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On Comparing Zagreb Indices of Graphs

Yıl 2012, Cilt: 41 Sayı: 2, 223 - 230, 01.02.2012

Öz

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. It is well-known that for connected or disconnected graphs, M2/m ≥ M1/n does not hold always. In K. C. Das (On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63, 433–440, 2010), it has been shown that the above relation holds for a special kind of graph. Here we continue our search for special kinds of graph for which the above relation holds.

Kaynakça

  • Aouchiche, M., Bonnefoy, J. M., Fidahoussen, A., Caporossi, G., Hansen, P. and Lacher´e, J. A. M. Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.) (Global Optimization: From Theory to Implementation, Springer, 2005).
  • Balaban, A. T., Motoc, I., Bonchev, D. and Mekenyan, O. Topological indices for structure- activity correlations, in: Steric Effects in Drug Design (M. Charton and I. Motoc, Eds.), Topics Curr. Chem. 114, 21–55, 1983.
  • Bonchev, D. and Trinajsti´c, N. Overall molecular descriptors 3. Overall Zagreb indices, Sar QSAR Environ. Res. 12, 213–236, 2001.
  • Caporossi, G. and Hansen, P. Variable neighborhood search for extremal graphs. 1. The AutoGraphiX system, Discrete Math. 212 (1-2), 29–44, 2000.
  • Caporossi, G. and Hansen, P. Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276, 81–94, 2004.
  • Das, K. C. Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285, –66, 2004.
  • Das, K. C. On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63, 433–440, 2010.
  • Das, K. C. and Gutman, I. Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52, 103–112, 2004.
  • Das, K. C., Gutman, I. and Horoldagva, B. Comparison between Zagreb indices and Zagreb coindices of trees, MATCH Commun. Math. Comput. Chem. 68, 189–198, 2012.
  • Das, K. C., Gutman, I. and Zhou, B. New upper bounds on Zagreb indices, J. Math. Chem. , 514–521, 2009.
  • Gutman, I. and Das, K. C. The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50, 83–92, 2004.
  • Gutman, I., Ruˇsˇci´c, B., Trinajsti´c, N. and Wilcox, C. F. Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62, 3399–3405, 1975.
  • Gutman, I. and Trinajstic, N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535–538, 1972.
  • Hansen, P. and Vukiˇcevi´c, D. Comparing the Zagreb indices, Croat. Chem. Acta 80, 165– , 2007.
  • Horoldagva, B. and Das, K. C. Comparing variable Zagreb indices for unicyclic graphs, MATCH Commun. Math. Comput. Chem. 62, 725–730, 2009.
  • Horoldagva, B. and Lee, S. -G. Comparing Zagreb indices for connected graphs, Discrete Appl. Math. 158, 1073–1078, 2010.
  • Khalifeh, M. H., Azari, H. Y-. and Ashrafi, A. R. The first and second Zagreb indices of some graph operations, Discrete Applied Math. 157, 804–811, 2009.
  • Mili´cevi´c, A., Nikoli´c, S. and Trinajsti´c, N. On reformulated Zagreb indices, Molecular Diversity 8, 393–399, 2004.
  • Nikoli´c, S., Kovaˇcevi´c, G., Mili´cevi´c, A. and Trinajsti´c, N. The Zagreb indices 30 years after, Croat. Chem. Acta 76, 113–124, 2003.
  • Nikoli´c, S., Mili´cevi´c, A., Trinajsti´c, N. and Juri´c, A. On the use of the variable Zagreb vM2 index in QSPR: Boiling points of benzenoid hydrocarbons, Molecules 9, 1208–1221, 2004.
  • Nikoli´c, S., Toli´c, I. M., Trinajsti´c, N. and Bauˇci´c, I. On the Zagreb indices as complexity indices, Croat. Chem. Acta 73, 909–921, 2000.
  • Sun, L. and Chen, T. Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees, Discrete Appl. Math. 157, 1650–1654, 2009.
  • Sun, L. and Wei, S. Comparing the Zagreb indices for connected bicyclic graphs, MATCH Commun. Math. Comput. Chem. 62, 699–714, 2009.
  • Todeschini, R. and Consonni, V. Handbook of Molecular Descriptors (Wiley–VCH, Wein- heim, 2000).
  • Vukiˇcevi´c, D. Talk on the meeting of International Academy of Mathematical Chemistry, Dubrovnik, 2007.
  • Vukiˇcevi´c, D. Comparing variable Zagreb indices, MATCH Commun. Math. Comput. Chem , 633–641, 2007.
  • Vukiˇcevi´c, D. and Graovac, A. Comparing Zagreb M1and M2indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57, 587–590, 2007.
  • Vukiˇcevi´c, D. and Graovac, A. Comparing variable Zagreb Mand M2indices for acyclic molecules, MATCH Commun. Math. Comput. Chem 60, 37–44, 2008.
  • Vukiˇcevi´c, D. and Graovac, A. Comparing Zagreb M1and M2indices: Overview of the results, personal communication. Vukiˇcevi´c, D. and Trinajsti´c, N. Modified Zagreb index M` u Comparison with the Randicc, N. Modified Zagreb index M` connectivity index for benzenoid systems, Croat. Chem. Acta 76, 183–187, 2003.
  • Vukiˇcevi´c, D. and Trinajsti´c, N. On the discriminatory power of the Zagreb indices for molecular graphs, MATCH Commun. Math. Comput. Chem 53, 111–138, 2005.
  • Yan, Z., Liu, H. and Liu, H. Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42, 565–574, 2007.
  • Zhang, L. and Wu, B. The Nordhaus-Goddum-Type inequalities for some chemical indices, MATCH Commun. Math. Comput. Chem. 54, 189–194, 2005.
  • Zhou, B. Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs, Int. J. Quantum Chem. 107, 875–878, 2007.
  • Zhou, B. and Gutman, I. Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54, 233–239, 2005.

