4. (-) Prove that a graph G is k-connected if and only -connected

Q: 4. Assume that k, t are positive integers with 2k+1 < 2t. Construct a simple (2k+1)-regular graph…

A: A graph with no loops and no parallel edges is called a simple graph.   In graph theory, a regular…

Q: Let G be the (3,4)-complete bipartite graph. (a) List all non-isomorphic subgraphs with 5 vertices…

A: Part(a) Solution:Let be the (3,4) complete bipartite graph K3,4      These are the 11…

Q: Prove that every connected graph with m vertices has at least (m-1) edges, for every m>1

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Q: Exhibit a perfect matching in the graph below or give a short proof that it has none.…

A: Given,A graph.Aim: To exhibit a perfect matching in the graph given or give a short proof that it…

Q: perfect graph if for every induced subgraph HC G, th e clique number of H i.c. x(H) = w(H). Prove…

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Q: 7. Suppose a graph G = (V, E) has n vertices and m edges (i.e., |V| = n and |E| = m). Further…

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Q: Q5. Suppose G is a graph of order 8 and suppose there are 2 vertices of order 7, 2 vertices of order…

A: Let G be a graph of order 8, in which there are 2 vertices of order 7, 2 vertices of order 6, 2…

Q: 4.1.3. (-) Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then…

A: Given: Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then G…

Q: 7. Prove : Every graph with radius r and girth with 2r +l is regular

A: To Determine :-  Every graph with radius r and girth with 2r +1 is regular.

Q: B\ Let G be a simple graph. Show that if G is not connected, then it's complement G is connected.

A: Let G be a simple graph.  Let G be not connected.  Now, we know that a graph is connected iff…

Q: 1. Determine whether R = {(x, y) | x + y = 0} on the set of all real numbers is reflexive,…

A: Since you have posted a multiple question according to guildlines I will solve first(Q1) question…

Q: Suppose that a graph G has n vertices, and all vertices have degree greater than or equal to“,.…

A: Let G be a graph with n vertices. Given that all vertices have degree greater than or equal to n-12.…

Q: 4. Prove: "The complement of the graph G is an Eulerian graph whenever G is also an Eulerian graph…

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Q: 2. Let G and H be a graphs. Prove that if G = H, then H= G.

A: Given: It is given that let G and H be a graphs , the objective is to prove that if and then

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: 5. Prove or disprove: Let l', and I; be two simple graphs; with V(T)| = m, E(T)=p.V(T)| =n, E(T2)| =…

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Q: (b) Give an example of a disconnected simple graph on n vertices in which every vertex has degree…

A: We need to give an example of a disconnected simple graph on n vertices in which every vertex has…

Q: (6) Let G be a graph such that for any u, v € V(G), there exists a unique (u, v)-path. Prove that G…

A: G is a graph such that for any u, v∈VG, there exists a unique u, v-path. To Prove: G is a tree.

Q: Determine whether G and H are isomorphic? Also find Euler circuit, Euler trail and Hamiltonian…

A: The isomorphism of graph is given below. To show graph contains  Euler circuit and path we use…

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

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: A K- regular graph is a graph where every vertex has degree K

A: Consider the given information. A graph is hamiltonian if it visits every vertex exactly once except…

Q: 3. (a) (i) State the Handshaking Lemma. (ii) In each of the following cases, either draw a picture…

A: We have to state the Handshaking Lemma and draw the graph  matching the given information.

Q: Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where DI

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Q: Show for every even integer n ≥ 2 that there exists a graph G of order n such that G has n/2 pairs…

A: According to the question, In graph theory, two vertices and u are similar if there is an…

Q: An independent set of a graph G is a subset I of the vertex set V such that no two vertices in I are…

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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: Prove or disprove: There exists some n > 1 such that there is a simple graph on n vertices, where…

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Q: Let G be a bipartite Prove that a(G) = graph of order n. if and only if G has a perfect matching.

A: Let G be a bipartite graph of order n. We need to prove that  if and only if G has a perfect…

Q: Prove that if G is a simple undirected graph with > 3, then G contains even cycle and the greatest…

A: G is a simple undirected graph with δ≥3.

Q: Determine if the following statement is true or false. If it is true, then prove it. If it is false,…

A: As per the question, we are asked to determine the truth value of a given statement about the…

Q: Prove that if a graph on n vertices is connected, and has n-1 edges, it must be acyclic.

A: A graph G is said to be a connected graph, if there is a path between any two vertices in that…

Q: 6. (In this problem, y denotes the domination umber of a graph.) Let G be a graph such that (G) > 3.…

A: Let G be a graph. Such that γ(G)≥3. Prove that diam(G)≤2.

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

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Q: 5 The tree graph of a connected graph G, denoted by T(G), is a simple undirected graph whose…

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Q: 4.1.10. (1) Find (with proof) the smallest 3-regular simple graph having connectivity 1.

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