 2) such that
G’n ≌ Kn−1
where Kn−1 is the complete graph on n − 1 vertices.
Give the vertex set and edge set of such a family, and prove that it has the stated property. (4 marks)
Let G be a graph.
(i) Prove that if G is connected then G’ is connected.
(ii) If G’ is connected, is G necessarily connected? Prove or give a counterexample. (5 marks)
Let G be a graph. Write down a formula expressing the degree of a vertex vxy ∈ V (G’ ), in terms of the degree of x, y ∈ V (G). (2 marks)
Prove that if a graph G is Eulerian, then G’ is Eulerian. (2 marks)
Is it true that if a graph G is connected and G’ is Eulerian, then G is Eulerian? Prove or give a counterexample. (2 marks)
Let G be a graph on n vertices {1, . . . , n}, where deg i = di . Prove that
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