Application of generalised hierarchical product of graphs for computing F-index of four operations on graphs

Nilanjan De

Abstract


The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph which was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced. In this paper we study the F-index of four operations on graphs which were introduced by Eliasi and Taeri, and hence using the derived results we find F-index of some particular and chemically interesting graphs.

Keywords


Topological index; vertex degree; Zagreb Index; F-Index; Generalised Hierarchical Product;Graph operations

Full Text:

PDF

References


I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. Total pi-electron energy of alternant hydrocarbons, Chemical Physics Letters, 17 (1972) 535-538.

K. Xu, K. Tang, H. Liu and J. Wang, The Zagreb indices of bipartite graphs with more edges, Journal of Applied Mathematics and Informatics, 33 (2015) 365-377.

K.C. Das, K. Xu and J. Nam, On Zagreb indices of graphs, Frontiers of Mathematics in China, 10 (2015) 567--582.

M.H. Khalifeha, H. Yousefi-Azaria and A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Applied Mathematics, 157(4) (2009) 804-811.

B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs, International Journal of Quantum Chemistry, 107 (2007) 875–878.

B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH. Communications in Mathematical and in Computer, 54 (2005) 233-239.

B. Furtula, I. Gutman, A forgotten topological index, Journal of Mathematical Chemistry, 53(4) (2015) 1184--1190.

N. De, S.M.A. Nayeem and A. Pal, F-index of some graph operations, Discrete Mathematics, Algorithms and Applications, 8(2) (2016), doi :10.1142/S1793830916500257.

N. De, S.M.A. Nayeem and A. Pal, The F-coindex of some graph operations, SpringerPlus, 5:221, (2016), doi: 10.1186/s40064-016-1864-7.

N. De, S.M.A. Nayeem, Computing the F-index of nanostar dendrimers, Pacific Science Review A: Natural Science and Engineering, doi:10.1016/j.psra.2016.06.001.

B. Furtula, I. Gutman, Ž.K. Vukicevic, G. Lekishvili and G. Popivoda, On an old/new degree-based topological index, textit{Bulletin de l'Académie Serbe des Sciences et des Arts (Cl. Math. Natur.)}, 40 (2015) 19–31.

H. Abdoa, D. Dimitrov and I. Gutman, On extremal trees with respect to the F-index, arXiv:1509.03574v2.

X. Li, J. Zheng, A unified approach to the extremal trees for different indices, MATCH. Communications in Mathematical and in Computer, 54 (2005) 195–-208.

S. Zhang, W. Wang and T.C.E. Cheng, Bicyclic graphs with the first three smalllest and largest values of the first general Zagreb index, MATCH. Communications in Mathematical and in Computer, 55 (2006) 579–-592.

H. Zhang, S. Zhang, Unicyclic graphs with the first three smallest and largest first general Zagreb index, MATCH. Communications in Mathematical and in Computer, 55 (2006) 427–-438.

I. Gutman, An exceptional property of the first Zagreb index, MATCH. Communications in Mathematical and in Computer, 72 (2014) 733–-740.

X. Li, H. Zhao, Trees with the first smallest and largest generalized topological indices, MATCH. Communications in Mathematical and in Computer, 50 (2004) 57–-62.

P.S. Ranjini, V. Lokesha and A. Usha, Relation between phenylene and hexagonal squeeze using harmonic index, International Journal of Graph Theory, 1 (2013) 116--121.

P.S. Ranjini, A. Usha, V. Lokesha and T. Deepika, Harmonic index, redefined Zagreb indices of dragon graph with complete graph, Asian Journal of Mathematics and Computer Research, 9(2) (2016) 161--166.

W. Gao, W. Wang and M.R. Farahani, Topological Indices Study of Molecular Structure in Anticancer Drugs, Journal of Chemistry, (2016), http://dx.doi.org/10.1155/2016/3216327.

X. Xu, Relationships between harmonic index and other topological indices, Applied Mathematical Sciences, 6(41) (2012) 2013--2018.

L. Barriere, F. Comellas, C. Dalfo and M. A. Fiol, The hierarchical product of graphs, Discrete Applied Mathematics, 157 (2009) 36--48.

L. Barri`{e}re, C. Dalf'{o}, M. A. Fiol and M. Mitjana, The generalized hierarchical product of graphs, textit{Discrete Mathematics}, 309 (2009) 3871--3881.

M. Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Applied Mathematics, 157 (2009) 794--803.

M. Arezoomand, B. Taeri, Applications of generalized hierarchical product of graphs in computing the Szeged index of chemical graphs, MATCH. Communications in Mathematical and in Computer, 64(3) (2010) 591--602.

M. Eliasi, A. Iranmanesh, Hosoya polynomial of hierarchical product of graphs, MATCH. Communications in Mathematical and in Computer, 69(1) (2013) 111--119.

M. Arezoomand, B. Taeri, Zagreb indices of the generalized hierarchical product of graphs, MATCH. Communications in Mathematical and in Computer, 69(1) (2013) 131--140.

Z. Luo, J. Wu, Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications, Journal of Applied Mathematics, 2014(2014), Article ID 241712, 8 pages, http://dx.doi.org/10.1155/2014/241712 2014.

N. De, S.M.A. Nayeem and A. Pal, Total eccentricity index of generalized hierarchical product of graphs, International Journal of Applied and Computational Mathematics, (2014), doi: 10.1007/s40819-014-0016-4.

S. Li, G. Wang, Vertex PI indices of four sums of graphs, Discrete Applied Mathematics, 159 (2011) 1601–-1607.

M. Metsidik, W. Zhang, F. Duan, Hyper and reverse Wiener indices of F-sums of graphs, Discrete Applied Mathematics, 158 (2010) 1433--1440.

B. Eskender, E. Vumar, Eccentric connectivity index and eccentric distance sum of some graph operations, Transactions on Combinatorics, 2(1) (2013) 103--111.

M. An, L. Xiong and K.C. Das, Two Upper Bounds for the Degree Distances of Four Sums of Graphs, Filomat, 28(3) (2014) 579–-590.

H. Deng, D. Sarala, S.K. Ayyaswamy and S. Balachandran, The Zagreb indices of four operations on graphs, Applied Mathematics and Computation, 275 (2016) 422-–431.

N. De, A. Pal, and S.M.A. Nayeem, The irregularity of some composite graphs, International Journal of Applied and Computational Mathematics, DOI 10.1007/s40819-015-0069-z.

N. De, A. Pal, and S.M.A. Nayeem, Total eccentricity index of some composite graphs, Malaya Journal of Matematik, 3(4)(2015) 523-529.

W. Yan, B.Y. Yang and Y.N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Applied Mathematics Letter, 20 (2007) 290--295.




DOI: http://dx.doi.org/10.19184/ijc.2018.2.2.5

Refbacks

  • There are currently no refbacks.


ISSN: 2541-2205

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View IJC Stats