A graph is connected if and only if one vertex is neighbor to all other vertices O True O False

Q: If a simple graph G is having 15 vertices and 5 edges then its complement graph G will have None of…

A: 15 vertices and 105 edges

Q: 1. Is there a connected graph whose degree are: a. 5, 3, 2, 2, 2, 1, 1

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Suppose a graph G has 15 vertices and 38 edges. Which of the following statements is true? O 1. G…

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Q: Prove that there must be at least two vertices with the same degree in a simple graph

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Q: If a simple graph G is having 10 vertices and 10 edges then its complement graph G will have 10…

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Q: 2. Show that there is no simple graph with six vertices of which the degrees of five vertices are 5,…

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Q: prove that any connected graph G in which every edge is contained in a cycle cannot be disconnected…

A: Want to prove:  If G is a connected graph in whichevery edge 'e' is contained in a cycle then g-{e}…

Q: Descrete math. Graph theory

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Q: A graph is Euler if * (d) It contains an Euler walk. (b) Its disconnected and every vertex has even…

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Q: Suppose m and n are integers that are not zero together. Show that there is not always a graph with…

A: For a graph we know that,        Sum of degree of vertices = 2 x Number of edges.     => Sum of…

Q: Show that every graph with n vertices and k edges, n> k has n – k components.

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Q: How many different labelled graphs are there on the vertex set [n]? Justify your claim.

A: Given Vertex set n Let vertex V=1, 2, 3, ........., n Each graph G with a vertex V is determined by…

Q: Suppose that a graph G has n vertices, and all vertices have degree greater than or equal to“,.…

A: Let G be a graph with n vertices. Given that all vertices have degree greater than or equal to n-12.…

Q: Use the Handshaking Lemma to show that a graph G always has an even number of vertices with odd…

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Q: If a simple graph Gis having 12 vertices and 9 edges then its complement graph G will have O 12…

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Q: An edge is called a bridge if the removal of the edge increases the number of connected components…

A: We draw a graph without loop, find number of bridges by removing edges.

Q: Construct a graph that is a tree with vertices P,Q,R, S,T such that the degree of P is 3. What is…

A: Given: The vertices are P, Q, R, S, T and the degree of P is 3 To find:  graph tree using the degree…

Q: 13. Determine whether the following graph is strongly connected. b.

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Q: How many different simple graphs are there om the vertex set [n]?

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Q: Prove that a graph with at least three vertices is connected if and only if at least two of the…

A: A graph G is a pair of sets V and E where the elements of the non empty set V are called the…

Q: If a simple graph G is having I1 vertices and 9 edges then its complement graph G will have O 11…

A: Given, number of edges in graph G, EG=9 Number of vertices in graph G, n=11 We know that,…

Q: (a) Prove that if a connected planar graph has 13 vertices and 12 edges then it must be a tree. (b)…

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Q: Suppose a graph G is connected and haas 106 vertices none of which have degree 1. What is the fewest…

A: We know that The minimum number of edges for undirected connected graph is (n-1) edges. But in this…

Q: If a simple graph G is having 13 vertices and 14 edges then its complement graph G will have O 13…

A: We have to solve given problem:

Q: A walk is closed if the first and last vertices of the walk are the same, that is, if you finish at…

A: A walk is closed if the first and last vertices of the walk are the same, that is, if you finish at…

Q: Suppose we add two edges in a tree T to form a graph G. How many cycles can be there in G.

A: When 1 edges are added to a tree,  a cycles are created When 2 edges are added to a tree, either 2…

Q: true or false  (d ) A Eulerian trail would begin at an odd vertex and end at another odd vertex .…

A: If there are no vertices of odd degree, all Eulerian trails are circuits. If there are exactly two…

Q: True or False: A graph in which every edge is a bridge is necessarily a tree.

A: True

Q: Which of the following is always true for a connected graph? Select one: O a. all vertices have even…

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Q: Consider the following statements: S: There exists a connected simple graph with 6 vertices and 5…

A: We have to verify the given statements.

Q: Complete bipartite graphs Km,n are trees if and only if

A: We need to find a necessary and sufficient condition for Complete bipartite graphs Km,n are trees.

Q: Let G be a simple, connected graph on n vertices (n≥3) Suppose that G has at least one cycle as a…

A: Given, G is a simple connected graph on n vertices with n≥3. G has at least one cycle as a subgraph…

Q: True or false: If a connected graph G can be decomposed into exactly k paths then exactly 2k…

A: To Prove: A connected graph can be decomposed into exactly k paths then exactly 2k vertices of G…

Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

Q: 4. Is the following graph strongly connected? If not, is it weakly connected? 1

A: Digraph: strongly and weakly connected

Q: Show that if G is a connected graph and e is an edge in G that is part of a cycle, then the graph G'…

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Q: How many non-isomorphic simple graphs edges? are there with four vertices and 3

A: Detailed explanation mentioned below

Q: for which values of m and n are km,n regular?

A: Given: A simple graph is called regular if every vertex of this graph has the same degree. A regular…

Q: Every simple graph is tree O True O False

A: Statement: Every simple graph is tree.

Q: If a simple graph G is having 13 vertices and 7 edges then its complement graph G will have O 13…

A: Formula: If a simple graph consists n vertices and m edges, then its complement graph will consist n…

Q: Draw an undirected graph with six vertices, each of degree 3, such that the graph is:(a)…

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Q: If a simple graph G is having 14 vertices and 13 edges then its complement graph G will have O 14…

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Q: 10. If we have two different non-crossing drawings of the same connected graph, is it possible for…

A: 10. For a connected planar graph, the below equation is always followed.        v - e + f = 2 Here,…

Q: Two simple graphs are if there is a bijection from the vertices of the first graph to the vertices…

A: To fill the correct answer in the blank space.

Q: Is it possible for a graph with 6 vertices to have degrees 2, 2, 2, 3, 5, and 9? (Loops are…

A: We know that numbers of vertices of odd degree in any graph is always even. We use this fact in step…

Q: Is it possible to have a connected graph with six (6) vertices of degrees 1, 1, 2, 2, 2, and 3? If…

A: in a connected graph, the sum of degree of all the vertices is twice the number of edges. therefore…

Q: Construct a simple graph that is a tree with vertices P, Q, R, S,T such that the degree of R is 3.…

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