A network is an abstract structure that consists of nodes that are connected by links. A bipartite network is a type of networks where the set of nodes can be divided into two disjoint sets in a way that each link connects a node from one partition with a node from the other partition. In this paper, we first determine the maximum $ H $-index of networks in the class of all $ n $-node connected bipartite network with matching number $ t $. We obtain that the maximum $ H $-index of a bipartite network with a given matching number is $ K_{t, n-t} $. Secondly, we characterize the network with the maximum $ H $-index in the class of all the $ n $-vertex connected bipartite network of given diameter. Based on our obtain results, we establish the unique bipartite network with maximum $ H $-index among bipartite networks with a given independence number and cover of a network.
Citation: Shahid Zaman, Fouad A. Abolaban, Ali Ahmad, Muhammad Ahsan Asim. Maximum $ H $-index of bipartite network with some given parameters[J]. AIMS Mathematics, 2021, 6(5): 5165-5175. doi: 10.3934/math.2021306
A network is an abstract structure that consists of nodes that are connected by links. A bipartite network is a type of networks where the set of nodes can be divided into two disjoint sets in a way that each link connects a node from one partition with a node from the other partition. In this paper, we first determine the maximum $ H $-index of networks in the class of all $ n $-node connected bipartite network with matching number $ t $. We obtain that the maximum $ H $-index of a bipartite network with a given matching number is $ K_{t, n-t} $. Secondly, we characterize the network with the maximum $ H $-index in the class of all the $ n $-vertex connected bipartite network of given diameter. Based on our obtain results, we establish the unique bipartite network with maximum $ H $-index among bipartite networks with a given independence number and cover of a network.
| [1] | R. B. Bapat, Graphs and Matrices, New York: Springer, 2010. |
| [2] | Q. Li, S. Zaman, W. Sun, J. Alam, Study on the normalized Laplacian of a penta-graphene with applications, Int. J. Quantum. Chem., 120 (2020), e26154. |
| [3] |
D. Plavšić, S. Nikolić, N. Trinajstić, Z. Mihalić, On the Harary index for the characterization of chemical graphs, J. Math. Chem., 12 (1993), 235-250. doi: 10.1007/BF01164638
|
| [4] |
O. Ivanciuc, T. S. Balaban, A. T. Balaban, Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem., 12 (1993), 309-318. doi: 10.1007/BF01164642
|
| [5] | D. Janežić, A. Milicevič, S. Nikolić, N. Trinajstić, Graph Theoretical Matrices in Chemistry, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, Serbia, 2007. |
| [6] |
H. Li, S. C. Li, H. Zhang, On the maximal connective eccentricity index of bipartite graphs with some given parameters, J. Math. Anal. Appl., 454 (2017), 453-467. doi: 10.1016/j.jmaa.2017.05.003
|
| [7] |
S. C. Li, Y. B. Song, On the sum of all distances in bipartite graphs, Discrete Appl. Math., 169 (2014), 176-185. doi: 10.1016/j.dam.2013.12.010
|
| [8] | S. C. Li, Y. Y. Wu, L. L. Sun, On the minimum eccentric distance sum of bipartite graphs with some given parameters, J. Math. Anal. Appl., 430 (2015), 1146-1162. |
| [9] | G. Wang, L. Yan, S. Zaman, M. Zhang, The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons, Int. J. Quantum. Chem., 120 (2020), e26334. Available from: https://doi.org/10.1002/qua.26334. |
| [10] | L. Feng, Z. Li, W. Liu, L. Lu, D. Stevanović, Minimal Harary index of unicyclic graphs with diameter at most 4, Appl. Math. Comput., 381 (2020), 125315. |
| [11] |
X. X. Li, Y. Z. Fan, The connectivity and the Harary index of a graph, Discrete Appl. Math., 181 (2015), 167-173. doi: 10.1016/j.dam.2014.08.022
|
| [12] | E. Egerváry, On combinatorial properties of matrices, Mat. Lapok, 38 (1931), 16-28. |
| [13] | D. König, Graphs and matrices, Mat. Fiz. Lapok, 38 (1931), 116-119. |
| [14] | I. Gutman, A property of the wiener number and its modifications, Indian J. Chem., 36A (1997), 128-132. |
| [15] | K. C. Das, B. Zhou, N. Trinajstić, Bounds on Harary index, J. Math. Chem., 46 (2009), 1377-1393. |
| [16] |
L. H. Feng, A. I. Zagreb, Harary and hyper-wiener indices of graphs with a given matching number, Appl. Math. Lett., 23 (2010), 943-948. doi: 10.1016/j.aml.2010.04.017
|
| [17] | K. Xu, K. C. Das, On Harary index of graphs, Discrete Appl. Math., 159 (2011), 1631-1640. |
| [18] | B. Zhou, X. Cai, N. Trinajstić, On the Harary index, J. Math. Chem., 44 (2008), 611-618. |
| [19] | A. Ilić, G. H. Yu, L. H. Feng, On the Harary index of trees, Utilitas Math., 87 (2012), 21-32. |
| [20] | S. Wagner, H. Wang, X. Zhang, Distance-based graph invariants of trees and the Harary index, Filomat, 27 (2013), 41-50. |
| [21] | K. Xu, K. C. Das, Extremal unicyclic and bicyclic graphs with respect to Harary index, Bull. Malay. Math. Sci. Soc., 36 (2013), 373-383. |
| [22] | G. H. Yu, L. H. Feng, On the maximal Harary index of a class of bicyclic graphs, Utilitas Math., 82 (2010), 285-292. |
| [23] | K. Xu, Trees with the seven smallest and the eight greatest Harary indices, Discrete Appl. Math., 160 (2012), 321-331. |
| [24] | H. Q. Liu, L. H. Feng, The distance spectral radius of graphs with given independence number, Ars Comb., 121 (2015), 113-123. |
| [25] | L. Feng, Y. Lan, W. Liu, X. Wang, Minimal Harary index of graphs with small parameters, MATCH Commun. Math. Comput. Chem., 76 (2016), 23-42. |
Figures(1)
Shahid Zaman, Fouad A. Abolaban, Ali Ahmad, Muhammad Ahsan Asim. Maximum $ H $-index of bipartite network with some given parameters[J]. AIMS Mathematics, 2021, 6(5): 5165-5175. doi: 10.3934/math.2021306
DownLoad: