# The zeroth-order general Randic index of graphs with a given clique number

## Main Article Content

## Abstract

The zeroth-order general Randi\'{c} index $^{0}R_{\alpha}(G)$ of the graph $G$ is defined as $\sum_{u\in V(G)}d(u)^{\alpha}$, where $d(u)$ is the degree of vertex $u$ and $\alpha$ is an arbitrary real number. In this paper, the maximum value of zeroth-order general Randi\'{c} index on the graphs of order $n$ with a given clique number is presented for any $\alpha\neq 0,1$ and $\alpha \notin (2,2n-1]$, where $n=|V(G)|$. The minimum value of zeroth-order general Randi\'{c} index on the graphs with a given clique number is also obtained for any $\alpha\neq 0,1$. Furthermore, the corresponding extremal graphs are characterized.

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## References

[1] H. Ahmeda, A. A. Bhattia and A. Ali, Zeroth-order general Randi c index of cac- tus graphs, AKCE Int. J. Graphs Comb. (2018), https://doi.org/10.1016/j.akcej. 2018.01.006. Google Scholar

[2] A. Ali, A. A. Bhatti and Z. Raza, A note on the zeroth-order general Randi c index of cacti and polyomino chains, Iranian J. Math. Chem. 5 (2014), 143-152. Google Scholar

[3] B. Bollob as and P. Erd os, Graphs of extremal weights, Ars Combin. 50 (1998), 225-233. Google Scholar

[4] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Elsevier, New York, 1976. Google Scholar

[5] S. Chen and H. Deng, Extremal (n, n + 1)-graphs with respected to zeroth-order general Randi c index, J. Math. Chem. 42 (2007), 555-564. Google Scholar

[6] H. Deng, A unified approach to the extremal Zagreb indices for trees, unicyclic graphs and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 57 (2007), 597–616. Google Scholar

[7] P. Erd os, On the graph theorem of Tur an, Mat. Lapok 21 (1970), 249-251. Google Scholar

[8] I. Gutman and N. Trinajsti c, Graph theory and molecular orbitals. III. Total p-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535-538. Google Scholar

[9] Y. Hu, X. Li, Y. Shi and T. Xu, Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi c index, Discrete Appl. Math. 155 (2007), 1044-1054. Google Scholar

[10] Y. Hu, X. Li, Y. Shi, T. Xu and I. Gutman, On molecular graphs with smallest and greatest zeroth-order general Randi c index, MATCH Commun. Math. Comput. Chem. 54 (2005), 425-434. Google Scholar

[11] H. Hua and H. Deng, On unicycle graphs with maximum and minimum zeroth-order genenal Randi c index, J. Math. Chem. 41 (2007), 173-181. Google Scholar

[12] L. B. Kier and L. H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976. Google Scholar

[13] L. B. Kier and L. H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press, Wiley, Chichester, UK, 1986. Google Scholar

[14] L. B. Kier and L. H. Hall, The nature of structure-activity relationships and their relation to molecular connectivity, Europ. J. Med. Chem. 12 (1977), 307–312. Google Scholar

[15] F. Li and M. Lu, On the zeroth-order general Randi c index of unicycle graphs with k pendant vertices, Ars Combin. 109 (2013), 229-237. Google Scholar

[16] S. Li and M. Zhang, Sharp bounds on the zeroth-order general Randi c indices of conjugated bicyclic graphs, Math. Comput. Model. 53 (2011), 1990-2004. Google Scholar

[17] X. Li and Y. Shi, A survey on the Randi c index, MATCH Commun. Math. Comput. Chem. 59 (2008), 127-156. Google Scholar

[18] X. Li and Y. Shi, (n, m)-graphs with maximum zeroth-order general Randi c index for a (-1, 0), MATCH Commun. Math. Comput. Chem. 62 (2009), 163-170. Google Scholar

[19] X. Li and H. Zhao, Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004), 57–62. Google Scholar

[20] X. Li and J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54 (2005), 195–208. Google Scholar

[21] X. Pan and S. Liu, Conjugated tricyclic graphs with the maximum zeroth-order general Randi c index, J. Appl. Math. Comput. 39 (2012), 511-521. Google Scholar

[22] L. Pavlovi c, Maximal value of the zeroth-order Randi c index, Discr. Appl. Math. 127 (2003), 615-626. Google Scholar

[23] M. Randi c, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), 6609-6615. Google Scholar

[24] G. Su, J. Tu and K. C. Das, Graphs with fixed number of pendent vertices and minimal zeroth-order general Randi c index, Appl. Math. Comput. 270 (2015), 705-710. Google Scholar

[25] G. Su, L. Xiong and X. Su, Maximally edge-connected graphs and zeroth-order general Randi c index for 0 < a < 1, Discrete Appl. Math. 167 (2014), 261-268. Google Scholar

[26] G. Su, L. Xiong, X. Su and G. Li, Maximally edge-connected graphs and zeroth- order general Randi c index for a <= -1, J. Comb. Optim. 31 (2016), 182-195. Google Scholar

[27] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley- VCH, Weinheim, 2000. Google Scholar

[28] R. Wu, H. Chen and H. Deng, On the monotonicity of topological indices and the connectivity of a graph, Appl. Math. Comput. 298 (2017), 188–200. Google Scholar

[29] K. Xu, The Zagreb indices of graphs with a given clique number, Appl. Math. Lett. 24 (2011), 1026–1030. Google Scholar

[30] S. Zhang and H. Zhang, Unicyclic graphs with the first three smallest and largest first general Zagreb index, MATCH Commun. Math. Comput. Chem. 55 (2006), 427–438. Google Scholar

[31] S. Zhang, W. Wang and T. C. E. Cheng, Bicyclic graphs with the first three smallest and largest values of the first general Zagreb Index, MATCH Commun. Math. Comput. Chem. 56 (2006), 579–592. Google Scholar