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Abstract
The complementary distance (CD) matrix of a graph $G$ is defined as $CD(G) = [c_{ij}]$, where $c_{ij} = 1+D-d_{ij}$ if $i \neq j$ 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.
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References
- bibitem{Bal} Balakrishnan, R., The energy of a graph, textit{Linear Algebra Appl.}, textbf{387} (2004), 287--295.
- bibitem{Bra} Brankov, V., Stevanovi'{c}, D. and Gutman, I., Equienergetic chemical trees, textit{J. Serb. Chem. Soc.}, textbf{69} (2004), 549--553.
- bibitem{Buc} Buckley, F. and Harary, F., textit{Distance in Graphs}, Addison--Wesley, Redwood, 1990.
- bibitem{Cve} Cvetkovi'{c}, D., Rowlinson, P. and Simi'{c}, S., textit{An Introduction to the Theory of Graph Spectra}, Cambridge Univ. Press, Cambridge, 2010.
- bibitem{Gut} Gutman, I., The energy of a graph, textit{Ber. Math. Stat. Sekt. Forschungsz. Graz}, textbf{103} (1978), 1--22.
- bibitem{Har} Harary, F., textit{Graph Theory}, Addison--Wesley, Reading, 1969.
- bibitem{Iva} Ivanciuc, O., Ivanciuc, T. and Balaban, A. T., The complementary distance matrix, a new molecular graph metric, textit{ACH-Models Chem.} {bf 137(1)} (2000), 57--82.
- bibitem{Jen} Jenev{z}i'{c}, D., Miliv{c}evi'{c}, A., Nikoli'{c}, S. and Trinajstic, N., Graph theoretical matrices in chemistry in: textit{Mathematical Chemistry Monographs}, Vol. 2, University of Kragujevac, Kragujevac, 2007.
- bibitem{Li} Li, X., Shi, Y. and Gutman, I., textit{Graph Energy}, Springer, New York, 2012.
- bibitem{Ram1} Ramane, H. S., Gutman, I. and Ganagi, A. B., On diameter of line graphs, textit{Iranian J. Math. Sci. Inf.}, textbf{8(1)} (2013), 105--109.
- bibitem{Ram2} Ramane, H. S., Revankar, D. S., Gutman, I. and Walikar, H. B., Distance spectra and distance energies
- of iterated line graphs of regular graphs, textit{Publ. Inst. Math. (Beograd)}, textbf{85} (2009), 39--46.
- bibitem{Ram3} Ramane, H. S. and Walikar, H. B., Construction of equienergetic graphs, textit{MATCH Commun. Math. Comput. Chem.}, textbf{57} (2007), 203--210.
- bibitem{Ram4} Ramane, H. S., Walikar, H. B., Rao, S. B., Acharya, B. D., Hampiholi, P. R., Jog, S. R. and Gutman, I., Equienergetic graphs, textit{Kragujevac J. Math.}, textbf{26} (2004), 5--13
- bibitem{Sac1} Sachs, H., "{U}ber selbstkomplementare Graphen, textit{Publ. Math. Debrecen}. textbf{9} (1962), 270--288.
- bibitem{Sac2} Sachs, H., "{U}ber Teiler, Faktoren und charakteristische Polynome von Graphen, Teil II, textit{Wiss. Z. TH Ilmenau}, textbf{13} (1967), 405--412.
- bibitem{Sen} Senbagamalar, J., Baskar Babujee, J. and Gutman, I., On Wiener index of graph complements, textit{Trans. Comb.}, textbf{3(2)} (2014), 11--15.
- bibitem{Xu} Xu, L. and Hou, Y., Equienergetic bipartite graphs, textit{MATCH Commun. Math. Comput. Chem.}, textbf{57} (2007), 363--370.
References
bibitem{Bal} Balakrishnan, R., The energy of a graph, textit{Linear Algebra Appl.}, textbf{387} (2004), 287--295.
bibitem{Bra} Brankov, V., Stevanovi'{c}, D. and Gutman, I., Equienergetic chemical trees, textit{J. Serb. Chem. Soc.}, textbf{69} (2004), 549--553.
bibitem{Buc} Buckley, F. and Harary, F., textit{Distance in Graphs}, Addison--Wesley, Redwood, 1990.
bibitem{Cve} Cvetkovi'{c}, D., Rowlinson, P. and Simi'{c}, S., textit{An Introduction to the Theory of Graph Spectra}, Cambridge Univ. Press, Cambridge, 2010.
bibitem{Gut} Gutman, I., The energy of a graph, textit{Ber. Math. Stat. Sekt. Forschungsz. Graz}, textbf{103} (1978), 1--22.
bibitem{Har} Harary, F., textit{Graph Theory}, Addison--Wesley, Reading, 1969.
bibitem{Iva} Ivanciuc, O., Ivanciuc, T. and Balaban, A. T., The complementary distance matrix, a new molecular graph metric, textit{ACH-Models Chem.} {bf 137(1)} (2000), 57--82.
bibitem{Jen} Jenev{z}i'{c}, D., Miliv{c}evi'{c}, A., Nikoli'{c}, S. and Trinajstic, N., Graph theoretical matrices in chemistry in: textit{Mathematical Chemistry Monographs}, Vol. 2, University of Kragujevac, Kragujevac, 2007.
bibitem{Li} Li, X., Shi, Y. and Gutman, I., textit{Graph Energy}, Springer, New York, 2012.
bibitem{Ram1} Ramane, H. S., Gutman, I. and Ganagi, A. B., On diameter of line graphs, textit{Iranian J. Math. Sci. Inf.}, textbf{8(1)} (2013), 105--109.
bibitem{Ram2} Ramane, H. S., Revankar, D. S., Gutman, I. and Walikar, H. B., Distance spectra and distance energies
of iterated line graphs of regular graphs, textit{Publ. Inst. Math. (Beograd)}, textbf{85} (2009), 39--46.
bibitem{Ram3} Ramane, H. S. and Walikar, H. B., Construction of equienergetic graphs, textit{MATCH Commun. Math. Comput. Chem.}, textbf{57} (2007), 203--210.
bibitem{Ram4} Ramane, H. S., Walikar, H. B., Rao, S. B., Acharya, B. D., Hampiholi, P. R., Jog, S. R. and Gutman, I., Equienergetic graphs, textit{Kragujevac J. Math.}, textbf{26} (2004), 5--13
bibitem{Sac1} Sachs, H., "{U}ber selbstkomplementare Graphen, textit{Publ. Math. Debrecen}. textbf{9} (1962), 270--288.
bibitem{Sac2} Sachs, H., "{U}ber Teiler, Faktoren und charakteristische Polynome von Graphen, Teil II, textit{Wiss. Z. TH Ilmenau}, textbf{13} (1967), 405--412.
bibitem{Sen} Senbagamalar, J., Baskar Babujee, J. and Gutman, I., On Wiener index of graph complements, textit{Trans. Comb.}, textbf{3(2)} (2014), 11--15.
bibitem{Xu} Xu, L. and Hou, Y., Equienergetic bipartite graphs, textit{MATCH Commun. Math. Comput. Chem.}, textbf{57} (2007), 363--370.