Polynomial Graphs

3 Key excerpts on "Polynomial Graphs"

• 

  eBook - ePub

  Chemical Graph Theory

  Introduction and Fundamentals

  • D Bonchev(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  Chapter 4 POLYNOMIALS IN GRAPH THEORY Ivan Gutman Faculty of Science, University of Kragujevac, Yugoslavia 1.    Why Polynomials in Graph Theory? 2.    On Chemical Applications of Graphic Polynomials 3.    Polynomials 4.    The Characteristic Polynomial 5.    The Matching Polynomial 6.    More Graphic Polynomials 7.    References 4.1  Why Polynomials in Graph Theory?

  A variety of problems in pure and applied graph theory can be treated and solved in a rather efficient manner by making use of polynomials. Polynomials provide both a convenient and powerful mathematical tool and a valuable proof technique in certain fields of graph theory. There are essentially three routes by which polynomials enter into the theory of graphs.

  First: Polynomials appear as generating functions of combinatorial graph invariants.

  For example, let m(G, k) be the number of ways in which k independent edges can be selected in the graph G. Then the sequence m(G, 1), m(G, 2), m(G, 3),… etc., can be presented by means of the power series:

  M

  (

  G , x

  )

  = 1 +   m

  (

  G , 1

  )

  x + m

  (

  G , 2

  )

  x 2

  + m

  (

  G .3

  )

  x 3

  + … ,

  with x being an auxiliary and meaningless variable. The above series is described as the generating function for the numbers m(G, k). It will be seen later that the function M(G, x) is in fact a polynomial.

  A generating function can be viewed merely as shorthand notation for the respective sequence. There are, however, many advantages to using a generating function instead of a sequence of graph invariants. For instance, one can easily demonstrate that M(G11 , x) =1 + 3x. The same generating function for the graph G12

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  eBook - ePub

  Algebra II Workbook For Dummies

  • Mary Jane Sterling(Author)
  • 2018(Publication Date)
  • For Dummies

    (Publisher)

  Chapter 7

  Plugging in Polynomials

  IN THIS CHAPTER

  Locating the intercepts in polynomials

  Figuring out where a polynomial function changes signs

  Assembling all the information you need to graph a polynomial curve

  Looking for roots with clues from Descartes — and others

  Doing division synthetically and taking other shortcuts

  Building up polynomials from their roots

  Apolynomial function is a well-mannered function with nice, predictable habits. The graph of a polynomial function is always a smooth curve with no abrupt changes in direction or shape. The equation of a polynomial function has only whole-number exponents on the variables.

  A polynomial function has the form , where n is a whole number and any is a real number. The x 0 is actually equal to 1, so that term is a constant; I include the final x and superscript here to complete the pattern. A polynomial function has at most n x -intercepts and at most turning points (where the curve of the graph changes direction).

  In this chapter, you graph polynomial functions after finding their intercepts and determining where the function is positive or negative. More-advanced tools used to find the x -intercepts (roots) of the polynomial equation include the rational root theorem and Descartes’s rule of sign. Finally, the remainder theorem, along with synthetic division, lets you quickly evaluate functions, and the factor theorem tells you how to reconstruct polynomials if you know only their roots.

  Finding Basic Polynomial Intercepts

  A polynomial function has exactly one y -intercept (where it crosses the y -axis). You find the y -intercept by replacing all the x ’s in the function formula with 0 and solving for y .

  Polynomial functions may or may not have x -intercepts. If a polynomial function has an odd degree (the highest power is an odd number), then the polynomial has to have at least one x -intercept; the curve has to cross the x -axis. A polynomial function with an even degree may or may not cross the x -axis. But in any case, a polynomial won’t have more x

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  eBook - ePub

  Algebra & Geometry

  An Introduction to University Mathematics

  • Mark Verus Lawson(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  quintic.

  For most of this book, we shall only deal with polynomials in one variable, but in a few places we shall also need the notion of a polynomial in several variables. Let x1 ,…,xn be n variables. When n = 1 we usually take just x, when n = 2, we take x, y and when n = 3, we take x, y, z. These variables are assumed to commute with each other. It follows that a product of these variables can be written

  x 1

  x 1

  …

  x n

  r n

  . This is called a monomial of degree r1 + … + rn . If ri = 0 then

  x i

  r i

  = 1

  and is omitted. A polynomial in n variables with coefficients in F , denoted by

  F [

  x 1

  , … ,

  x n

  ]

  , is any sum of a finite number of distinct monomials each multiplied by an element of F . The degree of such a non-zero polynomial is the maximum of the degrees of the monomials that appear.

  Example 7.1.1. Polynomials in one variable of arbitrary degree are the subject of this chapter whereas the theory of polynomials in two or three variables of degree 2 is described in Chapter 10 . The most general polynomial in two variables of degree 2 looks like this

  a

  x 2

  + b x y + c

  y 2

  + d x + e y + f .

  It consists of three monomials of degree 2: namely, x2 , xy and y2 ; two monomials of degree 1: namely, x and y; and a constant term arising from the monomial of degree 0. Polynomials of degree 1 give rise to linear equations which are discussed in Chapter 8

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