Prove that if G = (V, E) is a connected graph such that |E |V| – 1, then G is a tree. %3D

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

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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: Let G be a graph with n 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G?…

A: Given : G is a graph with n≥1 vertices We have to answer the following : (a) If χG=1, what can be…

Q: A) Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd length.

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Q: srepresentation issues in the given graph. Select all that apply. 100

A: 1. the horizontal axes is scale is appropriate where each point is at the distance of 4 units  2.…

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

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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: Let G = (V, E) be a connected graph. Use induction to show that if G contains exactly |G| = |V|.…

A: Suppose G=(V,E) be a connected graph, also let that G contains one cycle. Then we show that G=V

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

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Q: Draw a graph G with vertex and edge set  , where the endpoints of e1 are v1 and v2, the endpoints of…

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Q: Let G be a graph obtained from Kå after subdividing all edges of K6. So the graph G has 21 vertices.…

A: The graph for K6 is as follows:

Q: Let G =(V,E) be a finite graph on n ≥ 3

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

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Q: In 1 and 2, graphs are represented by drawings. Define each graph formally by specifying its vertex…

A: 1.Note that, a graph consists of two finite sets: a finite, a nonempty set of vertices and a finite…

Q: Let G be graph such that | VCG)l=IELG).° show that 8CG)<3

A: By handshaking lemma we see that if v1, v2,..., vn are n vertices and E is the number of edges then:…

Q: Determine if the graph shown below is Eulerian or Not  a. Eulerian b. Not Eulerian

A: The given graph:                 Eulerian graph: It is a closed walk(Starting and ending vertex…

Q: Show that if a simple graph G ha

A: To prove that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

Q: Let G be a connected graph. Prove: (a) If G contains a cycle C which contains an edge e, then G – e…

A: Part (a):   A connected graph is a graph in which there is a path joining each pair of vertices, in…

Q: Let G be a bipartite graph.(Give a sketch on each) a) Is the complement of G also bipartite?…

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Q: Find whether the graph below has an Euler trail, find such a trail. b f a he

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Q: Graphs G and H are said to be isomorphic if the vertices are in one-to-one correspondence and the…

A: Graph isomorphism: Two graphs G(V,E) and  G'(V',E') are said to be isomorphic if there exists an…

Q: 13. A) Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd…

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Q: The graph of g con

A: Consider the graph below

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

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Q: Let G = (V, E) be a finite and complete graph. For any set W of vertices and any edge e E E, define…

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Q: Let G = (V, E) be a graph and A = max,ev{deg(v)}. Show that x(G) <A+1. Give two examples of graphs…

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Q: pigeon hole principle

A: Answer (1) E = { (u,v) , u, v ∈ V | person u and V have met each other}(2) degree of a vertex u of G…

Q: c) Prove that if a plane graph G has n vertices, e edges, f faces and k components then n- e+f =k+1.

A: We can prove Euler's formula (n-e+ f = k+1) works generally for anyconnected simple planar graph.so,

Q: Which of the following statements are true. Do not show your explanations. a) A tree is a graph…

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Q: 1 Let G = (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > |V] –…

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Q: Let G be the simple graph on the vertex set V(G) = {a, b, c, d, e, f} with adjacency matrix: 011101…

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Q: prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

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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: Find the strongly connected components of this graph? * a d e

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Q: Determine if the graph shown below is Eulerian or Not A) B) C) G, o Eulerian o Not Eulerian o…

A: A connected graph G is said to be Eulerian if and only if degree of every is even .

Q: 4. Let n eN with n >1, V = Z and set E = {{x, y} : x, y E Z and |r – y| =n}. Define G, = (V, E). (a)…

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Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

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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: 8. If we remove an edge e from a graph G and G- e is still connected, then show that e lies along…

A: Given: Remove an edge e from a graph G and G-e is connected.

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…

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

Q: Let G be a simple graph with the vertex set V = {v1, v2, V3, v4, U5, U6}. Which of the following…

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Q: Let G = (V, E) be a simple undirected graph without loops, where |V| > 2. Let 8(v) denote the degree…

A: a) The statement is there does not exist u and v such that δu=0 and δv=V-1. First let us consider a…