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Bounds for Resistance–Distance Spectral Radius

Year 2013, Volume: 42 Issue: 1, 43 - 50, 01.01.2013

A. Dilek Güngör Maden İvan Gutman A. Sinan Çevik

Abstract

References

  • Balaban, A. T., Ciubotariu, D. and Medeleanu, M. Topological indices and real number vertex invariants based on graph eigenvalues or eigenvectors, J. Chem. Inf. Comput. Sci. 31, 517-523, 1991.
  • Bapat R. B. Resistance distance in graphs, Math. Student 68, 87-98, 1999.
  • Bapat R. B. Resistance matrix of a weighted graph, MATCH Commun. Math. Comput. Chem. 50, 73-82, 2004.
  • Bapat R. B., Gutman, I. and Xiao, W. A simple method for computing resistance distance, Z. Naturforsch. 58a, 494-498, 2003.
  • Bendito, E., Carmona, A., Encinas, A. M, Gesto, J. M. and Mitjana, M Kirchhoff indexes of a network, Lin. Algebra Appl. 432, 2278-2292, 2010.
  • Bonchev, D., Balaban, A. T., Liu, X. and Klein, D. J. Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50, 1-20, 1994.
  • Cao, D., Chvatal, V, Hoffman, A, J. and Vince, A. Variations on a theorem of Ryser, Lin. Algebra Appl. 260, 215-222, 1997.
  • Chen, H. and Zhang, F. Resistance distance and the normalized Laplacian spectrum, Discr. Appl. Math. 155, 654-661, 2007.
  • Das, K. C., G¨ ung¨ or, A. D. and C ¸ evik, A. S., On the Kirchoff index and the resistence distance energy of a graph, MATCH Commun. Math. Comput. Chem. 67(2), 541-556, 2012. Dobrynin, A. A., Entringer, R. and Gutman, I. Wiener index of trees: theory and applications, Acta Appl. Math. 66, 211-249, 2001.
  • Entringer, R. C., Jackson, D. E. and Snyder, D. A. Distance in graphs, Czech. Math. J. 26, 283-296, 1976.
  • Guo, Q., Deng, H. and Chen, D. The extremal Kirchhoff index of a class of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61, 713-722, 2009.
  • Gutman, I. and Medeleanu, M. On the structure–dependence of the largest eigenvalue of the distance matrix of an alkane, Indian J. Chem. 37A, 569-573, 1998.
  • Gutman, I. and Xiao, W. Generalized inverse of the Laplacian matrix and some applications, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur. 129, 15-23, 2004.
  • Gutman, I. and Xiao, W. Distance in trees and Laplacian matrix, Int. J. Chem. Model. 2, 327-334,2010.
  • Ivanciuc, O., Ivanciuc, T. and Balaban, A. T. Quantitative structure–property relationship evaluation of structural descriptors derived from the distance and reverse Wiener matrices, Internet El. J. Mol. Des. 1, 467-487,2002.
  • Klein, D. J. Graph geometry, graph metrics, & Wiener, MATCH Commun. Math. Comput. Chem. 35, 7-27, 1997.
  • Klein, D. J. and Randi´ c, M. Resistance distance, J. Math. Chem. 12, 81-95, 1993.
  • Lukovits, I. Nikoli´ c, S. and Trinajsti´ c, N. Resistance distance in regular graphs, Int. J. Quantum Chem. 71, 217-225, 1999.
  • Nordhaus, E. A. and Gaddum, J. W. On complementary graphs, Am. Math. Montly 63, 175-177, 1956.
  • Walikar, H. B., Misale, D. N., Patil, R. L. and Ramane, H. S. On the resistance distance of a tree, El. Notes Discr. Math. 15, 244-245, 2003.
  • Xiao, W. and Gutman, I. On resistance matrices, MATCH Commun. Math. Comput. Chem. 49, 67-81, 2003.
  • Xiao, W. and Gutman, I. Resistance distance and Laplacian spectrum, Theor. Chem. Acc. 110, 284-289, 2003.
  • Xiao, W. and Gutman, I. Relations between resistance and Laplacian matrices and their applications, MATCH Commun. Math. Comput. Chem. 51, 119-127, 2004.
  • Zhang, H., Jiang, X. and Yang, Y. Bicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 61, 697-712, 2009.
  • Zhang, H. and Yang, Y. Resistance distance and Kirchhoff index in circulant graphs, Int. J. Quantum Chem. 107, 330-339, 2007.
  • Zhang, H., Yang, Y. and Li, C. Kirchhoff index of composite graphs, Discr. Appl. Math. 157, 2918-2927, 2009.
  • Zhang, W. and Deng, H. The second maximal and minimal Kirchhoff indices of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61, 683-695, 2009.
  • Zhou, B. On the spectral radius of nonnegative matrices, Australas. J. Comb. 22, 301-306, 2000.
  • Zhou, B. and Liu, B. On almost regular matrices, Util. Math. 54, 151-155, 1998.
  • Zhou, B. and Trinajsti´ c, N. On the largest eigenvalue of the distance matrix of a connected graph, Chem. Phys. Lett. 447, 384-387, 2007.
  • Zhou, B, and Trinajsti´ c, N. On resistance–distance and Kirchhoff index, J. Math. Chem. 46, 283-289, 2009.

