15. Let m > 1 be an integer and let T be a tree with m edges. Prove that m+1 a(T) >

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

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Q: Can we draw a planar graph with n= 7 and e = 12 such that each region is bounded by exactly 3-edges…

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Q: Show that every graph with n vertices and k edges, n> k has n – k components.

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Q: on Find the largest number of paths from 4 to B that do not have any edges in common.

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Q: Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if…

A: Thanks for the question :)And your upvote will be really appreciable ;)

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Q: Show the results of inserting the keys C, R, A, U,J,S, F,T,0,P,M,L,N,W,Q in order into an empty…

A: Given: V={1,2,3,4,5,6,7,8,9}. and edges E(V)={e1=(1,3),e2=(2,4),e3=3,6 ,e4=7,8,…

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

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

Q: 4. Solve the travelling problem for the given graph by finding the total weight of all Hamilton…

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Q: 4. Use K-maps to find simpler circuits with the same output as of the circuits shown. Tyz Kydx +) +G…

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Q: Let K, be the complete graph on n vertices. a) Draw Kn for n = 1, 2, 3, 4, 5. For each, how many…

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Q: Prove or disprove that if G has m edges and n vertices, then the degeneracy of G is at least m

A: Given  be a graph with m edges and n vertices. Required to prove that the degeneracy of G is at…

Q: 3.1.6. (-) Let T be a tree with n vertices, and let k be the maximum size of an indepen- dent set in…

A: Given, Vertices of the tree, T =n Maximum size of independent set in T = k To find : α't

Q: 6. Prove that the number of vertices in an undirected graph with odd degree must be even. Hint.…

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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: Let n be a positive integer. Show that every connected graph on n vertices has at least n – 1 -…

A: This is a problem of Graph Theory.

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

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Q: Prove that if u is a vertex of odd degree in a graph, then there exists a path from u to another…

A: We have to prove :       If u is a vertex of odd degree in a graph, then there exists a path from…

Q: 9. Obtain the shortest distance and shortest path from vertex 1 to every other vertex in the network…

A: To find the shortest distance between the vertex 1 to every other vertex .

Q: Determine the number of distinct unlabeled simple graphs (not necessarily connected) up to 4 nodes.…

A: To determine the number of distinct unlabelled simple graphs upto 4 nodes.

Q: Prove or disprove (by giving a counterexample) that a graph G with n 2 4 vertices and ("5') +3 edges…

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Q: 3. A labeled Ks graph with labeled edges appears below. Which edge permutation is induced by the…

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Q: For every graph G on n vertices, either G or its complement with respect to Kn must be Eulerian…

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Q: 2. Prove that any graph has at least two vertices with the same degree. A complete bipartite graph…

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

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

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Q: 2. Suppose G is a planar graph having 67 edges. If each face of G has length at least 6, then what…

A: From the given information. The number of edges (e) = 67 The number of vertices at each face (v) = 6

Q: Suppose a graph G has 22 vertices. What is the minimum number of EDGES that G must contain if Diracs…

A: Dirac's theorem Let G = (V,E) be a graph with n vertices in which each vertex has degree at least…

Q: Consider a bipartite graph on the set A = {1,2,3,4,5,6} and B = {7,8,9,10,11,12}, where there is an…

A: Given: A bipartite graph is defined on set A={1,2,3,4,5,6} and    B={7,8,9,10,11,12}. Here there is…

Q: 2. Give an example of a directed graph with infinitely many vertices in which each verter has at…

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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: If a graph with six vertices is embedded in S?, what is the maximum number of edges it can have.…

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Q: Prove the following. 1. Prove that the number of odd vertices of a graph is even.

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Q: Let Xn be a the number of graphs with vertex set [2n], Every vertex has a degree of 1. What is the…

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Q: (4) For all n > 2, is it possible to partition the set of edges of K, into a single edge, a path of…

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

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

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Q: 2. Prove: "The size of every bipartite graph of order n is at most ". Provide an 4 illustration and…

A: Definition: A simple graph whose set of vertices can be partitioned into two sets V1 and V2 such…

Q: Draw a Cayley graph for Z/7Z with generating set {2¯,5¯}.

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Q: 2. Show that the graphs formed by deleting one edge from Ks and K33 are planar.

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Q: 2) Find, with a proof, the number of distinct graphs with vertex set [n].

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Q: Let K, be a graph obtained from K, by deleting one edge. How many subgraphs of K, are isomorphic to…

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Q: 1. Let C(G,k) denote the decision problem of whether the undirected graph G = subset of vertices V'…

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Q: Prove that if there is a circuit in a graph that starts and ends at a vertex v and if w is another…

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