(23) There exist some disconnected self-complementary graphs." (24) "For any even integer n ≥ 4, there exists a connected graph of order n such that all the vertices have odd degrees and that there is no bridge." (25) "For any even integer k ≥ 6, if C is an induced subgraph of a graph G and if the diameter of G is less than k/2, then G contains at least three cycles."

Q: .(a) Assume there is an adjacency matrix A of an undirected graph G. Further assume a 1 is indicated…

A: A graph is a mathematical representation of a network consisting of nodes and edges. The nodes…

Q: Under which conditions does: a) A cycle-free graph have an Hamiltonian circuit? b) A complete graph…

A: Euler:- 1) A graph is Euler if and only if the degree of every vertex is even.

Q: (a) Give an example of an acyclic graph (ie, a graph with no cycles) on seven vertices that is not…

A: A Graph is a non-linear data structure consisting of nodes and edges. The nodes are sometimes also…

Q: Problem 1: Determine whether graphs G and H are planar or not. To show planarity, give a planar…

A: Solution:

Q: (a) Recall that the path graph on n vertices has vertex set V = {1,.., n} and edge set E = {{i,i+1}…

A: According to the information given:- We have to recall that path graph on n vertices has vertex set…

Q: (h) Given an undirected graph G and weights w, for each vertex v, compute a vertex cover S…

A: Given that, An undirected graph G and the weight of each vertex v is wv.  Vertex cover of a graph is…

Q: in theory graph 3.14 Characterize graphs G for which Y(G) +Y₂(G) = p. Igung to ylimal

A: The question has been answered in step2

Q: a d b a) Give the vertex set of the G. b) Give the set of edge of G. c) Find the degree of each of…

A: In this question we have been given with the graph and we have to answer for the graph theory…

Q: a. Draw the directed graph corresponding to the adjacency matrix. [101] 2 1 0 Lo 2 1 b. Using the…

A: The above question is solved in step 2 :-

Q: E F B ប A Determine whether the graph contains an Euler path or an Euler circuit. Select the one…

A: Answer: We need to write the given graph is Euler path or Euler circuit. So we will see in the more…

Q: Is it true or false?  If it is true, include a (short, but clear) argument why it is true, and if it…

A: Given four statements and we need to find out if they are true or false.

Q: II. Solve the following problems applying the concept of Eulerian and Hamiltonian Graphs. Discuss…

A: The question is related to graph theory and finding route which will traverse all of the streets or…

Q: (d) Given a directed graph G, determine if there exists a collection of one or more cycles C1,...,…

A: The solution is given in step

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

A: We will prove the statement by induction on the number of vertices, v. Base case: If v = 3, then the…

Q: Give, with justification, a formula for the minimum number of edges that must be added to a general…

A: In order for an Euler Circuit to exist in the graph, every node must have an even degree because…

Q: Bipartite graph of order n A graph of order n which has an Euler circuit. A regular graph of order…

A: A graph  G=(V,E) is bipartite graph if : its vertices V can be partitioned into 2 subsets v1 and v2…

Q: Find a and b.

A: Given that  total vertices V = 18 Total edges E = 53 a vertices have degree 7  b vertices have…

Q: Select and discuss a maximal planar graph with 8 vertices   -- Show the graph and provide all of the…

A: Answer is given below-

Strongly Connected Components

 A directed graph in which a path exists between all its pairs of vertices is called a strongly connected graph. A portion of the strongly connected directed graph that contains a path to reach from each vertex to other vertices is called a strongly connected component (SCC).