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).

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