Consider a population that grows linearly following the recursive formula P N = P N − 1 + 125 , initial population P 0 = 80 . a. Find P 1 , P 2 , and P 3 . b. Give an explicit formula for P N . c. Find P 100 .
Consider a population that grows linearly following the recursive formula P N = P N − 1 + 125 , initial population P 0 = 80 . a. Find P 1 , P 2 , and P 3 . b. Give an explicit formula for P N . c. Find P 100 .
T2.2 Prove that a sequence s d₁, d₂,..., dn with n ≥ 3 of integers with 1≤d; ≤ n − 1 is the
degree sequence of a connected unicyclic graph (i.e., with exactly one cycle) of order n if and only
if at most n-3 terms of s are 1 and Σ di = 2n.
(i) Prove it by induction along the lines of the inductive proof for trees. There will be a special
case to handle when no d₂ = 1.
(ii) Prove it by making use of the caterpillar construction. You may use the fact that adding an
edge between 2 non-adjacent vertices of a tree creates a unicylic graph.
I need help with this problem and an explanation of the solution for the image described below. (Statistics: Engineering Probabilities)
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T2.1: Prove that the necessary conditions for a degree sequence of a tree are sufficient by showing
that if di 2n-2 there is a caterpillar with these degrees. Start the construction as follows: if
d1, d2,...,d2 and d++1 = d = 1 construct a path v1, v2, ..., vt and add d; - 2 pendent
edges to v, for j = 2,3,..., t₁, d₁ - 1 to v₁ and d₁ - 1 to v₁. Show that this construction results
vj
in a caterpillar with degrees d1, d2, ..., dn
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