Bounds for Resistance–Distance Spectral Radius

Year 2013, Volume: 42 Issue: 1, 43 - 50, 01.01.2013

A. Dilek Güngör Maden İvan Gutman A. Sinan Çevik

Abstract

Lower and upper bounds as well as Nordhauss-Gaddum-type results forthe resistance–distance spectral radius are obtained.

Keywords

Metric (in graph) Resistance Distance Resistance-Distance Spectral Radius

References

  • Balaban, A. T., Ciubotariu, D. and Medeleanu, M. Topological indices and real number vertex invariants based on graph eigenvalues or eigenvectors, J. Chem. Inf. Comput. Sci. 31, 517-523, 1991.
  • Bapat R. B. Resistance distance in graphs, Math. Student 68, 87-98, 1999.
  • Bapat R. B. Resistance matrix of a weighted graph, MATCH Commun. Math. Comput. Chem. 50, 73-82, 2004.
  • Bapat R. B., Gutman, I. and Xiao, W. A simple method for computing resistance distance, Z. Naturforsch. 58a, 494-498, 2003.
  • Bendito, E., Carmona, A., Encinas, A. M, Gesto, J. M. and Mitjana, M Kirchhoff indexes of a network, Lin. Algebra Appl. 432, 2278-2292, 2010.
  • Bonchev, D., Balaban, A. T., Liu, X. and Klein, D. J. Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50, 1-20, 1994.
  • Cao, D., Chvatal, V, Hoffman, A, J. and Vince, A. Variations on a theorem of Ryser, Lin. Algebra Appl. 260, 215-222, 1997.
  • Chen, H. and Zhang, F. Resistance distance and the normalized Laplacian spectrum, Discr. Appl. Math. 155, 654-661, 2007.
  • Das, K. C., G¨ ung¨ or, A. D. and C ¸ evik, A. S., On the Kirchoff index and the resistence distance energy of a graph, MATCH Commun. Math. Comput. Chem. 67(2), 541-556, 2012. Dobrynin, A. A., Entringer, R. and Gutman, I. Wiener index of trees: theory and applications, Acta Appl. Math. 66, 211-249, 2001.
  • Entringer, R. C., Jackson, D. E. and Snyder, D. A. Distance in graphs, Czech. Math. J. 26, 283-296, 1976.
  • Guo, Q., Deng, H. and Chen, D. The extremal Kirchhoff index of a class of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61, 713-722, 2009.
  • Gutman, I. and Medeleanu, M. On the structure–dependence of the largest eigenvalue of the distance matrix of an alkane, Indian J. Chem. 37A, 569-573, 1998.
  • Gutman, I. and Xiao, W. Generalized inverse of the Laplacian matrix and some applications, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur. 129, 15-23, 2004.
  • Gutman, I. and Xiao, W. Distance in trees and Laplacian matrix, Int. J. Chem. Model. 2, 327-334,2010.
  • Ivanciuc, O., Ivanciuc, T. and Balaban, A. T. Quantitative structure–property relationship evaluation of structural descriptors derived from the distance and reverse Wiener matrices, Internet El. J. Mol. Des. 1, 467-487,2002.
  • Klein, D. J. Graph geometry, graph metrics, & Wiener, MATCH Commun. Math. Comput. Chem. 35, 7-27, 1997.
  • Klein, D. J. and Randi´ c, M. Resistance distance, J. Math. Chem. 12, 81-95, 1993.
  • Lukovits, I. Nikoli´ c, S. and Trinajsti´ c, N. Resistance distance in regular graphs, Int. J. Quantum Chem. 71, 217-225, 1999.
  • Nordhaus, E. A. and Gaddum, J. W. On complementary graphs, Am. Math. Montly 63, 175-177, 1956.
  • Walikar, H. B., Misale, D. N., Patil, R. L. and Ramane, H. S. On the resistance distance of a tree, El. Notes Discr. Math. 15, 244-245, 2003.
  • Xiao, W. and Gutman, I. On resistance matrices, MATCH Commun. Math. Comput. Chem. 49, 67-81, 2003.
  • Xiao, W. and Gutman, I. Resistance distance and Laplacian spectrum, Theor. Chem. Acc. 110, 284-289, 2003.
  • Xiao, W. and Gutman, I. Relations between resistance and Laplacian matrices and their applications, MATCH Commun. Math. Comput. Chem. 51, 119-127, 2004.
  • Zhang, H., Jiang, X. and Yang, Y. Bicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 61, 697-712, 2009.
  • Zhang, H. and Yang, Y. Resistance distance and Kirchhoff index in circulant graphs, Int. J. Quantum Chem. 107, 330-339, 2007.
  • Zhang, H., Yang, Y. and Li, C. Kirchhoff index of composite graphs, Discr. Appl. Math. 157, 2918-2927, 2009.
  • Zhang, W. and Deng, H. The second maximal and minimal Kirchhoff indices of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61, 683-695, 2009.
  • Zhou, B. On the spectral radius of nonnegative matrices, Australas. J. Comb. 22, 301-306, 2000.
  • Zhou, B. and Liu, B. On almost regular matrices, Util. Math. 54, 151-155, 1998.
  • Zhou, B. and Trinajsti´ c, N. On the largest eigenvalue of the distance matrix of a connected graph, Chem. Phys. Lett. 447, 384-387, 2007.
  • Zhou, B, and Trinajsti´ c, N. On resistance–distance and Kirchhoff index, J. Math. Chem. 46, 283-289, 2009.
There are 31 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Mathematics
Authors

