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).