Is it necessarily the case that for a graph with n vertices and exactly (2¹) +1 edges be Hamiltonian? Either show this must be true of provide a counterex- ample.

Q: G = (V, E) has n > 2 vertices and no self-loops, show that there exist two vertices v w ch that…

A: Graph without self loop is called simple

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

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Q: Let V be the vector space P(t) of all polynomials in t over R, with inner product defined by (f(t),…

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Q: 10. Give an example for each of the following situations, or explain why they cannot happen: (a) A…

A: a) A simple graph on four vertices with total degree 2.b) A simple graph whose vertices all have…

Q: 16. The graph of y = x² is translated so that its vertex is (0, 4). What is the equation of the…

A:

Q: 9. Let G2 be depicted in the following diagram: b 9(a) What is the maximum length of a path in G2?…

A: Length of path: The number of edges in a path is called length of path. Length of cycle: The number…

Q: A Hamiltonian path in a network is a closed path thatpasses exactly once through each node in the…

A: Given: Travelling salesman problem (T.S.P.) and Hamiltonian path information.We need to explain why…

Q: "If G has a walk that uses all the edges, then G is connected." I know that this statement is false…

A: If G has a walk that uses all the edges, then G is connected Clearly this statement is false For…

Q: (7) Verify whether it is possible to have a connected Eulerian graph with 3 vertices and 6 edges,…

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Q: (4) Show that it is not possible to define a graph with 5 vertices of degree 3, and 2 vertices of…

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Q: If we have two different non-crossing drawings of the same connected graph, is it possible for the…

A: The given question is related with planar graphs. Suppose we have two different non-crossing…

Q: Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits…

A: A bipartite graph is a simple graph whose vertices can be partitioned into two sets V1​ and V2​ such…

Q: Suppose two Hamiltonian circuits are considered different if the collections of edges that they use…

A: Hamilton circuit: A circuit where every vertex must be passed only ones. Complete graph: A graph G,…

Q: 5. two. Show that every nontrivial acyclic graph has at least two vertices of degree less than

A: 7. ConclusionWe have proven that in every nontrivial acyclic graph (i.e., a tree or a forest), there…

Q: 3. (Exercise 3.4) Prove Lemma 3.3: "Let G be a 2-connected graph. If e and f are parallel edges in…

A: Let G be a 2-connected graph. e and f are parallel edges in G. Let H= G\e We have to prove that if…

Q: 2. Decide if the statement is True or False. Justify your answer. Explain why is it true or why not,…

A: i have solved the questions manually,see the attachments for explanation, i hope it helps, Thank…

Q: The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n >…

A: We need to match each statement (a) - (d) to and a graph (1)-(5) such that the chosen graph…

Q: 2.29. Is this graph planar (the edges cross each other in the middle but there is no vertex there)?

A: Planar graph: When a connected graph can be drawn without any edges crossing it is called planar.…

Q: (b) I had a go, and could see the following partial representation (b, A₁), (c, A2) and (d, A3).…

A: As per the question, we are given a partial representation consisting of three pairs, (b, A1), (c,…

Q: The Euler's formula v - e + f = 2 holds for all connected planar graphs. What if the graph is not…

A: The formula for multiple connected component c is, v +f = e + c + 1

Q: Suppose a graph G has 22 vertices. What is the minimum number of EDGES that G must contain if Diracs…

A: Dirac's theorem Let G = (V,E) be a graph with n vertices in which each vertex has degree at least…

Q: For both graphs drawn below, verify Euler's formula: (# of nodes) - (# of edges) + (# of loops) = 1.

A: Use the given information.

Q: If a graph with six vertices is embedded in S?, what is the maximum number of edges it can have.…

A: Consider the provided question, A graph G is embedable on S2 if it is embedable on a plane. Since,…

Q: Complete a path straight at vertex a that ends at vertex c.

A: A complete path is defined when all the vertices are passed through and none of the smaller paths…

Q: 39 Show that humber of connecad regular graph with ald edges and whose number of rertices is…

A: We solve this following by using pigeonhole principle

Q: 3) What is the adjacency matrix Ac for the following graph G based on the order of the vertices a,…

A: Note: Since you have asked multiple independent questions, we are only allowed to answer one. Thank…