(a) Give the adjauncy matrix of H. (b) Find three non-isomorp anninn trane u. Explain why your trees are not isomorphic. (c) Find a functionp: V(G) V(H) such that p is an isomorphism between G and H. Explain also why p is an isomorphism.
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- The parts (a) and (b) of this problem are independent of each other. G1 G2 a h 5 d (a) Prove that the graphs G1 and G2 are isomorphic by exhibiting an isomorphism from one to the other by concrete arguments and verify it by using adjacency matrices. Please take the ordering of the vertices as a, b, c, d, e, h while forming AG, , adjacency matrix of G.. Warning: One must stick to the labelings of the vertices as they are given, if one changes the labelings/orderings etc., the solution will not be taken into account. (b) Consider the complete graph K17 with vertex set V17 = {u1, U2, Uz, , U17}. Let H = (V, E) be the simple graph obtained from K17 by adding a new vertex u, i.e. V = V17 U {u} and deleting the edges {u1, u2} and {u2, Uz} and adding the edges {u1,u} and {u,u2} and keeping the remaining edges same. Determine whether H has an Euler circuit or not, an Euler path or not. One must validate any con- clusion by clear arguments.(a) Give the definition of an isomorphism from a graph G to a graph H. (b) Consider the graphs G and H below. Are G and H isomorphic?• If yes, give an isomorphism from G to H. You don’t need to prove that it isan isomorphism.• If no, explain why. If you claim that a graph does not have a certain feature,you must demonstrate that concretely. (c) Consider the degree sequence (1, 2, 4, 4, 5). For each of the following, ifthe answer is yes, draw an example. If the answer is no, explain why. (i) Does there exist a graph with this degree sequence?(ii) Does there exist a simple graph with this degree sequence?Let T:V→V be the adjacency operator of the Petersen graph as illustrated in the enclosed file. Here Vis the vector space of all formal real linear combinations of the vertices v1,..,v10 of the Petersen graph. Question: compute the spectrum of the adjacency operator Tof the Petersen graph. What do you observe?
- For each pair of graphs G1 = <V1, E1> and G2 = <V2, E2> a) determine if they are isomorphic or not. b) Determine a function that can be isomorphic between them if they are isomorphic. Otherwise you should justify why they are not isomorphic. c) is there an Euler road or an Euler bike in anyone graph? Is Hamilton available? You should draw if the answer is yes and reason if your answer is no.The 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.1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)
- . Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is Hamiltonian3.(a) Assume there is an adjacency matrix A of an undirected graph G. Further assume a 1 isindicated in the element, A(ij), if there is an edge between node i and node j, and a 0 is indicatedin the element A(ij) if there is no edge between node i and node j. Explain the characteristics ofthe matrix A when(i) the graph is complete,(ii) the graph has a loop (edge connecting a vertex to itself),(iii) the graph has an isolated vertex, i.e., a vertex with no edges incident on it.(b) Repeat (a) for adjacency list representation, each row takes a linked list like structure.In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…
- Show that the class of DCFLs is not closed under a) union, b) intersection.(a) Let G be a simple undirected graph with 18 vertices and 53 edges such that the degreesof G are only 3 and 7. Suppose there are a vertices of degree 7 and b vertices ofdegree 3. Find a and b. To receive any credit for this problem you must write completesentences, explain all of your work, and not leave out any details. problem 1, continued(b) Recall that a graph G is said to be k-regular if and only if every vertex in G has degreek. Draw all 3-regular simple graphs with 12 edges (mutually non-isomorphic). Hint:there are six of them. To receive credit for this problem, you should explain, as well aspossible, why all of your graphs are mutually non-isomorphic.(V, E) be a connected, undirected graph. Let A = V, B = V, and f(u) = neighbours of u. Select all that are true. Let G = a) f: AB is not a function Ob) f: A B is a function but we cannot always apply the Pigeonhole Principle with this A, B Odf: A B is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the above