Pure and Applied Mathematics Journal

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Pentacyclic Harmonic Graph

Received: 28 August 2016    Accepted: 08 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.

DOI 10.11648/j.pamj.20160505.15
Published in Pure and Applied Mathematics Journal (Volume 5, Issue 5, October 2016)
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), 2024. 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.
Author Information
  • Department of Mathematics, Parand Branch, Islamic Azad University, Parand, Iran

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

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    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://download.sciencepg.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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