1. Write the definition of isomorphism between two simple undirected graphs.
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Q: Are the ff. two graphs isomorphic? 一 一 一 1. 1. 1. 1. -
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Q: 20 - Determine whether the relation with the directed graph shown is an equivalence relation ? a d
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- (it is algoritma and graph theory class.) If G is a graph without loops, what can you say about the sum of the entries in (i) any row or column of the adjacency matrix of G? (ii) any row of the incidence matrix of G? (iii) any column of the incidence matrix of G?Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.Trace the graph below to determine whether or not it is Hamiltonian. If not, find the minimum number of edges to be removed to make it so. Mark the edge/s to be removed, and name one resulting Hamiltonian graph using the given letters.
- Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.III. Consider the directed graph described by the following:(a) Draw the graph.(b) Find a directed path from vertex 3 to vertex 6.(c) Find a directed cycle starting from and ending at vertex 4.(d) Find the adjacency matrix of the graph.(e) Does there exist a directed path from vertex 2 to vertex 6?Hey, The condensation of a graph G with k strong coherence components G1 =.(V1 , E1 ), . . . , Gk = (Vk , Ek )is the reduction of the original graphto its strong coherence components. In this case, the coherence components are combined into one node each in the condensation. The condensation to G is thus the graph G↓=({V1,...,Vk},E),where(Vi,Vj)∈E ⇔i̸=j∧∃u∈Vi,v∈Vj:(u,v)∈E holds. what is the Kondensation G↓ of the graph in the picture? Thank you in advance!
- Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.6. If G1 and G2 are isomorphic, must they have the same number of edges? If G1 and G2 have the same number of edges must they be isomorphic? If not, give an example.Draw a connected bipartite graph G = (V, E) such that its bipartition V1, V2 has |V1| = 5 and|V2| = 4 and deg(a) ≤ 3 for all a ∈ V .(a) Determine how many cut vertices and how many cut edges G has.(b) How many 1’s does the incidence matrix have in it?(c) Does your graph have an Euler path? Justify your answer
- Show that if an edge e is in a closed, trail of G, then e is in a cycle of GThe parts (a) and (b) of this problem are independentof each other.G1 G24 51 236sx yt u v(a) Prove that the graphs G1 and G2 are isomorphic byexhibiting an isomorphism from one to the other byconcrete arguments and verify it by using adjacencymatrices. Please take the ordering of the vertices as1, 2, 3, 4, 5, 6 while forming AG1, adjacency matrix ofG1.Warning: One must stick to the labelings ofthe vertices as they are given, if one changesthe labelings/orderings etc., the solution willnot be taken into account.(b) Consider the complete graph K13 with vertex setV13 = {u1, u2, u3, · · · , u13}.Let H = (V, E) be the simple graph obtained fromK13 by adding a new vertex u, i.e. V = V13 ∪ {u}and deleting the edges {u1, u2} and {u2, u3} andadding the edges {u1, u} and {u, u2} and keepingthe remaining edges same.Determine whether H has an Euler circuit or not,an Euler path or not. One must validate any conclusion by clear arguments.Which of these are paths in the directed graph shown? (Select all that apply.)