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The history of Tutte–Whitney polynomials
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Tutte closes the paper with the remark “[The number of spanning trees of a graph] has a simple expression as a determinant, and its properties are well known. Perhaps some of them will suggest new properties of the dichromate and hence of chromatic polynomials.” This reflects the inspiration that Tutte drew from the properties of numbers of spanning trees, since his earliest work. He used it in “squaring the square” [220], and it was the first member of his collection of deletion–contraction invariants [1106]; it was abstraction from this collection that led him to this polynomial [1098, 1108]. In more recent times, it has become clear that the number of spanning trees is quite atypical of Tutte polynomial evaluations, in that it is one of very few points where evaluating T(G;x,y) can be done in polynomial time [653] (see Chapter 9). This suggests that its mathematical theory can be expected to be richer and more tractable than that for other evaluations (including, for example, colorings with more than two colors).
Review of Basic Concepts
Published in Khalid Khan, Tony Lee Graham, Engineering Mathematics with Applications to Fire Engineering, 2018
The method of solving linear equations is to collect all the terms involving x on one side of the equation and everything else on the other side. The idea is to isolate the variable x to be on its own. The way this is achieved is through using certain operations like addition, subtraction, multiplication, division, and others (i.e., squaring and square rooting) to manipulate the equation so as to keep both sides of the equation the same. This is illustrated in the following examples.
The power of the snake: number theory with Python
Published in International Journal of Mathematical Education in Science and Technology, 2022
takes almost no time. I explain at a later time that the last computation is done using the Binary Exponentiation Algorithm (Burton, 2010, p. 70), also known as Fast Exponentiation, Exponentiation by Squaring, or Square and Multiply.