Research Article
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Year 2020, Volume: 49 Issue: 1, 87 - 95, 06.02.2020
https://doi.org/10.15672/HJMS.2019.664

Abstract

References

  • [1] A. Abiad, B. Brimkov, A. Erey, L. Leshock, X. Martínez-Rivera, S. O, S.-Y. Song and J. Williford, On the Wiener index, distance cospectrality and transmission-regular graphs, Discrete Appl. Math. 230, 1–10, 2017.
  • [2] Y. Alizadeh, Szeged dimension and $PI_v$ dimension of composite graphs, Iranian J. Math. Sci. Inform. 13, 45–57, 2018.
  • [3] Y. Alizadeh, V. Andova, S. Klavžar and R. Škrekovski, Wiener dimension: fundamen- tal properties and (5,0)-nanotubical fullerenes, MATCH Commun. Math. Comput. Chem. 72, 279–294, 2014.
  • [4] Y. Alizadeh and S. Klavžar, Complexity of topological indices: The case of connective eccentric index, MATCH Commun. Math. Comput. Chem. 76, 659–667, 2016.
  • [5] Y. Alizadeh and S. Klavžar, On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs, Appl. Math. Comput. 328, 113–118, 2018.
  • [6] V. Andova, T. Došlić, M. Krnc, B. Lužar and R. Škrekovski, On the diameter and some related invariants of fullerene graphs, MATCH Commun. Math. Comput. Chem. 68, 109–130, 2012.
  • [7] M. Arockiaraj, J. Clement and K. Balasubramanian, Analytical expressions for topo- logical properties of polycyclic benzenoid networks, J. Chemometrics, 30, 682–697, 2016.
  • [8] M. Arockiaraj, J. Clement and A.J. Shalini, Variants of the Szeged index in certain chemical nanosheets, Canad. J. Chem. 94, 608–619, 2016.
  • [9] D. Buset, Orbits on vertices and edges of finite graphs, Discrete Math. 57, 297–299, 1985.
  • [10] K.Ch. Das and M.J. Nadjafi-Arani, Comparison between the Szeged index and the eccentric connectivity index, Discrete Appl. Math. 186, 74–86, 2015.
  • [11] A.A. Dobrynin and I. Gutman, Solving a problem connected with distances in graphs, Graph Theory Notes N. Y. 28, 21–23, 1995.
  • [12] R.C. Entringer, D.E. Jackson and D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26, 283–296, 1976.
  • [13] M. Goubko and O. Miloserdov, Simple alcohols with the lowest normal boiling point using topological indices, MATCH Commun. Math. Comput. Chem. 75, 29–56, 2016.
  • [14] S. Gupta, M. Singh and A.K. Madan, Connective eccentricity index: a novel topo- logical descriptor for predicting biological activity, J. Mol. Graph. Model. 18, 18–25, 2000.
  • [15] I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes N. Y. 27, 9–15, 1994.
  • [16] I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors—Theory and Applications I, Univ. Kragujevac, Kragujevac, 2010.
  • [17] I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors—Theory and Applications II, Univ. Kragujevac, Kragujevac, 2010.
  • [18] I. Gutman, K. Xu and M. Liu, A congruence relation for Wiener and Szeged indices, Filomat, 29, 1081–1083, 2015.
  • [19] M. Imran, A.Q Baig and H. Ali, On molecular topological properties of hex-derived networks, J. Chemometrics, 30, 121–129, 2016.
  • [20] P.V. Khadikar, S. Karmarkar, V.K. Agrawal, J. Singh, A. Shrivastava, I. Lukovits and M.V. Diudea, Szeged index - applications for drug modeling, Lett. Drug Des. Discov. 2, 606–624, 2005.
  • [21] S. Klavžar, D. Azubha Jemilet, I. Rajasingh, P. Manuel and N. Parthiban, Gen- eral Transmission Lemma and Wiener complexity of triangular grids, Appl. Math. Comput. 338, 115–122, 2018.
  • [22] S. Klavžar and M.J. Nadjafi-Arani, Wiener index versus Szeged index in networks, Discrete Appl. Math. 161, 1150–1153, 2013.
  • [23] S. Klavžar and M.J. Nadjafi-Arani, Improved bounds on the difference between the Szeged index and the Wiener index of graphs, European J. Combin. 39, 148–156, 2014.
  • [24] S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9, 45-49, 1996.
  • [25] S. Li and H. Zhang, Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond, Discrete Math. 340, 311–324, 2017.
  • [26] M.J. Nadjafi-Arani, H. Khodashenas and A.R. Ashrafi, On the differences between Szeged and Wiener indices of graphs, Discrete Math. 311, 2233–2237, 2011.
  • [27] M.J. Nadjafi-Arani, H. Khodashenas and A.R. Ashrafi, Graphs whose Szeged and Wiener numbers differ by 4 and 5, Math. Comput. Modelling, 55, 1644–1648, 2012.
  • [28] F. Shafiei, M. Pashm Froush and F. Dialamehpour, QSPR study on benzene deriva- tives to some physicochemical properties by using topological indices, Iranian J. Math. Chem. 7, 93–110, 2016.
  • [29] S. Simić, I. Gutman and V. Baltić, Some graphs with extremal Szeged index, Math. Slovaca, 50, 1–15, 2000.
  • [30] H. Smith, L. Székely and H. Wang, Eccentricity sums in trees, Discrete Appl. Math. 207, 120–131, 2016.
  • [31] R. Todeschini and V. Consonni, Molecular Decriptors for Chemoinformatic, Wiley- VCH, Weinheim, 2009.
  • [32] K. Xu, K.Ch. Das and H. Liu, Some extremal results on the connective eccentricity index of graphs, J. Math. Anal. Appl. 433, 803–817, 2016.
  • [33] H. Zhang, S. Li and L. Zhao, On the further relation between the (revised) Szeged index and the Wiener index of graphs, Discrete Appl. Math. 206, 152–164, 2016.

Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes

Year 2020, Volume: 49 Issue: 1, 87 - 95, 06.02.2020
https://doi.org/10.15672/HJMS.2019.664

Abstract

Let $I$ be a summation-type topological index. The $I$-complexity $C_I(G)$ of a graph $G$ is the number of different contributions to $I(G)$ in its summation formula. In this paper the complexity $C_{Sz}(G)$ is investigated, where Sz is the well-studied Szeged index. Let $O_e(G)$ (resp. $O_v(G)$) be the number of edge (resp. vertex) orbits of $G$. While $C_{Sz}(G) \leq O_e(G)$ holds for any graph $G$, it is shown that for any $m\geq 1$ there exists a vertex-transitive graph $G_m$ with $C_{Sz}(G_m) = O_e(G_m) = m$. Also, for any $1\leq k\leq m+1$ there exists a graph $G_{m,k}$ with $C_{Sz}(G_{m,k}) = O_e(G_{m,k}) = m$ and $C_{W}(G_{m,k}) = O_v(G_{m,k}) = k$. The Sz-complexity is determined for a family of (5,0)-nanotubical fullerenes and the Szeged index is compared with the total eccentricity.

References

  • [1] A. Abiad, B. Brimkov, A. Erey, L. Leshock, X. Martínez-Rivera, S. O, S.-Y. Song and J. Williford, On the Wiener index, distance cospectrality and transmission-regular graphs, Discrete Appl. Math. 230, 1–10, 2017.
  • [2] Y. Alizadeh, Szeged dimension and $PI_v$ dimension of composite graphs, Iranian J. Math. Sci. Inform. 13, 45–57, 2018.
  • [3] Y. Alizadeh, V. Andova, S. Klavžar and R. Škrekovski, Wiener dimension: fundamen- tal properties and (5,0)-nanotubical fullerenes, MATCH Commun. Math. Comput. Chem. 72, 279–294, 2014.
  • [4] Y. Alizadeh and S. Klavžar, Complexity of topological indices: The case of connective eccentric index, MATCH Commun. Math. Comput. Chem. 76, 659–667, 2016.
  • [5] Y. Alizadeh and S. Klavžar, On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs, Appl. Math. Comput. 328, 113–118, 2018.
  • [6] V. Andova, T. Došlić, M. Krnc, B. Lužar and R. Škrekovski, On the diameter and some related invariants of fullerene graphs, MATCH Commun. Math. Comput. Chem. 68, 109–130, 2012.
  • [7] M. Arockiaraj, J. Clement and K. Balasubramanian, Analytical expressions for topo- logical properties of polycyclic benzenoid networks, J. Chemometrics, 30, 682–697, 2016.
  • [8] M. Arockiaraj, J. Clement and A.J. Shalini, Variants of the Szeged index in certain chemical nanosheets, Canad. J. Chem. 94, 608–619, 2016.
  • [9] D. Buset, Orbits on vertices and edges of finite graphs, Discrete Math. 57, 297–299, 1985.
  • [10] K.Ch. Das and M.J. Nadjafi-Arani, Comparison between the Szeged index and the eccentric connectivity index, Discrete Appl. Math. 186, 74–86, 2015.
  • [11] A.A. Dobrynin and I. Gutman, Solving a problem connected with distances in graphs, Graph Theory Notes N. Y. 28, 21–23, 1995.
  • [12] R.C. Entringer, D.E. Jackson and D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26, 283–296, 1976.
  • [13] M. Goubko and O. Miloserdov, Simple alcohols with the lowest normal boiling point using topological indices, MATCH Commun. Math. Comput. Chem. 75, 29–56, 2016.
  • [14] S. Gupta, M. Singh and A.K. Madan, Connective eccentricity index: a novel topo- logical descriptor for predicting biological activity, J. Mol. Graph. Model. 18, 18–25, 2000.
  • [15] I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes N. Y. 27, 9–15, 1994.
  • [16] I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors—Theory and Applications I, Univ. Kragujevac, Kragujevac, 2010.
  • [17] I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors—Theory and Applications II, Univ. Kragujevac, Kragujevac, 2010.
  • [18] I. Gutman, K. Xu and M. Liu, A congruence relation for Wiener and Szeged indices, Filomat, 29, 1081–1083, 2015.
  • [19] M. Imran, A.Q Baig and H. Ali, On molecular topological properties of hex-derived networks, J. Chemometrics, 30, 121–129, 2016.
