Main Article Content
Abstract
In the discipline of graph theory, topological indices are extremely important. The M-polynomial is a powerful tool for determining a graph's topological indices. The use of M-polynomials to describe macro-molecules and biochemical networking is a novel concept. Also, the M-polynomial of various micro-structural allows us to calculate a variety of topological indices. The chemical substances and biochemical networks are correlated with their chemical characteristics and bio-active compounds using these findings. In this research, we use the M-polynomial to create special essential topological indices of inverse graphs on finite cyclic groups, such as Randic, Zagreb, Augmented Zagreb, Harmonic, Inverse sum, and Symmetric division degree indices.
Keywords
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
References
- D. Vukicevic, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, Journal of Mathematical Chemistry, 46(4), (2009), 1369-1376.
- E. Deutsch S. Klavzar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry, 6(2), (2015), 93-102.
- H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1), (1947), 17-20.
- G. Liu, Z. Jia, and W. Gao, Ontology similarity computing based on stochastic primal dual coordinate technique, Open journal of mathematical sciences, 2(1), (2018), 221-227.
- L. Yan, M. R. Farahani, and W. Gao, Distance-based Indices computation of symmetry molecular structures, Open journal of mathematical sciences, 2(1), (2018), 323-337.
- M. Randic, Characterization of molecuar branching, Joumal of the American Chemicalsociety, 97(23), (1975), 6609-6615.
- B. Bollobas and P. Erdos, Graphs of extremal weights, Ars Combinatoria, 50, (1998), 225–233.
- D. Amic, D. Beslo, B. Lucic, S. Nikolic, and N. Trinajstic, The vertex-connectivity index revisited, Journal of Chemical Information and Computer Sciences, 38(5), (1998), 819–822.
- G. Caporossi, I. Gutman, P. Hansen, and L. Pavlovic, Graphs with maximum connectivity index, Computational Biology and Chemistry, 27(1), (2003), 85–90.
- Y. Hu, X. Li, Y. Shi, T. Xu, and I. Gutman, On molecular graphs with smallest and greatest zeroth-order general Randic index, Match-Communications in Mathematical and in Computer Chemistry, 54, (2005), 425–434.
- I. Gutman, M. Randic, and X. Li, Mathematical aspects of Randic type molecular structure descriptors, Mathematical Chemistry Monographs, No. 1, University of Kragujevac, Kragujevac, Serbia, 2006.
- M. Randic, On history of the Randic index and emerging hostility toward chemical graph theory, Match-Communications in Mathematical and in Computer Chemistry, 59, (2008), 5–124.
- H. M. U. Rehman, R. Sardar, and A. Raza, Computing topological indices of Hex Board and its line graph, Open journal of mathematical sciences, 1(1), (2017), 62-71.
- I. Gutman and K. C. Das, The first Zagreb indices 30 years after, MATCH communications in mathematical and in computer chemistry, 50, (2004), 83-92.
- N. De, Hyper Zagreb index of bridge and Chain graphs, Open journal of mathematical sciences, (2018), 1-17.
- J.B. Liu, C. Wang, S. Wang, and B. Wei, Zagreb indices and multiplicative Zagreb indices of Eulerian graphs, Bulletin of the Malaysian Mathematical Sciences Society, 42(1), (2017), 67-78.
- S. Noreen and A. Mahmood, Zagreb polynomials and redefined Zagreb indices for the line graph of carbon nanocones, Open journal ofmathematical analysis, 2(1), (2018), 66-73.
- M. S. Sardar, S. Zafar, and M. R. Farahani, The generalized Zagreb index of Capra-designed planar benzenoid series Cak (C6), Open journal of mathematical sciences, 1, (1), (2017), 44-51.
- H. Siddiqui and M. R. Farahani, Forgotten polynomial and forgotten index of certain interconnection networks, Open journal of mathematical analysis, 1, (1), (2017), 44-59.
- C. K. Gupta, V. Lokesha, B. S. Shetty, and P. S. Ranjini, On the symmetric division deg index of graph, Southeast Asian Bulletin of Mathmatics, 41(1), (2016), 1-23.
- V. Lokesha and T. Deepika, Symmetric division deg index of tricyclic and tetracyclic graphs, International Journal of Scientific and Engineering Research, 7(5), (2016), 53-55.
- 0. Favaron, M. Maheo, and J. F. Sacle, Some eigenvalue properties in graphs (conjectures of Graffiti - II), Discrete mathematics, 111(1-3), (1993), 197-220.
- K. Pattabiraman, Inverse sum indeg index of graphs, AKCE International Journal of Graphs and Combinatorics, 15, (2), (2018), 155-167.
- E. Estrada, L. Torres, L. Rodriguez, and I. Gutman, An atombond connectivity index: modelling the enthalpy of formation of alkanes, Indian Journal of Chemistry, 37A(10), (1998), 849-855.
- B. Furtula, A. Graovac, and D. Vukicevic, Augmented Zagreb index, Journal of Mathematical Chemistry, 48(2), (2010), 370-380.
