离散数学代做 数学作业代做 数学图论代写 MATH作业代写

MATH: Discrete Mathematics and Graph Theory

离散数学代做 Practice Class 7 1. Let V = {1, 2, 3, 4, 5, 6}. (a) The number of graphs with vertex set V is __ (b) The number of graphs with vertex set V and

Practice Class 7 离散数学代做

1. Let V = {1, 2, 3, 4, 5, 6}.

(a) The number of graphs with vertex set V is ________

(b) The number of graphs with vertex set V and 3 edges is ________

(c) The number of these 3-edge graphs which are connected is ________

(d) The minimum number of edges in a graph w/ vertex set V is ________

(e) Assuming connectedness, the minimum number of edges is ________

2. Complete the following definitions.

(a) Two graphs G = (V,E) and G′ = (V′ ,E′ ) are isomorphic if there is a bijection f : V → V′ such that ________

(b) A subgraph of G = (V,E) is a graph H = (W, F) where ________

(a) The number of edges in G is ________

(b) The number of connected components in G is ________

(c) The number of bridges in G is ________

(d) The number of 3-cycles in G is ________

(e) The number of subgraphs of G isomorphic to K₄ is ________

(f) The number of edges in G − a is ________

(g) The number of connected components in G − a is ________

(h) The number of edges in the complement of G is ________

(i) The number of walks of length 1 in G from a to c is ________

(j) The number of walks of length 2 in G from a to c is ________

(k) The number of walks of length 3 in G from a to c is  ________

(l) The number of walks of length 2 in G from a to itself is ________

(m) The number of walks of length 3 in G from a to itself is ________

4. 离散数学代做

In each of the following triples of graphs, two are isomorphic to each other, but the third graph belongs to a different isomorphism class. In the box, write which graph is in a different isomorphism class.

5. For each of the following statements, write T for true or F for false.

(a) Every graph is isomorphic to one with vertex set {1, 2, · · · , n} for some n.

(b) There are finitely many isomorphism classes of graphs.

(c) The complement of G has the same vertex set as G.

(d) The edge {v,w} of G is a bridge if and only if G − {v,w} has more connected components than G.

(e) If {v, w} is an edge of G, G − {v, w} never has fewer connected components than G.

(f) If v is a vertex of G, G − v never has more connected components than G.

(g) If v is a vertex of G, G − v never has fewer connected components than G.

(h) The complement of the complement of G is G itself.

Practice Class 8 离散数学代做

1. Complete the following definitions.

(a) A connected graph is Eulerian if it has a walk which returns to its starting vertex and ________

(b) A connected graph with ≥ 3 vertices is Hamiltonian if it has a walk which returns to its starting vertex and ________

(a) Its degree sequence is ________

(b) Is it regular?

(c) Is it Eulerian?

(d) Is it Hamiltonian?

3. Consider the graphs P₂,P₃, P₄,C3,C₄,K₄,K₅.

(a) Which of them are regular?

(b) Which of them are Eulerian?

(c) Which of them have an Eulerian trail?

(d) Which of them are Hamiltonian?

4. For each of the following sequences, say whether it is graphic or not graphic. 离散数学代做

(a) (0, 0, 0, 0)

(b) (0, 0, 1, 1)

(c) (0, 1, 1, 1)

(d) (0, 0, 2, 2)

(e) (3, 3, 3, 3)

(f) (2, 2, 2, 2, 2, 2, 2, 2)

(g) (3, 3, 3, 3, 3, 3, 3, 3, 3, 3)

(h) (0, 0, 1, 1, 2, 2, 3, 3, 3, 3)

(i) (1, 3, 3, 3, 5, 5)

5.For each of the following statements, write T for true or F for false.

(a) If a graph has degree sequence (1, 2, 2, 2, 3), the degree-3 vertex must be adjacent to all the degree-2 vertices.

(b) Any regular connected graph with ≥ 3 vertices is Hamiltonian.

(c) K_2,2 is isomorphic to C₄.

(d) A graph is Hamiltonian if and only if it contains a cycle.

(e) If v is a vertex of a Hamiltonian graph G, then G−v is connected.

(f) If G − v is connected for every vertex v, then G is Hamiltonian.

(g) If v and w are non-adjacent vertices of a Hamiltonian graph G with n vertices, then deg(v) + deg(w) ≥ n.

(h) If, for every pair of vertices v and w of a connected graph G with n ≥ 3 vertices, deg(v)+deg(w) ≥ n, then G is Hamiltonian.

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