# 1. Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

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

A: - We need to talk about matrix and adjacency list representation for the graph mentioned.

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: You have discovered an old paper on graph theory that discusses the viscosity of a graph (which for…

A: SOLUTION -1) Satisfying condition (V) is a necessary condition for a graph to be viscous.This is…

Q: 4.1.1. (-) Give a proof or a counterexample for each statement below. a) Every graph with…

A: Solution: We have to give a proof or a counterexample for each points below whether they are true…

Q: A simple graph G is called self-complementary if G and G are isomorphic. 50. Show that this graph is…

A: Find the self complementary graph below

Q: Problem 3: For each of the two graphs below, determine whether they are planar. Justify your answer:…

A: In graph theory, a planar graph is a graph that can be embedded in the plane, using the kuratowski's…

Q: 4.76. Suppose T is a minimal spanning tree found by Prim's algorithm in a connected graph G. (i)…

A: Let the graph G be connected, i.e. the answer exists. We denote by T the resulting graph found by…

Q: Prove whether the following graphs G and H are isomorphic or not? 2 3 f 6 5 G b. H

A: Graph: A graph is a set of vertices V that are connected using the edge E.   Isomorphic graph: In…

Q: Q 5 - Show that if a graph G has vertices of even degree, then the set of its edges can be…

A: We know the theorem which says an un-directed graph can have a cycle if and only if number of edges…

Q: Let G be a graph obtained from K12 after adding a vertex to the graph and connecting it to two…

A: The answer is given below for the above-given question:

Q: 1) Find the adjacency matrix M for the following graph (see attached) 1a) Find M^2,M^3,M^4. What do…

A: Adjacency matrix is a way to describe the relationships between nodes in a graph.Transitive closure…

Q: 5. a. State the conditions for an Euler Path and the conditions for an Euler Cycle b. Does the graph…

A: Euler path : it is a path, where every edge is visited exactly once Euler Cycle: it is an euler…

Q: Give an example of each of the following: (4) i) a nonsimple plane graph with its dual a simple…

A: In these questions, we are asked about nonsimple plane graph and its simple dual graph. And we also…

Q: Problem 8: Let G be a connected planar graph with n vertices and f faces. Prove that f≤ 2n-4.

A: Answer to the given question

Q: Elon Musk is running the graph construction'business. A client has asked for a special graph. A…

A: In this problem, we need to solve the problem as per the problem discussed in the details above.…

Q: A graph G = (V, E) satisfies |E| < 3 × |V| – 6. The min-degree of G is defined as…

A: Answer: For that, we are using handshaking Lamma: let K be min degree of the vertices then…

Q: Elon Musk is running the graph construction business. A client has asked for a special graph. A gra…

A: I have written code below:

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: Is tme use of Essentially, by processing the vertices in a random order, you can potentially avoid…

A: Explanation in step 1 we will add all the vertices in the graph. in step2 for every edge one vertex…

Q: • Theorem (Euler): A graph without isolated vertices is Eulerian iff it is connected and every…

A: - The question is to prove that a graph is a unit graph only if it is connected and the degree of…

Q: Prove that if v1 and v2 are distinct vertices of a graph G = (V,E) and a path exists in G from v1 to…

A: Since given graph G = (V, E) have v1 and v2  are distinct vertices and a path exists in G from v1 to…

Q: 35. "1 15 5 Us Uz VI V5 V2 V4 V3

A: SOLUTION -

Q: ) Show that a graph G with at least three vertices is 2-connected if and only if any vertex and any…

A: Find a proof below.

Q: (c) Consider graphs G₁, G₂ as follows. 2 山 6 Us Us G₂ G₁ u3 5 A (i) Find the vertex connectivity…

A: According to our guidelines i am solving first one:…

Q: a) Illustrate the sequence of vertices of this graph visited using depth-first search traversal…

A: In the given graph, we have vertices labeled from 'a' to 'p' with corresponding edges connecting…

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: (a)  Give the definition of an isomorphism from a graph G to a graph H. (b)  Consider the graphs G…

A: Two graphs are said to isomorphic if they contains same number of vertices that are connected in the…

Q: 4. Let G be a graph with n vertices. (a) State what it means for G to be a connected graph. (b)…

A: Dear Student, a) A graph G is said to be connected if we can traverse from any one edge of the graph…

Q: Prove that in a graph G, SCV(G) is an independent set if and only if S is a vertex cover and hence…

A: If S is an independent set of G, then every edge is incident on some vertex in S, otherwise there…

Q: For each pair of graphs G1 = and G2 = a) determine if they are isomorphic or not. b) Determine a…

A: The given graphs are isomorphic if they have: 1. Equal number of vertices.2. Equal number of…