- Y. Huang, B. Liu, and L. Gan, Augmented Zagreb index of connected graphs, Match-Communications in Mathematical and in Computer Chemistry, 67, (2012), 483-494.
- G. Caporossi, I. Gutman, P. Hansen, and L. Pavlovic, Graphs with maximum connectivity index, Computational biology and chemistry, 27(1), (2003), 85-90.
- E. Deutsch and S. Klavzar, M-Polynomial, and degree-based topological indices, Iranian journal of mathematical chemistry, 6(2), (2015), 93-102.
- J. B. Liu and X. F. Pan, Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, Applied mathematics and computation, 291, (2016), 84-88.
- M. Munir, W. Nazeer, Z.Shahzadi, and S. Kang, Some invariants of circulant graphs, Symmetry, 8(11), (2016), Art ID.134.
- M. Riaz, W. Gao, and A. Q. Baig, M-Polynomials and degree-based topological indices of some families of convex polytopes, Open journal of mathematical sciences, 2(1), (2018), 18-28.
- M. Ghorbani and N. Azimi, Notes on mulyiple Zagreb indices, Iranian journal of mathematical chemistry, 3(2), (2012), 137-143.
- M. Munir, W. Nazeer, S. Rafique, and S. Kang, M-Polynomial and degree-based topological indices of polyhex nanotubes, Symmetry, 8(12), (2016), Art ID. 149.
- S. M. Kang, W. Nazeer, W. Gao, D. Afzal, S.N. Gillani, M-polynomials and topological indices of dominating David derived networks, Open Chemistry, 16(1), (2018), 201-213.
- G. Kalaimurugan, K. Mageshwaran, Zk -Magic Labeling On Inverse Graphs from Finite Cyclic Group, American International Journal of Research in Science, Technology, Engineering and Mathematics, 23(1), (2018), 199-201.
- K. Mageshwaran, G. Kalaimurugan, B. Hammachukiattikul, V. Govindan, I.N. Cangul, On L(h, k)-Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups, Journal of Mathematics, 2021, (2021), 1-7.
- M. R. Alfuraidan, Y. F. Zakariyai, Inverse graphs associated with finite groups, Electronic Journal of Graph Theory and Applications, 5(1), (2017), 142-154.
- ¨Ozt¨urk S¨ozen, E., Alsuraiheed, T., Abdio˘glu, C., Ali, S., Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings, Symmetry, 15(12), 2023.
- S¨ozen, E. ¨O., Erya¸sar, E., Abdiolu, C., Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn, Current Organic Synthesis, 21(3), (2024), 239-245.
- K. Mageshwaran, Nazeek Alessa, S. Gopinath, and K. Loganathan, Topological Indices of Graphs from Vector Spaces, Mathematics, 11(2), (2023).
- S. Gopinath., A.R.P. Doss, G. Kalaimurugan, Topological Indices for Inverse Graphs Associated With Finite Cyclic Group, Communications in Mathematics and Applications, 14(1), (2023).
References
D. Vukicevic, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, Journal of Mathematical Chemistry, 46(4), (2009), 1369-1376.
E. Deutsch S. Klavzar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry, 6(2), (2015), 93-102.
H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1), (1947), 17-20.
G. Liu, Z. Jia, and W. Gao, Ontology similarity computing based on stochastic primal dual coordinate technique, Open journal of mathematical sciences, 2(1), (2018), 221-227.
L. Yan, M. R. Farahani, and W. Gao, Distance-based Indices computation of symmetry molecular structures, Open journal of mathematical sciences, 2(1), (2018), 323-337.
M. Randic, Characterization of molecuar branching, Joumal of the American Chemicalsociety, 97(23), (1975), 6609-6615.
B. Bollobas and P. Erdos, Graphs of extremal weights, Ars Combinatoria, 50, (1998), 225–233.
D. Amic, D. Beslo, B. Lucic, S. Nikolic, and N. Trinajstic, The vertex-connectivity index revisited, Journal of Chemical Information and Computer Sciences, 38(5), (1998), 819–822.
G. Caporossi, I. Gutman, P. Hansen, and L. Pavlovic, Graphs with maximum connectivity index, Computational Biology and Chemistry, 27(1), (2003), 85–90.
Y. Hu, X. Li, Y. Shi, T. Xu, and I. Gutman, On molecular graphs with smallest and greatest zeroth-order general Randic index, Match-Communications in Mathematical and in Computer Chemistry, 54, (2005), 425–434.
I. Gutman, M. Randic, and X. Li, Mathematical aspects of Randic type molecular structure descriptors, Mathematical Chemistry Monographs, No. 1, University of Kragujevac, Kragujevac, Serbia, 2006.
M. Randic, On history of the Randic index and emerging hostility toward chemical graph theory, Match-Communications in Mathematical and in Computer Chemistry, 59, (2008), 5–124.
H. M. U. Rehman, R. Sardar, and A. Raza, Computing topological indices of Hex Board and its line graph, Open journal of mathematical sciences, 1(1), (2017), 62-71.
I. Gutman and K. C. Das, The first Zagreb indices 30 years after, MATCH communications in mathematical and in computer chemistry, 50, (2004), 83-92.
