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Year 2021, Volume: 9 Issue: 1, 36 - 41, 01.03.2021

Harishchandra Ramane , B Parvathalu

https://doi.org/10.36753/mathenot.641660

Abstract

Project Number

F.510/3/ DRS-III /2016 (SAP-I)

References

  • [1] Buckley, F.: Iterated line graphs. Congr. Numer. 33, 390–394 (1981).
  • [2] Buckley, F.: The size of iterated line graphs. Graph Theory Notes of New York. 25, 33–36 (1993).
  • [3] Cvetkovic, D., Rowlinson, P., Simic, S.: Introduction to the Theory of Graph Spectra. Cambridge University Press. Cambridge (2010).
  • [4] Gutman, I.: The energy of a graph. Ber. Math. Stat. Sekt. Forschungsz. Graz. 103, 1–22 (1978).
  • [5] Harary, F.: Graph Theory. Addison-Wesley Publishing Co., Reading (1969).
  • [6] Ivanciuc, O., Ivanciuc, T., Balaban, A. T.: The complementary distance matrix, a new moleculargraph metric. ACHModels Chem. 137, 57–82 (2000).
  • [7] Indulal, G.: D-spectrum and D-energy of complements of iterated line graphs of regular graphs. J. Alg. Stru. Appl. 4, 51–56 (2017). https://doi.org/10.29252/asta.4.1.51
  • [8] Jenežic, D., Milicevic, A., Nikolic, S., Trinajstic, N.: Graph Theoretical Matrices in Chemistry. University of Kragujevac. Kragujevac (2007). https://doi.org/10.1021/ci700278s
  • [9] Li, X., Shi, Y., Gutman, I.: Graph Energy. Springer. New York (2012). https://doi.org/10.1007/978-1-4614-4220-2
  • [10] Ramane, H. S., Gudodagi, G. A.: Reciprocal complementary equienergetic graphs. Asian-European J. Math. 9, ID: 1650084, pages 15 (2016). https://doi.org/10.1142/S1793557116500844
  • [11] Ramane, H. S., Yalnaik,A. S.: Reciprocal complementary distance spectra and reciprocal complementary distance energy of line graphs of regular graphs. El. J. Graph Theory Appl. 3, 228–236 (2015). http://dx.doi.org/10.5614/ejgta.2015.3.2.10
  • [12] Sachs, H.: Über selbstkomplementare Graphen. Publ. Math. Debrecen. 9, 270–288 (1962).
  • [13] Sachs, H.: Über Teiler, Faktoren und charakteristische Polynome von Graphen, Teil II. Wiss. Z. TH Ilmenau. 13, 405–412 (1967).

Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs

Year 2021, Volume: 9 Issue: 1, 36 - 41, 01.03.2021

Harishchandra Ramane , B Parvathalu

https://doi.org/10.36753/mathenot.641660

Abstract

The reciprocal complementary distance ($RCD$) matrix of a graph $G$ is defined as $RCD(G) = [r_{ij}]$, where $r_{ij} = \frac{1}{1+D-d_{ij}}$ if $i \neq j$ and $r_{ij} = 0$, otherwise, where $D$ is the diameter of $G$ and $d_{ij}$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $RCD$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of $RCD$-matrix. Two graphs are said to be $RCD$-equienergetic if they have same $RCD$-energy. In this paper, the $RCD$-energy of the complement of line graphs of certain regular graphs in terms of the order and degree is obtained and as a consequence, pairs of $RCD$-equienergetic graphs of same order and having different $RCD$-eigenvalues are constructed.

Keywords

Reciprocal complementary distance ($RCD$) eigenvalues $RCD$-energy of a graph $RCD$-equienergetic graphs

Supporting Institution

University Grants Commission (UGC), New Delhi

Project Number

F.510/3/ DRS-III /2016 (SAP-I)

Thanks

The author HSR is thankful to the University Grants Commission (UGC), New Delhi, for support through UGC-SAP DRS-III, 2016-2021: F.510/3/ DRS-III /2016 (SAP-I).

