Let G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices of G

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

A:

Q: Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one…

A:

Q: Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3.…

A: Step 1: Consider the provided question, (17) (i)

Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: G – v3 G- v2 G – v4

A: A graph consist of Set of vertices and edges A subgraph W of a graph G consist of subset of edges…

Q: 11. Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

A:

Q: Let G be a simple graph on 9 vertices, and assume we know that the sum of all degrees in G is at…

A:

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

A:

Q: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Give an example of a graph G of order 6 and size 10 such that the minimum and maximum degree of a…

A:

Q: a) Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex…

A: Part(a): G is a simple graph with n vertices and degree of each vertex in G is at least n-12. We…

Q: Prove that if the graph G is discontinuous, then the complementary graph G is continuous.

A:

Q: Let G be an undirected graph with n vertices. Let A(G) be the maximum degree of any vertex in G,…

A:

Q: Let Gbe a simple graph with at least 11l vertices, and let G be its complement. (i) Prove that Gand…

A: Given: G is a simple graph with at least 11 vertices and Ḡ be its complement.

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: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

A:

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: Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in G…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

A: Consider the equation

Q: Suppose that G is a k-regular graph of n vertices, and m edges. The vertex connectivity of G is…

A: Sol

Q: Find the smallest 3-regular graph G such that K(G) = 1.

A:

Q: True/False: If the graphs G1 and G2 are isomorphic, and G, is connected, then G2 must be connected…

A: If G1 and G2 are isomorphic , and G1 is connected

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that…

A:

Q: For any connected graph G of order 55, if x is a vertex of G, then the number of components of G-x…

A: A connected graph is graph that is connected in the sense of a topological space, i.e., there is a…

Q: 2. Prove that the size of every k-chromatic graph is at least (

A: Consider the provided question, According to you we need to solve only (2). (2) We need to prove…

Q: 11. > Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

A:

Q: Draw all nonisomorphic simple graphs with four vertices and no more than two edges.

A: Given: The objective is to draw all the nonisomorphic simple graphs with the four vertices and no…

Q: 2. Determine if the following two graphs are isomorphic. а) b)

A: (a) let us suppose  graph 1 graph 2 we have to find if graph is isomorphic or not

Q: State whether the following statement is true or false. Justify your answer with a short proof or a…

A: Consider the general case, Let G be k-edge connected graph then we shall rove that G is k connected.…

Q: If G is a simple graph with at least two vertices. prove that G must contain two or more vertices of…

A:

Q: Let G be a graph with vertex set VG and edge set EG. Let V C VG and EC EG. Is there always a…

A: Consider the given information.

Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

A:

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. Prove that if…

A:

Q: Prove that every graph G on {1, ...n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For a graph…

Q: Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced…

A:

Q: Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible…

A: Please check the next step for details

Q: (!) Let G be a graph whose odd cycles are pair wo odd cycles in G have a common vertex. Prove

A: Given : Graph G has odd cycles that are pair-wise intersecting, i.e, every two odd cycles in G has a…

Q: draw all nonisomorphic simple graphs with four vertices and no more than two edges

A: Given: Four vertices and two edges. To sketch: All non-isomorphic simple graphs no more than two…

Q: Give an example of a non-planar graph that is not homeomorphic to K5or K3,3.

A: We can solve this using definition of planar graph.

Q: Let G be a simple graph with 20 vertices. Suppose that G h ponents, and every pair of distinct…

A: Given: Simple graph G with 20 vertices. G has two-component with distinct vertices that satisfy…

Q: Prove that if G is a graph of order p and has size greater than ¹) then G is connected. 2

A:

Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: What is a(G) if G is a complete graph on n vertices?

A: It is given that G is a complete graph on n vertices. Also provided are some basic definitions about…

Q: Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for…

A: If we are confident with our definition we can easily attempt like these questions, G is connected…