Use the Handshaking Lemma to show that a graph G always has an even number of vertices with odd degrees.

Q: Let G be a simple undirected graph (without multiple edges and loops) with  n vertices and m edges.…

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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: 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: The of a vertex is the number of edges along the unique path between it and the root. Height Degree…

A: The level of a vertex is the number of edges along the unique path between uand the root.

Q: A trail is a walk that does not repeat an edge. Prove that a trail that repeats a vertex must…

A: Assume that vertex Vk is repeated Let the trail be W: V1V2.....Vk........Vn....Vi.....Vk.......Vj…

Q: Can the following graphs drawn in plane without edge crossings? a

A: Given: Graphs, To determine: Whether the given graphs can be drawn without crossing.

Q: Let G be a simple connected planaı 13 vertices and 19 edges. Then, the number of faces in the planar…

A: Given that, G is a simple connected planar graph with 13 vertices and 19 edges.

Q: (i) Verify the Hand-Shaking Theorem for the graph Go. a

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Q: Determine whether the graph is connected. Which vertices are odd? Which vertices are even? А

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Q: Show that the graph is 2-colorable by finding a 2-coloring. If the graph is not 2-colorable, explain…

A: A graph is said to be 2-colorable if we can color the vertices in such a way that its vertices are…

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: Give an example of a graph G of order 6 and size 10 such that the minimum and maximum degree of a…

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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: d. Add up all the indegrees, and add up all the outdegrees. What do you notice? Can you explain why…

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Q: Suppose a planar graph G is not connected and has exactly 2 components. What is the value of n – m+…

A: To obtain the result, use Euler's formula for planar graphs. Let G be a connected planar graph with…

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: Suppose that we have a graph with at least two vertices. Show that it is not possible that all…

A: Suppose, we have a graph with at least two vertices.

Q: hich graph appears to contain all the orderad pairs in this rolation?

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Q: Show that a finite simple graph with more than one vertex has at least two vertices with the same…

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Q: The of a vertex is the number of edges along the unique path between it and the root. Height

A: The level of a vertex v in a rooted tree is the length of the unique path from the root to this…

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: Verify the Handshaking theorem of the following graphs

A: To verify : The Handshaking theorem for the following graphs: (a) (b)

Q: what is K-connected of a graph ? And give an example

A: Definition: A graph is said to be K- connected graph or K edge connected graph if after removed less…

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

A: In the given question we have find the number of the distict graph with n verex.it's answer is "The…

Q: Out(4) G3. List the set of vertices for the graph above:

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Q: Give an example of obtaining the complete graph?

A: A graph is a complete graph if  each pair of graph vertices is connected by an edge. The complete…

Q: Which of the following graphs are bipartite? I

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

Q: If G is a graph, then the sum of its vertex degrees is nonnegative and even.

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Q: Draw all nonisomorphic simple graphs with four vertices and no more than two edges.

A: Given: The objective is to draw all the nonisomorphic simple graphs with the four vertices and no…

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: True or false: If a connected graph G can be decomposed into exactly k paths then exactly 2k…

A: To Prove: A connected graph can be decomposed into exactly k paths then exactly 2k vertices of G…

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: If G is a simple graph with at least two vertices. prove that G must contain two or more vertices of…

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Q: (c) A complete graph with two connected components.

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Q: B: Fined the dual graph for Jollowing graphs the

A: Given

Q: State whether the following statement is true or false. Justify your answer with a short proof or a…

A: Given:- Every even graph is regular.

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

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Q: Show that it is not possible to creak graph the" degree with g vertices such that every a of vertex…

A: Given that  The no of vertices = 9 degree of each vertices =3 To show such graph is not possible.

Q: G is a graph on 6 vertices with exactly

A: (b) G is a graph on 6 vertices with exactly two biconnected components:

Q: Draw all subgraphs of this graph

A: Given: There are 3 vertices, hence there will be 23=8 sub graphs Graph nodes: Nodes is defined as…

Q: If G is a (not necessarily simple) graph with n vertices where each vertex has degree greater than…

A: Definitions: Simple graph: A simple graph is an undirected graph with no parallel edges and…

Q: Prove that in any simple graph G with n vertices and m edges, 2m ≤ n^2 − n.

A: We need to prove that in any simple graph G with n vertices and m edges, 2m≤n2−n

Q: Two simple graphs are if there is a bijection from the vertices of the first graph to the vertices…

A: To fill the correct answer in the blank space.

Q: 3. Find the degree of the vertices of the following graphs and verify the handshaking theorem.

A: Degree of a vertex is the number of edges incident on it.

Q: draw all nonisomorphic simple graphs with four vertices and no more than two edges

A: Given: Four vertices and two edges. To sketch: All non-isomorphic simple graphs no more than two…

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: The graph with edges AC and BD is a connected graph. True False

A: We have to find , the graph with edges AC and BD is a connected graph or not.