A. Dilek Güngör Maden This is me

İvan Gutman This is me

A. Sinan Çevik This is me

Publication Date January 1, 2013
Published in Issue Year 2013 Volume: 42 Issue: 1

Cite

APA Maden, A. D. G., Gutman, İ., & Çevik, A. S. (2013). Bounds for Resistance–Distance Spectral Radius. Hacettepe Journal of Mathematics and Statistics, 42(1), 43-50.
AMA Maden ADG, Gutman İ, Çevik AS. Bounds for Resistance–Distance Spectral Radius. Hacettepe Journal of Mathematics and Statistics. January 2013;42(1):43-50.
Chicago Maden, A. Dilek Güngör, İvan Gutman, and A. Sinan Çevik. “Bounds for Resistance–Distance Spectral Radius”. Hacettepe Journal of Mathematics and Statistics 42, no. 1 (January 2013): 43-50.
EndNote Maden ADG, Gutman İ, Çevik AS (January 1, 2013) Bounds for Resistance–Distance Spectral Radius. Hacettepe Journal of Mathematics and Statistics 42 1 43–50.
IEEE A. D. G. Maden, İ. Gutman, and A. S. Çevik, “Bounds for Resistance–Distance Spectral Radius”, Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, pp. 43–50, 2013.
ISNAD Maden, A. Dilek Güngör et al. “Bounds for Resistance–Distance Spectral Radius”. Hacettepe Journal of Mathematics and Statistics 42/1 (January 2013), 43-50.
JAMA Maden ADG, Gutman İ, Çevik AS. Bounds for Resistance–Distance Spectral Radius. Hacettepe Journal of Mathematics and Statistics. 2013;42:43–50.
MLA Maden, A. Dilek Güngör et al. “Bounds for Resistance–Distance Spectral Radius”. Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, 2013, pp. 43-50.
Vancouver Maden ADG, Gutman İ, Çevik AS. Bounds for Resistance–Distance Spectral Radius. Hacettepe Journal of Mathematics and Statistics. 2013;42(1):43-50.

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