The Matching Energy of Random Multipartite Graphs

Caibing Chang; Bo Deng; Haizhen Ren; Feng Fu

doi:doi:10.11648/j.ajmcm.20190404.12

| Peer-Reviewed

The Matching Energy of Random Multipartite Graphs

Caibing Chang, Bo Deng, Haizhen Ren, Feng Fu

Published in American Journal of Mathematical and Computer Modelling (Volume 4, Issue 4)

Received: 22 September 2019 Accepted: 2 December 2019 Published: 7 December 2019

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Let G_n be a simple graph with n vertices. Gutman and Wagner founded the theory of random graphs, they introduced the matching energy of the graph G_n, which was defined as the sum of the absolute values of the eigenvalues of the matching polynomial of the graph Gn. For the Erdös-Rényi type random graph G_n,p of order n with a fixed probability p, where p is a real number greater than zero and less than 1, that is, the graph G on n vertices by connecting two vertices with probability p(e), and each edge is independent of other one. Chen, Li and Lian solved a conjecture proposed by Gutman and Wagner, that is, the expectation of the matching energy of G_n,p converges to a certain number associated with n and p almost surely. But they only did the result for random bipartite graphs. In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al’s method to generalize this conclusion to any random multipartite graphs.

Published in	American Journal of Mathematical and Computer Modelling (Volume 4, Issue 4)
DOI	10.11648/j.ajmcm.20190404.12
Page(s)	94-98
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), 2019. Published by Science Publishing Group

Keywords

Random Graphs, Matching Energy, Multipartite Graphs

References

[1]	J. A. Bondy, U. S. R. Murty, Graph Theory, GTM 244, Springer, 2008.
[2]	X. Chen, X. Li, H. Lian, The matching energy of random graphs, Discrete Applied Mathematics 193 (2015), 102–109.
[3]	P. Erdös, A. Rényi, On random graphs I, Publ. Math. Debrecen 6 (1959), 290–297.
[4]	E. J. Farrell, An introduction to matching polynomial, J. Combin. Theory Ser. B 27 (1979), 75–86.
[5]	I. Gutman, S. Wagner, The matching energy of a graph, Discr. Appl. Math. 160 (2012), 2177–2187.
[6]	I. Gutman, Acyclic systems with extremal H¨uckel π-electron energy, Theor. Chim. Acta. 45 (1977), 79–87.
[7]	I. Gutman, Partial ordering of forests according to their characteristic polynomials, in: A. Hajnal, V. T. Sos (Eds.), Combinatorics, North-Holland, Amsterdam, 1978, pp. 429–436.
[8]	I. Gutman, X. Li, Energies of Graphs – Theory and Applications, Math. Chem. Monogr. Vol. 17 (2016). ISBN 978-86-6009-033-3, Kragujevac, Serbia.
[9]	I. Gutman, M. Milun, N. Trinajstić, Graph theory and molecular orbitals 19, non-parametric resonance energies of arbitrary conjugated systems, J. Amer. Chem. Soc. 99 (1977), 1692–1704.
[10]	I. Gutman, The matching polynomial, MATCH Commun. Math. Comput. Chem. 6 (1979), 75–91.
[11]	C. Godsil, Matchings and walks in graphs, J. Graph Theory 5 (1981), 285–297.
[12]	C. Godsil, I. Gutman, On the theory of the matching polynomial, J. Graph Theory 5 (1981), 137–144.
[13]	C. Godsil, Algebraic Combinatorics, Chapman Hall, New York, 1993.
[14]	O. J. Heilman, E. H. Lieb, Theory of monomerCdimer systems, Comm. Math. Phys. 25 (1972).
[15]	I. Gutman, Graphs with greatest number of matchings, Publ. Inst. Math. (Beagrad) 27 (1980) 581-586.
[16]	I. Gutman, Correction of the paper “Graphs with greatest number of matchings”, Publ. Inst. Math. (Beagrad) 32 (1982) 61-63.
[17]	I. Gutman, D. Cvetkovic, Finding tricyclic graphs with a maximal number of matchins-another example of computer aided research aided research in graph theory, Publ. Inst. Math. (Beagrad) 35 (1984) 33-40.
[18]	I. Gutman, F. Zhang, On a quasiordering of bipartite graphs, Publ. Inst. Math. (Beagrad) 35 (1984) 33-40.
[19]	I. Gutman, F. Zhang, On the ordering of graphs with respect to their matching numbers, Discr. Appl. Math. 15 (1986) 25-33.

Cite This Article

Plain Text BibTeX RIS

APA Style

Caibing Chang, Bo Deng, Haizhen Ren, Feng Fu. (2019). The Matching Energy of Random Multipartite Graphs. American Journal of Mathematical and Computer Modelling, 4(4), 94-98. https://doi.org/10.11648/j.ajmcm.20190404.12

Copy | Download

ACS Style

Caibing Chang; Bo Deng; Haizhen Ren; Feng Fu. The Matching Energy of Random Multipartite Graphs. Am. J. Math. Comput. Model. 2019, 4(4), 94-98. doi: 10.11648/j.ajmcm.20190404.12

Copy | Download

AMA Style

Caibing Chang, Bo Deng, Haizhen Ren, Feng Fu. The Matching Energy of Random Multipartite Graphs. Am J Math Comput Model. 2019;4(4):94-98. doi: 10.11648/j.ajmcm.20190404.12

