Let G be a simple undirected graph (without multiple edges and loops) with  n vertices and m edges.  What is the maximal m?

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: G – v3 G- v2 G – v4

A: A graph consist of Set of vertices and edges A subgraph W of a graph G consist of subset of edges…

Q: Draw a loop from vertex G. What is now the degree of G?

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Q: Prove that every maximal matching in a graph G has at least a'(G)/2 edges.

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Q: 1) An undirected graph G has 40 vertices and the degree of each vertex is at least 5. What is the…

A: Given:    Vertices = 40       degree = 5

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: Show that the maximum number of edges in a graph with n-vertices is *C2 (d)

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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: Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

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Q: Suppose G is a simple graph (no loops; no parallel edges) with n = 14 vertices and m = 7 edges. What…

A: GIVEN  vertices n=14 edges m=7 EXPLANATION as we know  every graph has atleast n-m component…

Q: Let G be a graph without loop. If G has 6 vertices and 20 edges then the minimum number of edges…

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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: 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: Let G be a simple graph with n vertices, which is regular of degree d. By considering the number of…

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

A: We have to Show that if a simple graph G has n vertices and at least n edges, then G must contain a…

Q: (b) Give an example of a disconnected simple graph on n vertices in which every vertex has degree…

A: We need to give an example of a disconnected simple graph on n vertices in which every vertex has…

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: Let G be a graph with vertex set V and edge set E. If |V] = 7, what is the largest value that |E||…

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A: To find- Which of the following relations represented by the directed graph is symmetric?   Concept…

Q: (a) Prove that if a connected planar graph has 13 vertices and 12 edges then it must be a tree. (b)…

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Q: A K- regular graph is a graph where every vertex has degree K

A: Consider the given information. A graph is hamiltonian if it visits every vertex exactly once except…

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: 5. Determine whether the given pair of graphs is isomorphic. u2 v2 V3 V4

A: The number of vertices in graph 1= 8 The number of vertices in graph 2 = 8  The number of vertices…

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: The set of all finite connected simple graphs is countable. O true O false

A: Here the given statement is 'the set of all finite connected simple ghaph is countable'

Q: Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in G…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: 4a Let n > 4. What is the maximum possible number of edges in a graph with n vertices and n-2…

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

A: Given

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: Show that every simple graph on n vertices is isomorphic to a subgraph of Kn.

A: We need to prove that every simple graph on n vertices is isomorphic to a subgraph of Kn (complete…

Q: Let G be a connected graph with at least two nodes. Prove that it has a node such that if this node…

A: Consider a connected graph with at least two vertices. choose vertices u,v so that du,v=maximum if…

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: Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible…

A: Please check the next step for details

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: 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: Select all the graphs that are trees. B.

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Q: What is a(G) if G is a complete graph on n vertices?

A: It is given that G is a complete graph on n vertices. Also provided are some basic definitions about…

Q: Are the following directed graphs strongly connected or not? Explain why in detail? 3 4 A

A: A directed graph is said to be strongly connected if there is a path between any pair of vertices in…

Q: State whether each of the following statements is TRUE or FALSE. . Every maximal matching is a…

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