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Journal > Journal of the Indonesian Mathematical Society > COMPLEMENTARY DISTANCE SPECTRA AND COMPLEMENTARY DISTANCE ENERGY OF LINE GRAPHS OF REGULAR GRAPHS

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Journal of the Indonesian Mathematical Society
Volume 22 Number 1 (April 2016)
COMPLEMENTARY DISTANCE SPECTRA AND COMPLEMENTARY DISTANCE ENERGY OF LINE GRAPHS OF REGULAR GRAPHS
Ramane, Harishchandra S. ( Karnatak University, Dharwad - 580003, India)
Nandeesh, K.C. ( Karnatak University, Dharwad - 580003, India)
 Article Info ABSTRACT Published date: 03 May 2016 The complementary distance (CD) matrix of a graph \$G\$ is defined as \$CD(G) = [c_{ij}]\$, where \$c_{ij} = 1+D-d_{ij}\$ if \$i eq j\$ and \$c_{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 \$CD\$-energy of \$G\$ is defined as the sum of the absolute values of the eigenvalues of \$CD\$-matrix. Two graphs are said to be \$CD\$-equienergetic if they have same \$CD\$-energy. In this paper we show that the complement of the line graph of certain regular graphs has exactly one positive \$CD\$-eigenvalue. Further we obtain the \$CD\$-energy of line graphs of certain regualr graphs and thus constructs pairs of \$CD\$-equienergetic graphs of same order and having different \$CD\$-eigenvalues. Copyrights © 2016