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All Journal Electronic Journal of Graph Theory and Applications (EJGTA) Indonesian Journal of Combinatorics
Ashwini S. Yalnaik
Karnatak University, Dharwad
Computer Science & IT Decision Sciences, Operations Research & Management Electrical & Electronics Engineering

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Reciprocal complementary distance spectra and reciprocal complementary distance energy of line graphs of regular graphs Harishchandra S. Ramane; Ashwini S. Yalnaik
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 3, No 2 (2015): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2015.3.2.10

Abstract

The reciprocal complementary distance (RCD) matrix of a graph $G$ is defined as $RCD(G) = [rc_{ij}]$ where $rc_{ij} = \frac{1}{1+D-d_{ij}}$ if $i \neq j$ and $rc_{ij} = 0$, 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 $RCD(G)$. Two graphs are said to be $RCD$-equienergetic if they have same $RCD$-energy. In this paper we show that the line graph of certain regular graphs has exactly one positive $RCD$-eigenvalue. Further we show that $RCD$-energy of line graph of these regular graphs is solely depends on the order and regularity of $G$. This results enables to construct pairs of $RCD$-equienergetic graphs of same order and having different $RCD$-eigenvalues.
Hamming index of graphs with respect to its incidence matrix Harishchandra S. Ramane; Ishwar B. Baidari; Raju B. Jummannaver; Vinayak V. Manjalapur; Gouramma A. Gudodagi; Ashwini S. Yalnaik; Ajith S. Hanagawadimath
Indonesian Journal of Combinatorics Vol 6, No 2 (2022)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/ijc.2022.6.2.4

Abstract

Let B(G) be the incidence matrix of a graph G. The row in B(G)corresponding to a vertex v, denoted by s(v) is the string which belongs to ℤm2, a set of m-tuples over a field of order two. The Hamming distance between the strings s(u) and s(v) is the number of positions in which s(u) and s(v) differ. In this paper we obtain the Hamming distance between the strings generated by the incidence matrix of a graph. The sum of Hamming distances between all pairs of strings, called Hamming index of a graph is obtained.
Co-Authors Ajith S. Hanagawadimath Gouramma A. Gudodagi Harishchandra S. Ramane Ishwar B. Baidari Raju B. Jummannaver Vinayak V. Manjalapur