Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity

Abdulazeez Idris; Bashir Ismail; Mustapha Usman Jama’are; Dahiru Zaharaddeen

doi:doi:10.11648/j.mlr.20180302.13

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Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity

Abdulazeez Idris, Bashir Ismail, Mustapha Usman Jama’are, Dahiru Zaharaddeen

Published in Machine Learning Research (Volume 3, Issue 2)

Received: 28 July 2018 Accepted: 21 August 2018 Published: 19 September 2018

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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.

Published in	Machine Learning Research (Volume 3, Issue 2)
DOI	10.11648/j.mlr.20180302.13
Page(s)	28-32
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Adjacent Vertices, Chromatic Number, Chromatic Polynomials

References

[1]	Bondy J. A. (1976). Graphs theory with applications, the MACMILAN PRESS LTD London, pp. 131-133.
[2]	David G. (2013). An introduction to combinatorics and graph theory, California, USA: Creative Commons publisher, Pp. 85 -86.
[3]	Gary C. (2009). Chromatic graph theory, U.S.A: Taylor & Francis Group, Pp. 14-167.
[4]	Kalamazoo R. (2003). Graph theory, Yugoslau journal of Operations Research, vol. 13, pp. 208-210.
[5]	Srinivas B. R. (2014). Chromatic polynomials, International Journal of Scientific and Innovative Mathematical Research IJSIMR) Volume 2, Issue 11, November, PP 918-920.
[6]	Tamas H. (2009). the chromatic polynomials, (journal) EOTVOS LO RAND University, vol 12, pp. 12-18.
[7]	Thuslasiraman K. (1992). Graphs; theory and algorithm, New York: John Wiley and sons, Inc. 1992, pp. 253- 254.
[8]	Oystein O. (1962). Theory of graphs, USA: American mathematical society, pp. 2-22.
[9]	Sabyasach M. eta’l (2013). Application of Vertex coloring in a particular triangular closed path structure and in Kraft’s inequality. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS). Vol. 1. Arxiv.org/pdf/1309.3513.pdf.
[10]	Ridhi J. Eta’l (2016). Graph coloring and its implementation, international Journal of Advance Research and Innovation in technology. (vol. 2, issue 5).

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Abdulazeez Idris, Bashir Ismail, Mustapha Usman Jama’are, Dahiru Zaharaddeen. (2018). Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity. Machine Learning Research, 3(2), 28-32. https://doi.org/10.11648/j.mlr.20180302.13

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ACS Style

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. Mach. Learn. Res. 2018, 3(2), 28-32. doi: 10.11648/j.mlr.20180302.13

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AMA Style

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. Mach Learn Res. 2018;3(2):28-32. doi: 10.11648/j.mlr.20180302.13

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@article{10.11648/j.mlr.20180302.13,
  author = {Abdulazeez Idris and Bashir Ismail and Mustapha Usman Jama’are and Dahiru Zaharaddeen},
  title = {Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity},
  journal = {Machine Learning Research},
  volume = {3},
  number = {2},
  pages = {28-32},
  doi = {10.11648/j.mlr.20180302.13},
  url = {https://doi.org/10.11648/j.mlr.20180302.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mlr.20180302.13},
  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.},
 year = {2018}
}

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TY  - JOUR
T1  - Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity
AU  - Abdulazeez Idris
AU  - Bashir Ismail
AU  - Mustapha Usman Jama’are
AU  - Dahiru Zaharaddeen
Y1  - 2018/09/19
PY  - 2018
N1  - https://doi.org/10.11648/j.mlr.20180302.13
DO  - 10.11648/j.mlr.20180302.13
T2  - Machine Learning Research
JF  - Machine Learning Research
JO  - Machine Learning Research
SP  - 28
EP  - 32
PB  - Science Publishing Group
SN  - 2637-5680
UR  - https://doi.org/10.11648/j.mlr.20180302.13
AB  - 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.
VL  - 3
IS  - 2
ER  -

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Author Information

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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APA Style

Abdulazeez Idris, Bashir Ismail, Mustapha Usman Jama’are, Dahiru Zaharaddeen. (2018). Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity. Machine Learning Research, 3(2), 28-32. https://doi.org/10.11648/j.mlr.20180302.13

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ACS Style

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. Mach. Learn. Res. 2018, 3(2), 28-32. doi: 10.11648/j.mlr.20180302.13

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AMA Style

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. Mach Learn Res. 2018;3(2):28-32. doi: 10.11648/j.mlr.20180302.13

Copy | Download

@article{10.11648/j.mlr.20180302.13,
  author = {Abdulazeez Idris and Bashir Ismail and Mustapha Usman Jama’are and Dahiru Zaharaddeen},
  title = {Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity},
  journal = {Machine Learning Research},
  volume = {3},
  number = {2},
  pages = {28-32},
  doi = {10.11648/j.mlr.20180302.13},
  url = {https://doi.org/10.11648/j.mlr.20180302.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mlr.20180302.13},
  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.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Vertex Colorings of Graph and Some of Their Applications in Promoting Global Competitiveness for National Growth and Productivity
AU  - Abdulazeez Idris
AU  - Bashir Ismail
AU  - Mustapha Usman Jama’are
AU  - Dahiru Zaharaddeen
Y1  - 2018/09/19
PY  - 2018
N1  - https://doi.org/10.11648/j.mlr.20180302.13
DO  - 10.11648/j.mlr.20180302.13
T2  - Machine Learning Research
JF  - Machine Learning Research
JO  - Machine Learning Research
SP  - 28
EP  - 32
PB  - Science Publishing Group
SN  - 2637-5680
UR  - https://doi.org/10.11648/j.mlr.20180302.13
AB  - 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.
VL  - 3
IS  - 2
ER  -

Copy | Download