  • [20] P.V. Khadikar, S. Karmarkar, V.K. Agrawal, J. Singh, A. Shrivastava, I. Lukovits and M.V. Diudea, Szeged index - applications for drug modeling, Lett. Drug Des. Discov. 2, 606–624, 2005.
  • [21] S. Klavžar, D. Azubha Jemilet, I. Rajasingh, P. Manuel and N. Parthiban, Gen- eral Transmission Lemma and Wiener complexity of triangular grids, Appl. Math. Comput. 338, 115–122, 2018.
  • [22] S. Klavžar and M.J. Nadjafi-Arani, Wiener index versus Szeged index in networks, Discrete Appl. Math. 161, 1150–1153, 2013.
  • [23] S. Klavžar and M.J. Nadjafi-Arani, Improved bounds on the difference between the Szeged index and the Wiener index of graphs, European J. Combin. 39, 148–156, 2014.
  • [24] S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9, 45-49, 1996.
  • [25] S. Li and H. Zhang, Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond, Discrete Math. 340, 311–324, 2017.
  • [26] M.J. Nadjafi-Arani, H. Khodashenas and A.R. Ashrafi, On the differences between Szeged and Wiener indices of graphs, Discrete Math. 311, 2233–2237, 2011.
  • [27] M.J. Nadjafi-Arani, H. Khodashenas and A.R. Ashrafi, Graphs whose Szeged and Wiener numbers differ by 4 and 5, Math. Comput. Modelling, 55, 1644–1648, 2012.
  • [28] F. Shafiei, M. Pashm Froush and F. Dialamehpour, QSPR study on benzene deriva- tives to some physicochemical properties by using topological indices, Iranian J. Math. Chem. 7, 93–110, 2016.
  • [29] S. Simić, I. Gutman and V. Baltić, Some graphs with extremal Szeged index, Math. Slovaca, 50, 1–15, 2000.
  • [30] H. Smith, L. Székely and H. Wang, Eccentricity sums in trees, Discrete Appl. Math. 207, 120–131, 2016.
  • [31] R. Todeschini and V. Consonni, Molecular Decriptors for Chemoinformatic, Wiley- VCH, Weinheim, 2009.
  • [32] K. Xu, K.Ch. Das and H. Liu, Some extremal results on the connective eccentricity index of graphs, J. Math. Anal. Appl. 433, 803–817, 2016.
  • [33] H. Zhang, S. Li and L. Zhao, On the further relation between the (revised) Szeged index and the Wiener index of graphs, Discrete Appl. Math. 206, 152–164, 2016.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Yaser Alizadeh This is me 0000-0002-8533-0425

Sandi Klavzar 0000-0002-1556-4744

Publication Date February 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 1

Cite

APA Alizadeh, Y., & Klavzar, S. (2020). Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes. Hacettepe Journal of Mathematics and Statistics, 49(1), 87-95. https://doi.org/10.15672/HJMS.2019.664
AMA Alizadeh Y, Klavzar S. Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes. Hacettepe Journal of Mathematics and Statistics. February 2020;49(1):87-95. doi:10.15672/HJMS.2019.664
Chicago Alizadeh, Yaser, and Sandi Klavzar. “Complexity of the Szeged Index, Edge Orbits, and Some Nanotubical Fullerenes”. Hacettepe Journal of Mathematics and Statistics 49, no. 1 (February 2020): 87-95. https://doi.org/10.15672/HJMS.2019.664.
EndNote Alizadeh Y, Klavzar S (February 1, 2020) Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes. Hacettepe Journal of Mathematics and Statistics 49 1 87–95.
IEEE Y. Alizadeh and S. Klavzar, “Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, pp. 87–95, 2020, doi: 10.15672/HJMS.2019.664.
ISNAD Alizadeh, Yaser - Klavzar, Sandi. “Complexity of the Szeged Index, Edge Orbits, and Some Nanotubical Fullerenes”. Hacettepe Journal of Mathematics and Statistics 49/1 (February 2020), 87-95. https://doi.org/10.15672/HJMS.2019.664.
JAMA Alizadeh Y, Klavzar S. Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes. Hacettepe Journal of Mathematics and Statistics. 2020;49:87–95.
MLA Alizadeh, Yaser and Sandi Klavzar. “Complexity of the Szeged Index, Edge Orbits, and Some Nanotubical Fullerenes”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, 2020, pp. 87-95, doi:10.15672/HJMS.2019.664.
Vancouver Alizadeh Y, Klavzar S. Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):87-95.