N. De, Hyper Zagreb index of bridge and Chain graphs, Open journal of mathematical sciences, (2018), 1-17.
J.B. Liu, C. Wang, S. Wang, and B. Wei, Zagreb indices and multiplicative Zagreb indices of Eulerian graphs, Bulletin of the Malaysian Mathematical Sciences Society, 42(1), (2017), 67-78.
S. Noreen and A. Mahmood, Zagreb polynomials and redefined Zagreb indices for the line graph of carbon nanocones, Open journal ofmathematical analysis, 2(1), (2018), 66-73.
M. S. Sardar, S. Zafar, and M. R. Farahani, The generalized Zagreb index of Capra-designed planar benzenoid series Cak (C6), Open journal of mathematical sciences, 1, (1), (2017), 44-51.
H. Siddiqui and M. R. Farahani, Forgotten polynomial and forgotten index of certain interconnection networks, Open journal of mathematical analysis, 1, (1), (2017), 44-59.
C. K. Gupta, V. Lokesha, B. S. Shetty, and P. S. Ranjini, On the symmetric division deg index of graph, Southeast Asian Bulletin of Mathmatics, 41(1), (2016), 1-23.
V. Lokesha and T. Deepika, Symmetric division deg index of tricyclic and tetracyclic graphs, International Journal of Scientific and Engineering Research, 7(5), (2016), 53-55.
0. Favaron, M. Maheo, and J. F. Sacle, Some eigenvalue properties in graphs (conjectures of Graffiti - II), Discrete mathematics, 111(1-3), (1993), 197-220.
K. Pattabiraman, Inverse sum indeg index of graphs, AKCE International Journal of Graphs and Combinatorics, 15, (2), (2018), 155-167.
E. Estrada, L. Torres, L. Rodriguez, and I. Gutman, An atombond connectivity index: modelling the enthalpy of formation of alkanes, Indian Journal of Chemistry, 37A(10), (1998), 849-855.
B. Furtula, A. Graovac, and D. Vukicevic, Augmented Zagreb index, Journal of Mathematical Chemistry, 48(2), (2010), 370-380.
Y. Huang, B. Liu, and L. Gan, Augmented Zagreb index of connected graphs, Match-Communications in Mathematical and in Computer Chemistry, 67, (2012), 483-494.
G. Caporossi, I. Gutman, P. Hansen, and L. Pavlovic, Graphs with maximum connectivity index, Computational biology and chemistry, 27(1), (2003), 85-90.
E. Deutsch and S. Klavzar, M-Polynomial, and degree-based topological indices, Iranian journal of mathematical chemistry, 6(2), (2015), 93-102.
J. B. Liu and X. F. Pan, Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, Applied mathematics and computation, 291, (2016), 84-88.
M. Munir, W. Nazeer, Z.Shahzadi, and S. Kang, Some invariants of circulant graphs, Symmetry, 8(11), (2016), Art ID.134.
M. Riaz, W. Gao, and A. Q. Baig, M-Polynomials and degree-based topological indices of some families of convex polytopes, Open journal of mathematical sciences, 2(1), (2018), 18-28.
M. Ghorbani and N. Azimi, Notes on mulyiple Zagreb indices, Iranian journal of mathematical chemistry, 3(2), (2012), 137-143.
M. Munir, W. Nazeer, S. Rafique, and S. Kang, M-Polynomial and degree-based topological indices of polyhex nanotubes, Symmetry, 8(12), (2016), Art ID. 149.
S. M. Kang, W. Nazeer, W. Gao, D. Afzal, S.N. Gillani, M-polynomials and topological indices of dominating David derived networks, Open Chemistry, 16(1), (2018), 201-213.
G. Kalaimurugan, K. Mageshwaran, Zk -Magic Labeling On Inverse Graphs from Finite Cyclic Group, American International Journal of Research in Science, Technology, Engineering and Mathematics, 23(1), (2018), 199-201.
K. Mageshwaran, G. Kalaimurugan, B. Hammachukiattikul, V. Govindan, I.N. Cangul, On L(h, k)-Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups, Journal of Mathematics, 2021, (2021), 1-7.
M. R. Alfuraidan, Y. F. Zakariyai, Inverse graphs associated with finite groups, Electronic Journal of Graph Theory and Applications, 5(1), (2017), 142-154.
¨Ozt¨urk S¨ozen, E., Alsuraiheed, T., Abdio˘glu, C., Ali, S., Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings, Symmetry, 15(12), 2023.
S¨ozen, E. ¨O., Erya¸sar, E., Abdiolu, C., Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn, Current Organic Synthesis, 21(3), (2024), 239-245.
K. Mageshwaran, Nazeek Alessa, S. Gopinath, and K. Loganathan, Topological Indices of Graphs from Vector Spaces, Mathematics, 11(2), (2023).
S. Gopinath., A.R.P. Doss, G. Kalaimurugan, Topological Indices for Inverse Graphs Associated With Finite Cyclic Group, Communications in Mathematics and Applications, 14(1), (2023).