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A directed graph is simple if each ordered pair of vertices is the head and tail of at most one edge; one loop may be present at each vertex. For each n ≥ 1, prove or disprove the following statement. Every simple directed graph with n vertices has two vertices with the same outdegree or two vertices with the same indegree.
(a) How many words of length 2 are there over the alphabet A = { 0; 1; 2; 3}?
(b) Construct the de Bruijn digraph D4,2 and use this digraph to nd an appropriate de Bruijn sequence.
(c) For b = 4 and n = 2, how many de Bruijn sequences can be formed?
Suppose [S, T] and [X, Y ] are two minimal cuts in a network N. Prove that both [S [X U T ∩ Y ] and [S ∩ X, T U Y ] are also minimal cuts in N.
1.Sam Lucarelli, owner of Lucarelli Products, is evaluating whether to produce a new product line. After thinking through the production process and the costs of raw materials and
Canonical correlation analysis : A process of analysis for investigating the relationship between the two groups of variables, by ?nding the linear functions of one of the sets of
In the network shown below, the rst of the two numbers on each arc indicates the arc capacity and the second (in parentheses) of the two numbers indicates the current flow. Use t
You have probably noticed by now that some of the statements of necessary and sufficient conditions sound more natural than others. For example it seems more natural to express "We
The regression analysis is used to fit a model describing the relationship of a dependent variable with independent variable(s). Here we have fitted three regression models:
Biplots: It is the multivariate analogue of the scatter plots, which estimates the multivariate distribution of the sample in a few dimensions, typically two and superimpose on th
Ordinal variable is a measurement which allows a sample of the individuals to be ranked with respect to some characteristic but where differences at different points of the scale
Bayes factor : A summary of evidence for the modelM1 against the another modelM0 provided by the set of data D, which can be used in the model selection. Given by the ratio of post
Different approaches to the study of early indian history
The skewness is a measure of asymmetry and as it is positive at 4.29, it is greater than zero which reveals that the tail extends to the right indicating the distribution to be mor
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