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- Suppose S is the unit cube in the first octant of uvw-space withone vertex at the origin. What is the image of the transformationT: x = u/2, y = v/2, z = w/2?(Show your work) compute the girth of all generalized Coxeter graphs with parameter Pn,u,v where n is less or equal to12Prove that for every natural number v ≥ 4 there exists a planar graph with v vertices which has all the areas bounded by C4 cycles.
- Given the tent map T(x) = {2x for x<= 1/22-2x for x> 1/2}Prove that the set of all periodic points of T is dense in [0, 1] and determine the number of points with least periods and their distinct orbits.Suppose A is a bipartite graph that has color classes V and W. So if for all v∈V and w∈W, then d(v)≥d(w). Prove that A has a perfect matching of V into W.(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.
- 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.If x and y are 3−cycles in Sn, prove that ⟨x, y⟩ is isomorphic to Z3, A4, A5 or Z3 × Z3.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 connected
- (Show your work) prove that the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are.(a) Consider coordinates in the XY-plane, (x, y) with x, y ≥ 0 and a binary relation ≺ where(x1, y1) ≺ (x2, y2) if and only if x1 ≤ x2 and y1 ≤ y2.i. Verify that ≺ is a partial order.ii. Provide an example set A ⊂ R2 such that the following Hasse diagram corresponds with thepartial order ≺ on A (from i.) and label the vertices accordingly.iii. What element(s), if any, would you need to add to your poset in order for it to be a lattice?Explain why.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.