a) Prove that, for all integers K > 5 P(K + 1,5) + P(K,5) = 5P(K,4) b) What is graph, Explain its all terminology and give exampleof each.

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

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Q: How many pairwise non-isomorphic connected simple graphs are there on 6 vertices which contain two…

A: We have to find that how many pairwise non-isomorphic connected simple graphs are there on 6…

Q: Determine the number of non-isomorphic 10-regular graphs of order 13. Briefly explain your…

A: It is known that G and H is isomorphic iff Gc and Hc are isomorphic. Given that any regular  graph…

Q: 9.9 If G is a connected graph, a centre of G is a vertex v with the property that the maximum of the…

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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: 10. A policeman wants to patrol a neighbarhood, but anly wants to walk down the street once. Based…

A: This is a problem of Graph Theory.

Q: 2. Show that there is no simple graph with six vertices of which the degrees of five vertices are 5,…

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Q: prove that any connected graph G in which every edge is contained in a cycle cannot be disconnected…

A: Want to prove:  If G is a connected graph in whichevery edge 'e' is contained in a cycle then g-{e}…

Q: Prove on the number of vertices that every graph G with n edges has a bipartite subgraph with at…

A: We will first assume that  P(n) : every graph G with n edges has a bipartite subgraph with atleast…

Q: How many different labelled graphs are there on the vertex set [n]? Justify your claim.

A: Given Vertex set n Let vertex V=1, 2, 3, ........., n Each graph G with a vertex V is determined by…

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: How many non-isomorphic simple graphspre there with 11 vertices, 18 edges, minimum degree 3, maximum…

A: Given, Number of vertices V = 11 Number of edges E = 18 Minimum degree = 3 Maximum degree = 6 Number…

Q: 7. Check whether the given graph is Bipartite or not. Explain. Also construct a proper subgraph from…

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Q: 2. Is it possible for a connected graph with 7 vertices and 11 edges to be drawn so that no edges…

A: (2) Consider the provided question, Graph with no edges crossing means that it should be a planar…

Q: Draw all non-isomorphic graphs of order n with 1<n<4.

A: As per the question we have to draw all the non-isomorphic graphs of order 1, 2, 3, 4 But as there…

Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

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: The most famous theorem in graph theorem is the following: Theorem 1. The vertices of a planar graph…

A: No, it is not true that all graphs with χG&lt;4 are planar. it is known that Peterson graph is not…

Q: Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must be…

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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: How many different simple graphs are there om the vertex set [n]?

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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: (16) Give an example of two nonisomorphic 2-regular graphs of the same order and show that they are…

A: Isomorphism of two graph:Let, G1 and G2 be two graph . ThenG1 and G2 are said to be isomorphismif ∃…

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

A: As per Bartleby's answering policy, we can answer only one question, so kindly post the remaining…

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

A: Given: To prove the given statement as follows:

Q: How can one prove a graph is connected?

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Q: Plot the Polan graph Or=-6 los O in interval Lo,1603 interval Lorz6o 3 36c in

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Q: The complete tripartite graph K,, consists of three sets of vertices (of sizes r, s and t), with an…

A: Given that The complete tripartite graph Kr,s,t The graph of K2,2,2 in the following graph Hence,…

Q: An outerplanar graph is a graph that admits a planar embedding with all vertices on the boundary of…

A: Given that an outerplanar graph is a graph that admits a planar embedding with all vertices on the…

Q: 1. Create a connected graph having 6 vertices and 11 edges. 2. Label vertices with A,B,C,D,E and F.…

A: Given: 1) Create a connected graph having 6 vertices and 11 edges 2) Lable vertices with A, B, C, D,…

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

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Q: Show that the petersen graph is nonplanar.( Hint :show that the graph is contradictible to k5.

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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: Let G be a simple, connected graph on n vertices (n≥3) Suppose that G has at least one cycle as a…

A: Given, G is a simple connected graph on n vertices with n≥3. G has at least one cycle as a subgraph…

Q: a. Draw an acyclic graph with 7 vertices and 4 components. b. Prove that if G is an acyclic graph…

A: Given G is an acyclic graph with v vertices and 4 components.

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

A: For each Xn, we count graphs that satisfy two conditions: The graph must have a vertex set of 2n…

Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

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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…

A: The objective is to prove or provided a counterexample for the following situations, For all n≥2, is…

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: 1. Does there exist a disconnected graph of order 10 having 36 edges? Explain.

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Q: 1. (a) What is the maximum number of edges that a bipartite planar graph with 10 vertices can have?…

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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: A graph contains 21 edges, 3 vertices of degree 4 and all other vertices of degree 2. Find total…

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Q: 11. Determine if the following graph is bipartite graph. If it is, show how to divide its vertices…

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Q: complement.

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Q: Prove that the sum of the degrees of the vertices of any finite graph is even.

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Q: Construct a 3-regular graph with 10 vertices. Is it possible to have a 3-regular graph with 15…

A: 3-Regular graph with 10 vertices

Q: A graph is called d-regular if every vertex has a degree of d. Let k and n be odd numbers, and prove…

A: Here we have to know the basic concept or the basic theorem before answering the question  THEOREM :…

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: . Which complete bipartite graphs Km,n, where m and n positive integers, are trees? are

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