Some Results on Forgotten Topological Coindex

Document Type : Research Paper


1 Kazerun Branch, Islamic Azad University

2 Safadasht Branch, Islamic Azad University


The forgotten topological coindex (also called Lanzhou index) is defined for a simple connected graph G as the sum of the terms du2+dv2 over all non-adjacent vertex pairs uv of G, where du denotes the degree of the vertex u in G. In this paper, we present some inequalities for the forgotten topological coindex in terms of some graph parameters such as the order, size, number of pendent vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree. We also study the relation between this invariant and some well-known graph invariants such as the Zagreb indices and coindices, multiplicative Zagreb indices and coindices, Zagreb eccentricity indices, eccentric connectivity index and coindex, and total eccentricity. Exact formulae for computing the forgotten topological coindex of double graphs and extended double cover of a given graph are also proposed.



    1. M. V. Diudea, QSPR/QSAR Studies by Molecular Descriptors, NOVA, New York, 2001.
    2. V. Sharma, R. Goswami and A. K. Madan, Eccentric connectivity index: A novel highly discriminating topological descriptor for structure–property and structure–activity studies, J. Chem. Inf. Comput. Sci. 37 (1997) 273–282.
    3. M. Azari, Further results on non-self-centrality measures of graphs, Filomat 32 (14) (2018) 5137–5148.
    4. H. Hua and Z. Miao, The total eccentricity sum of non-adjacent vertex pairs in graphs, Bull. Malays. Math. Sci. Soc. 42 (3) (2019) 947–963.
    5. D. Vukičević and A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov. 57 (2010) 524–538.
    6. K. Xu, K. C. Das and A. D. Maden, On a novel eccentricity-based invariant of a graph, Acta Math. Sin. (Engl. Ser.)32 (1) (2016) 1477–1493.
    7. I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
    8. I. Gutman, B. Ruščić, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 3399–3405.
    9. A. Ali, I. Gutman, E. Milovanović and I. Milovanović, Sum of powers of the degrees of graphs: Extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018) 5–84.
    10. M. Azari and A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Commun. Math. Comput. Chem. 70 (2013) 901–919.
    11. M. Azari, A. Iranmanesh and I. Gutman, Zagreb indices of bridge and chain graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 921–938.
    12. T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (1) (2008) 66–80.
    13. G. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 65 (2011) 79–84.
    14. M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997) 219–225.
    15. M. Veylaki and M. J. Nikmehr, The third hyper–Zagreb coindices of some graph operations, J. Appl. Math. Comput. 50 (2016) 315–325.
    16. R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010) 359–372.
    17. I. Gutman, Multiplicative Zagreb indices of trees, Bull. Int. Math. Virt. Inst. 1 (2011) 13–19.
    18. K. Xu, K. C. Das and K. Tang, On the multiplicative Zagreb coindex of graphs, Opuscula Math. 33 (1) (2013) 191–204.
    19. G. H. Shirdel, H. Rezapour and A. M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4 (2) (2013) 213–220.
    20. I. Gutman, On hyper-Zagreb index and coindex, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur. 150 (2017) 1–8.
    21. B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) 1184–1190.
    22. N. De, S. M. A. Nayeem and A. Pal, The F-coindex of some graph operations, Springer Plus (2016) 5:221.
    23. D. Vukičević, Q. Li, J. Sedlar and T. Došlić, Lanzhou index, MATCH Commun. Math. Comput. Chem. 80 (2018) 863–876.
    24. Z. Che and Z. Chen, Lower and upper bounds of the forgotten topological index, MATCH Commun. Math. Comput. Chem. 76 (2016) 635–648.
    25. N. De, F-index and coindex of some derived graphs, Bull. Int. Math. Virt. Inst. 8 (2018) 81–88.
    26. I. Gutman, A. Ghalavand, T. Dehghan-Zadeh and A. R. Ashrafi, Graphs with smallest forgotten index, Iranian J. Math. Chem. 8 (2017) 259–273.
    27. A. Yousefi, A. Iranmanesh, A. A. Dobrynin and A. Tehranian, The F–index for some special graphs and some properties of the F–index, Iranian J. Math. Chem. 9 (3) (2018) 213–225.
    28. M. Azari, Sharp lower bounds on the Narumi-Katayama index of graph operations, Appl. Math. Comput. 239 (2014) 409–421.
    29. F. Falahati-Nezhad and M. Azari, Bounds on the hyper-Zagreb index, J. Appl. Math. Inform. 34 (3−4) (2016) 319–330.
    30. F. Falahati-Nezhad, M. Azari and T. Došlić, Sharp bounds on the inverse sum indeg index, Discrete Appl. Math. 217 (2017) 185–195.
    31. A. R. Ashrafi, T. Došlić and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010) 1571–1578.
    32. M. Azari, On eccentricity version of Zagreb coindices, submitted.
    33. A. Ilić, G. Yu and L. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl. 381 (2011) 590–600.