1. Let C(G,k) denote the decision problem of whether the undirected graph G = subset of vertices V' C V such that |V'| = k and there is an edge connecting every pair of vertices in V'. Prove that C(G, k) is NP-Complete. (V, E) has a
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A: Given:
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Q: (b) Find the Wronskian W[y1,Y2](t) of y1 and y2-
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Q: 3.9 Suppose that the second-order system i = f(x), with a locally Lipschitz f(z) has a limit cycle.…
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- 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.(Show your work) prove that the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are.(a) Find a conjugacy C between G(x) = 4x(1-x) and g(x)=2-x^2 . (b) Show that g(x) has chaotic orbits.
- (Show your work) compute the girth of all generalized Coxeter graphs with parameter Pn,u,v where n is less or equal to12a. The general continuous-time random walk is defined by g_(ij)={[mu_(i)",",j=i-1","],[-(lambda_(i)+mu_(i))",",j=i","],[lambda_(i)",",j=i+1","],[0","," otherwise "]:} Write out the forward and backward equations. b. The continuous-time queue with infinite buffer can be obtained by modifying the general random walk in the preceding problem to include a barrier at the origin. Put g_(0j)={[-lambda_(0)",",j=0","],[lambda_(0)",",j=1","],[0","," otherwise ".]:} Find the stationary distribution assuming sum_(j=1)^(oo)((lambda_(0)cdotslambda_(j-1))/(mu_(1)cdotsmu_(j))) < oo. If lambda_(i)=lambda and mu_(i)=mu for all i, simplify the above condition to one involving only the relative values of lambda and mu. c. Repeat a and b assuming there is a boundary at j=N. Comments.need explicit calculation process, step by stepA(n)............ is the collection of all points in the plane the sum of whose distances from two fixed points is a constant
- Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced subgraph isomorphic to k1,r+1 has a 1-factor. Show that this is not true if you replace r connected by r edge connectedShow that there is no Hamiltonian path inCay({(1, 0), (0, 1)}:Z3 ⊕ Z2)from (0, 0) to (2, 0).Extend the proof of the Schmidt decomposition to the case where two parties A and B may have state spaces of different dimensionality.