On Comparing Zagreb Indices of Graphs

Yıl 2012, Cilt: 41 Sayı: 2, 223 - 230, 01.02.2012

Öz

Kaynakça

  • Aouchiche, M., Bonnefoy, J. M., Fidahoussen, A., Caporossi, G., Hansen, P. and Lacher´e, J. A. M. Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.) (Global Optimization: From Theory to Implementation, Springer, 2005).
  • Balaban, A. T., Motoc, I., Bonchev, D. and Mekenyan, O. Topological indices for structure- activity correlations, in: Steric Effects in Drug Design (M. Charton and I. Motoc, Eds.), Topics Curr. Chem. 114, 21–55, 1983.
  • Bonchev, D. and Trinajsti´c, N. Overall molecular descriptors 3. Overall Zagreb indices, Sar QSAR Environ. Res. 12, 213–236, 2001.
  • Caporossi, G. and Hansen, P. Variable neighborhood search for extremal graphs. 1. The AutoGraphiX system, Discrete Math. 212 (1-2), 29–44, 2000.
  • Caporossi, G. and Hansen, P. Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276, 81–94, 2004.
  • Das, K. C. Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285, –66, 2004.
  • Das, K. C. On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63, 433–440, 2010.
  • Das, K. C. and Gutman, I. Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52, 103–112, 2004.
  • Das, K. C., Gutman, I. and Horoldagva, B. Comparison between Zagreb indices and Zagreb coindices of trees, MATCH Commun. Math. Comput. Chem. 68, 189–198, 2012.
  • Das, K. C., Gutman, I. and Zhou, B. New upper bounds on Zagreb indices, J. Math. Chem. , 514–521, 2009.
  • Gutman, I. and Das, K. C. The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50, 83–92, 2004.
  • Gutman, I., Ruˇsˇci´c, B., Trinajsti´c, N. and Wilcox, C. F. Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62, 3399–3405, 1975.
  • Gutman, I. and Trinajstic, N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535–538, 1972.
  • Hansen, P. and Vukiˇcevi´c, D. Comparing the Zagreb indices, Croat. Chem. Acta 80, 165– , 2007.
  • Horoldagva, B. and Das, K. C. Comparing variable Zagreb indices for unicyclic graphs, MATCH Commun. Math. Comput. Chem. 62, 725–730, 2009.
  • Horoldagva, B. and Lee, S. -G. Comparing Zagreb indices for connected graphs, Discrete Appl. Math. 158, 1073–1078, 2010.
  • Khalifeh, M. H., Azari, H. Y-. and Ashrafi, A. R. The first and second Zagreb indices of some graph operations, Discrete Applied Math. 157, 804–811, 2009.
  • Mili´cevi´c, A., Nikoli´c, S. and Trinajsti´c, N. On reformulated Zagreb indices, Molecular Diversity 8, 393–399, 2004.
  • Nikoli´c, S., Kovaˇcevi´c, G., Mili´cevi´c, A. and Trinajsti´c, N. The Zagreb indices 30 years after, Croat. Chem. Acta 76, 113–124, 2003.
  • Nikoli´c, S., Mili´cevi´c, A., Trinajsti´c, N. and Juri´c, A. On the use of the variable Zagreb vM2 index in QSPR: Boiling points of benzenoid hydrocarbons, Molecules 9, 1208–1221, 2004.
  • Nikoli´c, S., Toli´c, I. M., Trinajsti´c, N. and Bauˇci´c, I. On the Zagreb indices as complexity indices, Croat. Chem. Acta 73, 909–921, 2000.
  • Sun, L. and Chen, T. Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees, Discrete Appl. Math. 157, 1650–1654, 2009.
  • Sun, L. and Wei, S. Comparing the Zagreb indices for connected bicyclic graphs, MATCH Commun. Math. Comput. Chem. 62, 699–714, 2009.
  • Todeschini, R. and Consonni, V. Handbook of Molecular Descriptors (Wiley–VCH, Wein- heim, 2000).
  • Vukiˇcevi´c, D. Talk on the meeting of International Academy of Mathematical Chemistry, Dubrovnik, 2007.
  • Vukiˇcevi´c, D. Comparing variable Zagreb indices, MATCH Commun. Math. Comput. Chem , 633–641, 2007.
  • Vukiˇcevi´c, D. and Graovac, A. Comparing Zagreb M1and M2indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57, 587–590, 2007.
  • Vukiˇcevi´c, D. and Graovac, A. Comparing variable Zagreb Mand M2indices for acyclic molecules, MATCH Commun. Math. Comput. Chem 60, 37–44, 2008.
  • Vukiˇcevi´c, D. and Graovac, A. Comparing Zagreb M1and M2indices: Overview of the results, personal communication. Vukiˇcevi´c, D. and Trinajsti´c, N. Modified Zagreb index M` u Comparison with the Randicc, N. Modified Zagreb index M` connectivity index for benzenoid systems, Croat. Chem. Acta 76, 183–187, 2003.
  • Vukiˇcevi´c, D. and Trinajsti´c, N. On the discriminatory power of the Zagreb indices for molecular graphs, MATCH Commun. Math. Comput. Chem 53, 111–138, 2005.
  • Yan, Z., Liu, H. and Liu, H. Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42, 565–574, 2007.
  • Zhang, L. and Wu, B. The Nordhaus-Goddum-Type inequalities for some chemical indices, MATCH Commun. Math. Comput. Chem. 54, 189–194, 2005.
  • Zhou, B. Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs, Int. J. Quantum Chem. 107, 875–878, 2007.
  • Zhou, B. and Gutman, I. Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54, 233–239, 2005.
Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular İstatistik
Bölüm Matematik
Yazarlar

