1. Draw pictures of all the isomorphism classes of trees with 5 vertices.

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

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Q: 3. Find the tree and tree cover of Petersen graph. ***** *****

A: The Petersen Graph

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: 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: 1. Write the definition of isomorphism between two simple undirected graphs.

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Q: Consider the trees with 7 vertices which have exactly 4 leaves. (a) Find a representative of each…

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Q: Prove that for every coloring (n colors) of a complete graph on 2n+1 vertices there's a…

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Q: A graph G has the following vertex and edge sets. V = {1,2,3, ..., n} E = {{1,2}, {1,3}, {1,4},...,…

A: Solu

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

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Q: (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is…

A: (d)  To prove that the pair of graphs are not isomorphic by showing that there is        a property…

Q: 3. Determine whether the given pair of graphs is isomorphic. Exhibit an isomorphism (by showing a…

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Q: 2. Find three spanning trees representing two isomorphism classes of graph

A: Given,

Q: Here are two 3-regular graphs, both with six vertices and nine edges. If they are isomorphic, give…

A: Given ,two graphs are G & HConsider , 3 vertices v1,v2,v3     & 3 edges     e1 , e2  &…

Q: Show all isomorphism classes of trees on 6 vertices (i.e. all unlabelled trees on 6 vertices)

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Q: 7 How many isomorphisms are there between the following two graphs? Explain. (If you do not know…

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Q: Prove that every complete graph K, with n>3 has a Hamilton circuit. Write in complete sentences in…

A: A complete graph on n vertices is denoted by Kn and is defined as a simple undirected graph in which…

Q: (a) Find the number of the isomorphism classes of connected graphs with 5 vertices and 5 edges. Draw…

A: Write the isomorphism of 5 vertices and 5 edges. Isomorphism is mapping of one-to-one of two sets.

Q: Show that there is a graph G with exactly 5 nodes where both G and its complement, have chromatic…

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Q: What would a non-transitive graph of three groups, lets say (A, B, and C) look like?

A: Given non transitive graph of three groups

Q: find the number of the isomorphism classes of graphs with 5 vertices and 6 edges which have (i)…

A: Given the a graphs with 5 vertices and 6 edges. We have to find the number of the isomorphism…

Q: 1. Let F be a forest with 50 vertices and 35 edges. How many trees does have F.

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Q: 3. Determine whether the given pair of graphs is isomorphic. Exhibit an isomorphism (by showing a…

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Q: Let L be the set of undirected graphs. Prove that L is in NL.

A: Let L be the set of undirected graphs. A undirected graph is bipartite if its nodes can be divided…

Q: Draw all non-isomorphic unrooted trees with exactly five vertices. b. Draw all non-isomorphic…

A: We Know that  Two trees are called isomorphic if one of them can be obtained from other by a series…

Q: If (A, B) is a Dedekind cut of Q, then B always has a smallest element. True False

A: Dedekind cut of Q :  Dedekind cut is a partition of the rational numbers into two sets A and B

Q: What is the number of isomorphism classes of graphs with 6 vertices and 3 edges

A: It is known that sum of degree in a graph must be even because sum of degrees is  twice the edges.…

Q: Prove the following theorem: If the graph G is a tree then there is a unique path connecting any two…

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Q: 2. (a) Find the number of the isomorphism classes of connected graphs with 5 vertices and 5 edges.…

A: Isomorphism is the one-to-one mapping of connected graphs. Check the number of vertices of the graph…

Q: Pick a set of five vertices of G and draw the subgraph induced by that set of vertices G:

A: Let GV,E be a graph with vertex set V and edge set E.  A vertex-induced subgraph HV1,E1 is a…

Q: Justify the following statement: the total degree (sum of all degrees of all vertices) of a finite…

A: Isomorphic Graphs Two graphs G1 and G2 are said to be isomorphic if − There exists a function ‘f’…

Q: ii. Let A be a set of six elements and σ an equivalence relation on A such that the resulting…

A: A graph G is bipartite if its vertex set can be partitioned into two sets X and Y in such a way that…

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: Show that if G is a connected graph and e is an edge in G that is part of a cycle, then the graph G'…

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Q: Draw all forests of order 6.

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Q: Show that every simple graph on n vertices subgraph of K.

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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: A graph is connected if and only if one vertex is neighbor to all other vertices O True O False

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Q: 4. (a) (i) List all non-isomorphic trees (not rooted) on 6 vertices with no vertex of degree larger…

A: Consider the given information. (a) Now, draw all the non-isomorphic tress on 6 vertices with no…

Q: 5.1.7. (-) Construct a graph G that is neither a clique nor an odd cycle but has a vertex ordering…

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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…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For a graph…

Q: Prove or disprove: Every tree has at most one perfect matching.

A: We use mathematical induction to solve this problem. The detailed solution is typesetted in latex…

Q: 4. Draw every simple graph with five vertices and four edges (isolated vertices are allowed) up to…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 5. Are the following directed graphs isomorphic? Provide an isomorphism if they are or an argument…

A: To find an directed isomorphic we should first check if the given graphs are isomorphic or not, if…

Q: Show all isomorphism classes of trees on 5 vertices (i.e. all unlabelled trees on 5 vertices)

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Q: 2. Consider the trees with 7 vertices which have exactly 4 leaves. (a) Find a representative of each…

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Q: 1.36. Show that if G is a tree then for any pair of vertices a, b there is exactly one path from a…

A: Given that, G be a tree. As, G be a tree that implies that, G is connected and G has no cycle. Since…

Q: Prove: “A graph G is a tree if and only if every two vertices of G are connected by a unique path."…

A: Let G is the tree. If part: We have to prove, "if G is tree then every two vertices of G connected…

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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