Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one component with at least 4 vertices. What is the largest number of vertices that a component of G have? Compute the maximum number of edges of G

Q: Let G be a simple undirected graph (without multiple edges and loops) with  n vertices and m edges.…

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Q: 1) An undirected graph G has 40 vertices and the degree of each vertex is at least 5. What is the…

A: Given:    Vertices = 40       degree = 5

Q: Let G be a simple connected planaı 13 vertices and 19 edges. Then, the number of faces in the planar…

A: Given that, G is a simple connected planar graph with 13 vertices and 19 edges.

Q: If a simple disconnected graph has 7 vertices and 2 connected components what is the maximum number…

A: Given question is If a simple disconnected graph has 7 vertices and 2 connected components what is…

Q: How many non-isomorphic simple graphspre there with 11 vertices, 18 edges, minimum degree 3, maximum…

A: Given, Number of vertices V = 11 Number of edges E = 18 Minimum degree = 3 Maximum degree = 6 Number…

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: . Let G be a graph with V = {1,2, 3, 4, 5, 6} and no edge between an odd and even vertex (a) Draw…

A: Given that V={1,2,3,4,5,6} a) draw the graph G with the maximum number of edges b) How many…

Q: Show that any graph with two or more nodes and no self-loops contains two nodes that have equal…

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Q: Suppose a connected planar graph G has 27 vertices. How many edges must G have if there are 16…

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Q: Show that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

A: We have to Show that if a simple graph G has n vertices and at least n edges, then G must contain a…

Q: show that a simple planar connected graph with 17 edges and ten vertices cannot be properly colored…

A: We can solve this using Euler theorem

Q: Can you help me find the least number of vertices that make up the secure total dominating set of…

A: Graph theory: secure total dominating set

Q: Draw an undirected graph that has exactly 11 edges and at least 5 vertices, in which two of the…

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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: Let G be a simple graph with n vertices and k connected components. Find the maximum and the minimum…

A: The objective of this question is to find the maximum and minimum possible number of edges in a…

Q: e maximum edge count for 4 loop G graph

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Q: Prove that every cycle in the bipartite graph have even number of edges.

A: A bipartite graph is a graph which contains two vertex sets. Let G be a bipartite graph and let  X…

Q: An outerplanar graph is a graph that admits a planar embedding with all vertices on the boundary of…

A: Given that an outerplanar graph is a graph that admits a planar embedding with all vertices on the…

Q: Draw every simple graph with five vertices and four edges (isolated vertices are allowed) p to…

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Q: Compute the betweenness centrality of all four vertices

A: Betweenness centrality measures the extent to which a vertex lies on paths between other vertices.…

Q: Draw a directed graph with as few edges as possible that is strongly-connected, has 8 vertices of…

A: Strongly connected directed 8 vertices one has in-degree 3 two have cut degree 3 rest have samller…

Q: Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a…

A: First, we need to understand the problem. We are given two paths P1 and P2 of maximum length in a…

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: Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it…

A: A complete bipartite graph K(m,n) has two sets of vertices, with m vertices in one set and n…

Q: t will be the domina ?

A: Given: Z is a complete graph with n vertices. To find : Domination number of R.

Q: All cycle graphs are also K-regular graphs. True or False

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Q: show that every simple finite graph has two vertices of the same degree

A: Solution: Suppose G is a finite simple graph with n vertices.  Since G is simple, no vertex is…

Q: 2 3

A: For a vertex x, the pair a,b under consideration are such that  i) a≠x,b≠x and  ii) there is a…

Q: Can a simple graph exist with 9 vertices all of degree 5?

A: Handshaking lemma: In any graph, sum of degree of all the vertices is twice the number of edges…

Q: The graph G is a planar connected graph. It has 26 edges, and 19 vertices. How many faces does G…

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Q: Let G be a directed graph with 10 vertices. Let a and b be vertices that are connected by at least…

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Q: Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a…

A: To prove that two paths P1 ​ and P2 ​ of maximum length in a connected graph GGG have a common…

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…