Pentacyclic Harmonic Graph

Ahmad Salehi Zarrin Ghabaei; Shahroud Azami

doi:doi:10.11648/j.pamj.20160505.15

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Pentacyclic Harmonic Graph

Ahmad Salehi Zarrin Ghabaei, Shahroud Azami

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 5)

Received: 28 August 2016 Accepted: 8 September 2016 Published: 11 October 2016

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Abstract

Let be a graph on n vertices and let be the degree of vertex A graph is defined to be harmonic if is an eigenvector of the -adjacency matrix of We now show that there are 4 regular and 45 non-regular connected pentacyclic harmonic graphs and determine their structure. In the end we conclude that all of c-cyclic harmonic graphs for are planar graphs.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 5)
DOI	10.11648/j.pamj.20160505.15
Page(s)	165-173
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), 2016. Published by Science Publishing Group

Keywords

Harmonic Graph, Eigenvalue, Spectra

References

[1]	Borovicanin B., Grunewald S., Gutman I., Petrovic M. 2003. Harmonic graphs with small number of cycles, Discrete Mathematics 265, 31-44.
[2]	Cvetkovi'c D., Doob M., Sachs H. 1995. Spectra of Graphs: Theory and Applications (3rd ed.), Johann Ambrosius Barth, Heidelberg.
[3]	Cvetkovi'c D., Rowlinson P., Simi'c S. 1997. Eigenspaces of Graphs, Cambridge University Press, Cambridge.
[4]	Dress A., Gutman I. 2003. Asymptotic results regarding the number of walks in a graph, Appl. Math. Lett., 16(3), 389-393.
[5]	Dress A., Gutman I. 2003. On the number of walks in a graph, Appl. Math. Lett. 16(5), 797-801.
[6]	Favaron O., Mah'eo M., Sacl'e J.F., 1993. Some eigenvalue properties in graphs (conjectures of GraLti—II), Discrete Math. 111, 197–220.
[7]	Godsil C., Royle G. 2001. Algebraic Graph Theory, Springer-Verlag, New York.
[8]	Grunewald S. 2002. Harmonic tree, App. Math. Lett. 15, 1001-1004.
[9]	Petrovi´c M., Borovi´canin B., Radosavljevi´c Z. 2006. The integral 3-harmonic graphs, Linear Algebra and its Applications 416, 298–312.
[10]	Mahmoodi A. 2013. Some Results on Harmonic Graphs, Gen. Math. Notes, 19(1), 53-59.
[11]	Salehi Zarrin Ghabaei A., Azami S. 2014. Some properties of harmonic graphs, Advances in Environmental Biology, 8(11), 597-605.

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APA Style

Ahmad Salehi Zarrin Ghabaei, Shahroud Azami. (2016). Pentacyclic Harmonic Graph. Pure and Applied Mathematics Journal, 5(5), 165-173. https://doi.org/10.11648/j.pamj.20160505.15

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ACS Style

Ahmad Salehi Zarrin Ghabaei; Shahroud Azami. Pentacyclic Harmonic Graph. Pure Appl. Math. J. 2016, 5(5), 165-173. doi: 10.11648/j.pamj.20160505.15

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AMA Style

Ahmad Salehi Zarrin Ghabaei, Shahroud Azami. Pentacyclic Harmonic Graph. Pure Appl Math J. 2016;5(5):165-173. doi: 10.11648/j.pamj.20160505.15

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@article{10.11648/j.pamj.20160505.15,
  author = {Ahmad Salehi Zarrin Ghabaei and Shahroud Azami},
  title = {Pentacyclic Harmonic Graph},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {5},
  pages = {165-173},
  doi = {10.11648/j.pamj.20160505.15},
  url = {https://doi.org/10.11648/j.pamj.20160505.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160505.15},
  abstract = {Let  be a graph on n vertices  and let  be the degree of vertex  A graph  is defined to be harmonic if  is an eigenvector of the -adjacency matrix of  We now show that there are 4 regular and 45 non-regular connected pentacyclic harmonic graphs and determine their structure. In the end we conclude that all of c-cyclic harmonic graphs for  are planar graphs.},
 year = {2016}
}

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TY  - JOUR
T1  - Pentacyclic Harmonic Graph
AU  - Ahmad Salehi Zarrin Ghabaei
AU  - Shahroud Azami
Y1  - 2016/10/11
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160505.15
DO  - 10.11648/j.pamj.20160505.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 165
EP  - 173
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160505.15
AB  - Let  be a graph on n vertices  and let  be the degree of vertex  A graph  is defined to be harmonic if  is an eigenvector of the -adjacency matrix of  We now show that there are 4 regular and 45 non-regular connected pentacyclic harmonic graphs and determine their structure. In the end we conclude that all of c-cyclic harmonic graphs for  are planar graphs.
VL  - 5
IS  - 5
ER  -

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Author Information

Ahmad Salehi Zarrin Ghabaei

Department of Mathematics, Parand Branch, Islamic Azad University, Parand, Iran
Shahroud Azami

Department of Mathematics, Parand Branch, Islamic Azad University, Parand, Iran

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APA Style

Ahmad Salehi Zarrin Ghabaei, Shahroud Azami. (2016). Pentacyclic Harmonic Graph. Pure and Applied Mathematics Journal, 5(5), 165-173. https://doi.org/10.11648/j.pamj.20160505.15

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ACS Style

Ahmad Salehi Zarrin Ghabaei; Shahroud Azami. Pentacyclic Harmonic Graph. Pure Appl. Math. J. 2016, 5(5), 165-173. doi: 10.11648/j.pamj.20160505.15

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AMA Style

Ahmad Salehi Zarrin Ghabaei, Shahroud Azami. Pentacyclic Harmonic Graph. Pure Appl Math J. 2016;5(5):165-173. doi: 10.11648/j.pamj.20160505.15

Copy | Download

@article{10.11648/j.pamj.20160505.15,
  author = {Ahmad Salehi Zarrin Ghabaei and Shahroud Azami},
  title = {Pentacyclic Harmonic Graph},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {5},
  pages = {165-173},
  doi = {10.11648/j.pamj.20160505.15},
  url = {https://doi.org/10.11648/j.pamj.20160505.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160505.15},
  abstract = {Let  be a graph on n vertices  and let  be the degree of vertex  A graph  is defined to be harmonic if  is an eigenvector of the -adjacency matrix of  We now show that there are 4 regular and 45 non-regular connected pentacyclic harmonic graphs and determine their structure. In the end we conclude that all of c-cyclic harmonic graphs for  are planar graphs.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Pentacyclic Harmonic Graph
AU  - Ahmad Salehi Zarrin Ghabaei
AU  - Shahroud Azami
Y1  - 2016/10/11
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160505.15
DO  - 10.11648/j.pamj.20160505.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 165
EP  - 173
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160505.15
AB  - Let  be a graph on n vertices  and let  be the degree of vertex  A graph  is defined to be harmonic if  is an eigenvector of the -adjacency matrix of  We now show that there are 4 regular and 45 non-regular connected pentacyclic harmonic graphs and determine their structure. In the end we conclude that all of c-cyclic harmonic graphs for  are planar graphs.
VL  - 5
IS  - 5
ER  -

Copy | Download