3. Let T, = (v,e,) and T, = (v,e,) be two trees with e, = 12 and v, = 3 + 5v,. Find: and Remember: e = v - 1. Show your work in the space provided below.
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- Sketch a diagram such that the vertices of △ABC lie respectively on the sides of △XYZ so that AX, BY, CZ are concurrent.Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits of length three, then e ≤ 2v − 4. (Show work)Show that R3 is spanned by S = {(1, 1, 1), (1, −1, 0), (0, 1, −1)}.
- Let F, F ′ be forests on the same set of vertices, with ∥F ∥ < ∥F ′∥. Show that F ′ has an edge e such that F + e is again a forest.Prove that the following conditions on a connected graph Γ are equivalent. Γ is a tree. Given any two vertices v and w in Γ, there is a unique reduced edge path from v to w. For every edge e # E(Γ), removing e from Γ disconnects the graph. (Note: Removing e does not remove its associated vertices.) If G is finite then #V (Γ) = #E(Γ) + 1.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.
- I need proof this thm ASAP I vll upvote for u If u prove within 30 minutes Prove that if there exists at most one path between any two vertices of a simple graph G, then G is a forest and conversely.Find c such that (9,4), (10,3), and (c,6) lie on a line.Show that the Grundy index Γ′(T) exists for every tree T and that ∆(T)≤Γ′(T)≤2∆(T)−1
- 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?Assume a directed acyclic graph is encoded as a relation R. Give a concise proposition stating that R is irreflexive (i.e., that no nodes have edges to themselves).Use the function to find the image of v and the preimage of w.