References

  • [1] Buckley, F.: Iterated line graphs. Congr. Numer. 33, 390–394 (1981).
  • [2] Buckley, F.: The size of iterated line graphs. Graph Theory Notes of New York. 25, 33–36 (1993).
  • [3] Cvetkovic, D., Rowlinson, P., Simic, S.: Introduction to the Theory of Graph Spectra. Cambridge University Press. Cambridge (2010).
  • [4] Gutman, I.: The energy of a graph. Ber. Math. Stat. Sekt. Forschungsz. Graz. 103, 1–22 (1978).
  • [5] Harary, F.: Graph Theory. Addison-Wesley Publishing Co., Reading (1969).
  • [6] Ivanciuc, O., Ivanciuc, T., Balaban, A. T.: The complementary distance matrix, a new moleculargraph metric. ACHModels Chem. 137, 57–82 (2000).
  • [7] Indulal, G.: D-spectrum and D-energy of complements of iterated line graphs of regular graphs. J. Alg. Stru. Appl. 4, 51–56 (2017). https://doi.org/10.29252/asta.4.1.51
  • [8] Jenežic, D., Milicevic, A., Nikolic, S., Trinajstic, N.: Graph Theoretical Matrices in Chemistry. University of Kragujevac. Kragujevac (2007). https://doi.org/10.1021/ci700278s
  • [9] Li, X., Shi, Y., Gutman, I.: Graph Energy. Springer. New York (2012). https://doi.org/10.1007/978-1-4614-4220-2
  • [10] Ramane, H. S., Gudodagi, G. A.: Reciprocal complementary equienergetic graphs. Asian-European J. Math. 9, ID: 1650084, pages 15 (2016). https://doi.org/10.1142/S1793557116500844
  • [11] Ramane, H. S., Yalnaik,A. S.: Reciprocal complementary distance spectra and reciprocal complementary distance energy of line graphs of regular graphs. El. J. Graph Theory Appl. 3, 228–236 (2015). http://dx.doi.org/10.5614/ejgta.2015.3.2.10
  • [12] Sachs, H.: Über selbstkomplementare Graphen. Publ. Math. Debrecen. 9, 270–288 (1962).
  • [13] Sachs, H.: Über Teiler, Faktoren und charakteristische Polynome von Graphen, Teil II. Wiss. Z. TH Ilmenau. 13, 405–412 (1967).
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Harishchandra Ramane 0000-0003-3122-1669

B Parvathalu This is me 0000-0002-5151-8446

Project Number F.510/3/ DRS-III /2016 (SAP-I)
Publication Date March 1, 2021
Submission Date November 1, 2019
Acceptance Date November 9, 2020
Published in Issue Year 2021 Volume: 9 Issue: 1

Cite

APA Ramane, H., & Parvathalu, B. (2021). Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs. Mathematical Sciences and Applications E-Notes, 9(1), 36-41. https://doi.org/10.36753/mathenot.641660
AMA Ramane H, Parvathalu B. Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs. Math. Sci. Appl. E-Notes. March 2021;9(1):36-41. doi:10.36753/mathenot.641660
Chicago Ramane, Harishchandra, and B Parvathalu. “Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs”. Mathematical Sciences and Applications E-Notes 9, no. 1 (March 2021): 36-41. https://doi.org/10.36753/mathenot.641660.
EndNote Ramane H, Parvathalu B (March 1, 2021) Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs. Mathematical Sciences and Applications E-Notes 9 1 36–41.
IEEE H. Ramane and B. Parvathalu, “Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs”, Math. Sci. Appl. E-Notes, vol. 9, no. 1, pp. 36–41, 2021, doi: 10.36753/mathenot.641660.
ISNAD Ramane, Harishchandra - Parvathalu, B. “Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs”. Mathematical Sciences and Applications E-Notes 9/1 (March2021), 36-41. https://doi.org/10.36753/mathenot.641660.
JAMA Ramane H, Parvathalu B. Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs. Math. Sci. Appl. E-Notes. 2021;9:36–41.
MLA Ramane, Harishchandra and B Parvathalu. “Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 1, 2021, pp. 36-41, doi:10.36753/mathenot.641660.
Vancouver Ramane H, Parvathalu B. Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs. Math. Sci. Appl. E-Notes. 2021;9(1):36-41.
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