Let G be a (p, q) graph. Prove that the sum of the degrees of the vertices of G equals twice the size of G, i.e. Evev(G) deg (v) = 2q.

Q: Let G = (V, E) be a planar graph such that |V |< 19. Show that if the length of the shortest cycle…

A: For a given graph G=V,E where V≤19 that shortest cycle in G is 5, then we need to prove that G has a…

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: 3. Show that the two drawings represent the same graph by labeling the vertices and edges of the…

A: Give figure is

Q: Let H be a graph If H has at least one vertex of degree greater than zero then there are two of even…

A: Given a graph representing H. Statement: If H has at least one vertex of degree greater than zero…

Q: Show that For n > 1 let G, be the simple graph with vertex set V(Gn) = {1,2,., n} in which two…

A: A simple connected graph G is said to be planar if we can draw the graph in the plane with no two…

Q: use Kuratowski's theorem to determine whether the given graph is planar. 23. а h 24. a he P 25. a

A: Planar Graph: A graph that can be drawn in the plane without any edges that cross each other where…

Q: 29. Let G be a graph with vertex set VG and edge set Ec. Let V C VG and EC EG. Is there always a…

A:

Q: Let G be a graph There are always two vertices of the same order verify that the following graphs…

A: We have  to verify  that in each graph   is there  exists  any two  vertices  of same order  .

Q: Find a pair, G and H, of isomorphic graphs among the four graphs below, and give the one-to-one…

A: Isomorphism in graphs depends upon the number of edges and vertices present in the graphs. The…

Q: to draw a graph for y=2x+5, a person can draw a point at x of 0 and y of _____, a second point by…

A: Given:

Q: For a given graph G having n vertices and e edges which is connected and has no cycles, which of the…

A: In the question it is provided that a graph G with n vertices and e edges is connected and has no…

Q: Consider the following directed graph. a. Give the indegree of each vertex. b. Give the outdegree of…

A: The vertices are {a , b , c , d , e} (a) The indegree of the vertex is the number of edges coming to…

Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A: Let v0 and v1 be distinct vertices of a graph G=V,E. Assume that there exists path in G from v0 to…

Q: Let G be a planar graph in which every face is either a triangle or a pentagon. If G has 12…

A: It is given that Let G be a planar graph in which every face is either a triangle or a pentagon. If…

Q: 9. Verify that Euler's formula does not hold for a graph that is not connected: count the vertices,…

A: The given question is related with graph theory. We have to verify that Euler's formula does not…

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: Which of the following graphs are isomophic? (Choose two) F E E G D D H J J I A В E E G D H J J To…

A:

Q: Use Kuratowski's theorem to prove that the following graph is not planar.

A: To Prove- Use Kuratowski's theorem to prove that the following graph is not planar.   Theorem Used-…

Q: В. С. D. А.

A:

Q: The complement of a graph G is a new graph formed by removing all the edges of G and replacing them…

A:

Q: Show that for any planar graph, v-e+f=2. v=vertices e=edges f=faces

A:

Q: how that the following graphs are planar: 1.b. 1.c.

A: To prove the graphs are to be planar, draw the graphs on the plane such that their edges intersect…

Q: Let G be a graph with 10 vertice and 5 components. If a vertex is removed from G, the number of…

A: Answer: 9 and 4

Q: Let G = (V, E) be a k-connected graph, let æ E V, and let B C V\{x} such that |B| > k. Show that…

A: Fan: Let v∈V(G) and let A∈V(G)\{v}. A v,A-fan of size k is a collection of k paths P1,P2,...,Pk so…

Q: Which of the following graphs are bipartite? I

A: A graph is bipartite iff graph does not contains any odd cycle.

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. 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: Consider the graph G in Fig. 8-58. Find: (a) degree of each vertex (and verify Theorem 8.1); (b) all…

A:

Q: 9. Verify that Euler's formula does not hold for a graph that is not connected: count the vertices,…

A: Euler's formula for v vertices, e edges and f faces is: v-e+f=2 For the given graph we have vertices…

Q: Prove that every simple graph G= (V, E) with |V] > 2 has at least two vertices of equal degree. %3D…

A: We have a simple graph, with V≥2, that is, the number of vertices is greater than or equal to 2.…

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: [1] Write down the vertex-connectivity K(G) and the edge-connectivity 2(G) for each of the following…

A: Vertex connectivity: The minimum number of vertices required to disconnect the graph G is called…

Q: d. e

A: A graph is said to be strongly connected if we can find a path from each vertex to every other…

Q: Suppose the graph is planar but not connected, and has 5 components. Derive a variant of Euler's…

A: As per the question we have to derive an alternate Euler's formula in case of planar but not…

Q: 12. If G is a simple graph with vertices a, b, c, d, e and d(a) = d(b) = d(c) = 3 (d = degree), then…

A: Here we apply Hand-shaking lemma for simple graph.

Q: b) Determine whether there is an Euler trail from vzto V6 in the following graph. If there is, find…

A:

Q: For the given graph G, which of the following statements is false? A B) (F a) G is a simple directed…

A:

Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A:

Q: Suppose G is a simple undirected graph with n vertices. If G is self-complementary, prove that…

A: ↦A Simple undirected graph G is meant by a graph which have no multiple loop. there exist at most…

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

A: Solution :-

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: Which of the following graphs contain a subdivision of K3,3 as a subgraph

A: The solution is given below in detail in which we found no graph contains a subdivision of K3, 3 1)

Q: Let n e N with n > 2 and let G, = (V, E) be the graph with vertex set V = {v1, v2, . . . , v2n} and…

A: Since you have posted a question with multiple subparts, we will solve the first three subparts…

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: Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and…

A: We are given a G graph G with v number of vertices and e number of edges in the graph. we are also…

Q: ! Consider the graph G below to answer the following questions E A C F (a) Draw a subgraph of G that…

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

A: To find: A graph which is non planar and not homomorphic to K5 or K3,3. If the edges of the graph…

Q: Consider the graph G : bh is an edge in G. The order in G-bh is

A: Graph G is given, with vertices and edges.order of G=number of vertices of G