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.
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