Copy | Download

@article{10.11648/j.ajmcm.20190404.12,
  author = {Caibing Chang and Bo Deng and Haizhen Ren and Feng Fu},
  title = {The Matching Energy of Random Multipartite Graphs},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {4},
  number = {4},
  pages = {94-98},
  doi = {10.11648/j.ajmcm.20190404.12},
  url = {https://doi.org/10.11648/j.ajmcm.20190404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190404.12},
  abstract = {Let Gn be a simple graph with n vertices. Gutman and Wagner founded the theory of random graphs, they introduced the matching energy of the graph Gn, which was defined as the sum of the absolute values of the eigenvalues of the matching polynomial of the graph Gn. For the Erdös-Rényi type random graph Gn,p of order n with a fixed probability p, where p is a real number greater than zero and less than 1, that is, the graph G on n vertices by connecting two vertices with probability p(e), and each edge is independent of other one. Chen, Li and Lian solved a conjecture proposed by Gutman and Wagner, that is, the expectation of the matching energy of Gn,p converges to a certain number associated with n and p almost surely. But they only did the result for random bipartite graphs. In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al’s method to generalize this conclusion to any random multipartite graphs.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - The Matching Energy of Random Multipartite Graphs
AU  - Caibing Chang
AU  - Bo Deng
AU  - Haizhen Ren
AU  - Feng Fu
Y1  - 2019/12/07
PY  - 2019
N1  - https://doi.org/10.11648/j.ajmcm.20190404.12
DO  - 10.11648/j.ajmcm.20190404.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 94
EP  - 98
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20190404.12
AB  - Let Gn be a simple graph with n vertices. Gutman and Wagner founded the theory of random graphs, they introduced the matching energy of the graph Gn, which was defined as the sum of the absolute values of the eigenvalues of the matching polynomial of the graph Gn. For the Erdös-Rényi type random graph Gn,p of order n with a fixed probability p, where p is a real number greater than zero and less than 1, that is, the graph G on n vertices by connecting two vertices with probability p(e), and each edge is independent of other one. Chen, Li and Lian solved a conjecture proposed by Gutman and Wagner, that is, the expectation of the matching energy of Gn,p converges to a certain number associated with n and p almost surely. But they only did the result for random bipartite graphs. In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al’s method to generalize this conclusion to any random multipartite graphs.
VL  - 4
IS  - 4
ER  -

Copy | Download

Author Information

Caibing Chang

School of Mathematics and Statistics, Qinghai Normal University, Xining, China
Bo Deng

School of Mathematics and Statistics, Qinghai Normal University, Xining, China
Haizhen Ren

School of Mathematics and Statistics, Qinghai Normal University, Xining, China
Feng Fu

School of Mathematics and Statistics, Qinghai Normal University, Xining, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Caibing Chang, Bo Deng, Haizhen Ren, Feng Fu. (2019). The Matching Energy of Random Multipartite Graphs. American Journal of Mathematical and Computer Modelling, 4(4), 94-98. https://doi.org/10.11648/j.ajmcm.20190404.12

Copy | Download

ACS Style

Caibing Chang; Bo Deng; Haizhen Ren; Feng Fu. The Matching Energy of Random Multipartite Graphs. Am. J. Math. Comput. Model. 2019, 4(4), 94-98. doi: 10.11648/j.ajmcm.20190404.12

Copy | Download

AMA Style

Caibing Chang, Bo Deng, Haizhen Ren, Feng Fu. The Matching Energy of Random Multipartite Graphs. Am J Math Comput Model. 2019;4(4):94-98. doi: 10.11648/j.ajmcm.20190404.12

Copy | Download

@article{10.11648/j.ajmcm.20190404.12,
  author = {Caibing Chang and Bo Deng and Haizhen Ren and Feng Fu},
  title = {The Matching Energy of Random Multipartite Graphs},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {4},
  number = {4},
  pages = {94-98},
  doi = {10.11648/j.ajmcm.20190404.12},
  url = {https://doi.org/10.11648/j.ajmcm.20190404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190404.12},
  abstract = {Let Gn be a simple graph with n vertices. Gutman and Wagner founded the theory of random graphs, they introduced the matching energy of the graph Gn, which was defined as the sum of the absolute values of the eigenvalues of the matching polynomial of the graph Gn. For the Erdös-Rényi type random graph Gn,p of order n with a fixed probability p, where p is a real number greater than zero and less than 1, that is, the graph G on n vertices by connecting two vertices with probability p(e), and each edge is independent of other one. Chen, Li and Lian solved a conjecture proposed by Gutman and Wagner, that is, the expectation of the matching energy of Gn,p converges to a certain number associated with n and p almost surely. But they only did the result for random bipartite graphs. In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al’s method to generalize this conclusion to any random multipartite graphs.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - The Matching Energy of Random Multipartite Graphs
AU  - Caibing Chang
AU  - Bo Deng
AU  - Haizhen Ren
AU  - Feng Fu
Y1  - 2019/12/07
PY  - 2019
N1  - https://doi.org/10.11648/j.ajmcm.20190404.12
DO  - 10.11648/j.ajmcm.20190404.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 94
EP  - 98
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20190404.12
AB  - Let Gn be a simple graph with n vertices. Gutman and Wagner founded the theory of random graphs, they introduced the matching energy of the graph Gn, which was defined as the sum of the absolute values of the eigenvalues of the matching polynomial of the graph Gn. For the Erdös-Rényi type random graph Gn,p of order n with a fixed probability p, where p is a real number greater than zero and less than 1, that is, the graph G on n vertices by connecting two vertices with probability p(e), and each edge is independent of other one. Chen, Li and Lian solved a conjecture proposed by Gutman and Wagner, that is, the expectation of the matching energy of Gn,p converges to a certain number associated with n and p almost surely. But they only did the result for random bipartite graphs. In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al’s method to generalize this conclusion to any random multipartite graphs.
VL  - 4
IS  - 4
ER  -

Copy | Download