A tournament is a digraph whose underlying graph is a complete graph. A root of a digraph is a vertex from which every vertex is reachable.  A king of a digraph is a vertex u such that d(u,v)2 for every vertex v. Prove that every tournament has a root. Prove that every tournament has a king.

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: If G1 and G2 are isomorphic, must they have the same number of vertices? If G1 and G2 have the same…

A:

Q: A trail is a walk that does not repeat an edge. Prove that a trail that repeats a vertex must…

A: Assume that vertex Vk is repeated Let the trail be W: V1V2.....Vk........Vn....Vi.....Vk.......Vj…

Q: this graph connected? O Connected ONot Connected

A: Note that the graph is said to be connected if the pair of any two vertices is connected by a path.

Q: by using python  TODO: add an undirected edge between node1 and node2

A: To add an undirected edge between node1 and node2 in Python, you need to first understand how to…

Q: 4. A simple undirected graph has 17 edges, and each vertex is at least of degree 3. What is the…

A:

Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

Q: (a) Show that if a graph has an Euler tour, then the degree of each of its vertices is even. In the…

A: To prove the theorem that a connected graph has an Euler tour if and only if every vertex has even…

Q: 2 3 4 8 7 The root of the Tree above is node 1. A neighbor of node X is any node which shares an…

A: For the given tree, express the set of all nodes which meet each of the following criteria.

Q: Draw any graph with one connected component and at least five nodes which is not a tree.

A:

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: 4a Let n > 4. What is the maximum possible number of edges in a graph with n vertices and n-2…

A:

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: If v is an endpoint of a cut edge. Show that v is a cut vertex if and only if this vertex is not…

A: Let suppose that v is a cut vertex, i.e., that removing v and all of its incident edges increases…

Q: Prove the following theorem: If the graph G is a tree then there is a unique path connecting any two…

A:

Q: If graph G₁ is planar and isomorphic to graph G₂, then G₂ must be planar. True False If the nodes of…

A:

Q: Prove that cycles are bipartite for an even number of vertexes. But they are not bipartite for the…

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: Is the graph below planar? Is it maximal planar? if it is not a maximal planar either remove or add…

A: Given: Graph G To find: To check whether given graph G is a planar or not.

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…