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Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity
Machine Learning Research
Volume 3, Issue 2, June 2018, Pages: 28-32
Received: Jul. 28, 2018; Accepted: Aug. 21, 2018; Published: Sep. 19, 2018
Authors
Abdulazeez Idris, Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria
Bashir Ismail, Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria
Mustapha Usman Jama’are, Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria
Dahiru Zaharaddeen, Department of Mathematics, Federal University, Dutsinma, Nigeria
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Abstract
This paper studies various results on vertex colorings of simple connected graphs, chromatic number, chromatic polynomials and some Algebraic properties of chromatic polynomials. Results were obtained on the roots of chromatic polynomials of simple connected graphs based on Read’s conjecture. The chromatic number of every graph is the minimum number of colors to properly color the graph. Chromatic polynomial of a graph is a polynomial in integer and the leading coefficient of chromatic polynomial of a graph of order n and size m is always 1, whose coefficient alternate in sign. Through the application of famous graph theorem (the hand shaking lemma) by whiskey which states that: “the order of a graph twice its size”. Hence, every graph has a chromatic polynomial but not all polynomials are chromatic. For example, the polynomial λ5 − 11 λ4 + 14 λ3 − 6 λ2 + 2 λ is a polynomial for a graph on five vertices and eleven edges which does not exists. Because the maximum number size for a graph of order five is ten. The paper equally gave some practical applications of Vertex coloring in real life situations such as scheduling, allocation of channels to television and radio stations, separation of chemicals and traffic light signals.
Keywords
Adjacent Vertices, Chromatic Number, Chromatic Polynomials
Abdulazeez Idris, Bashir Ismail, Mustapha Usman Jama’are, Dahiru Zaharaddeen, Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity, Machine Learning Research. Vol. 3, No. 2, 2018, pp. 28-32. doi: 10.11648/j.mlr.20180302.13
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