Batmend Horoldagva Bu kişi benim

Kinkar Ch. Das Bu kişi benim

Yayımlanma Tarihi 1 Şubat 2012
Yayımlandığı Sayı Yıl 2012 Cilt: 41 Sayı: 2

Kaynak Göster

APA Horoldagva, B., & Das, K. C. (2012). On Comparing Zagreb Indices of Graphs. Hacettepe Journal of Mathematics and Statistics, 41(2), 223-230.
AMA Horoldagva B, Das KC. On Comparing Zagreb Indices of Graphs. Hacettepe Journal of Mathematics and Statistics. Şubat 2012;41(2):223-230.
Chicago Horoldagva, Batmend, ve Kinkar Ch. Das. “On Comparing Zagreb Indices of Graphs”. Hacettepe Journal of Mathematics and Statistics 41, sy. 2 (Şubat 2012): 223-30.
EndNote Horoldagva B, Das KC (01 Şubat 2012) On Comparing Zagreb Indices of Graphs. Hacettepe Journal of Mathematics and Statistics 41 2 223–230.
IEEE B. Horoldagva ve K. C. Das, “On Comparing Zagreb Indices of Graphs”, Hacettepe Journal of Mathematics and Statistics, c. 41, sy. 2, ss. 223–230, 2012.
ISNAD Horoldagva, Batmend - Das, Kinkar Ch. “On Comparing Zagreb Indices of Graphs”. Hacettepe Journal of Mathematics and Statistics 41/2 (Şubat 2012), 223-230.
JAMA Horoldagva B, Das KC. On Comparing Zagreb Indices of Graphs. Hacettepe Journal of Mathematics and Statistics. 2012;41:223–230.
MLA Horoldagva, Batmend ve Kinkar Ch. Das. “On Comparing Zagreb Indices of Graphs”. Hacettepe Journal of Mathematics and Statistics, c. 41, sy. 2, 2012, ss. 223-30.
Vancouver Horoldagva B, Das KC. On Comparing Zagreb Indices of Graphs. Hacettepe Journal of Mathematics and Statistics. 2012;41(2):223-30.