## 2 Preliminaries
Any two distinctive links of the set that are not incident with a common node is called a link independent set of any network \(\aleph\). Similarly, a link independence number of any network \(\aleph\) is the maximum cardinalities among entire link independent sets. It is indicated as \(\alpha^{\prime}(\aleph)\). The set of nodes (links) in which every link (node) of \(\aleph\) is incident with at least one node (link) of the set is called a node (link) cover of a network \(\aleph\). The minimum of the cardinalities among entire node (link) covers is said to be the node (link) cover number of a given network \(\aleph\) and is indicated as \(\beta(\aleph)(\beta^{\prime}(\aleph))\). In any connected network \(\aleph\) with order \(n\), has a matching number \(\alpha^{\prime}(\aleph)\) must fulfill \(1\leq\alpha^{\prime}(\aleph)\leq\lfloor\frac{n}{2}\rfloor\). Meanwhile, in the case of a link cover of any network, one can constantly suppose that the network should consists no isolated node. It can easily be observed that for a network \(\aleph\) of order \(n\), \(\alpha(\aleph)+\beta(\aleph)=n\). Additionally, if \(\aleph\) has no isolated node, then one has \(\alpha^{\prime}(\aleph)+\beta^{\prime}(\aleph)=n\). For a bipartite network \(\aleph\), one has \(\alpha^{\prime}(\aleph)=\beta(\aleph)\), and \(\alpha(\aleph)=\beta^{\prime}(\aleph)\). For the sake of simplicity, we assume that \(\alpha^{q}_{n}\) is the class of all bipartite networks with order \(n\) having matching number \(q\). Whereas, \(\mathcal{B}^{q}_{n}\) indicate the class of all bipartite networks with order \(n\) having diameter \(d\). Similarly, \(\mathcal{C}^{s}_{n}\) (resp. \(\mathcal{D}^{t}_{n}\)) be the class of all \(n\)-node bipartite networks with connectivity \(s\) (resp. link connectivity \(t\)). We define \(M_{1}(\aleph)=\sum\limits_{u\in V(\aleph)}d^{2}_{\aleph}(u)\) as the first Zagreb index of a network \(\aleph\). Similarly, \(M_{2}(\aleph)=\sum\limits_{uv\in E(\aleph)}d_{\aleph}(u)d_{\aleph}(v)\) is the second Zagreb index of a network \(\aleph\), for further detail one can see [5]. Inspired from the above definitions [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7] invented another similar kinds of network invariant called the first (resp. second) Zagreb eccentricity index. The first (resp. second) Zagreb eccentricity index of \(\aleph\) is denoted by \(\xi_{1}(\aleph)\) (resp. \(\xi_{2}(\aleph)\)) and defined as  
\[\xi_{1}(\aleph)=\sum_{x_{1}\in V(\aleph)}\varepsilon^{2}(x_{1}),\ \ \ \ \ \ \ \xi_{2}(\aleph)=\sum_{x_{1}x_{2}\in E(\aleph)} \varepsilon(x_{1})\varepsilon(x_{2}).\]  
Some extremal problems related to first and second Zagreb eccentricity indices are presented by [PERSON] and [PERSON] in [8]. In [9] the authors obtained trees having sharp lower bound of Zagreb eccentricity indices with given domination number, maximum degree, and bipartition size. Some extremal problems of unicyclic networks which minimize and maximize the first and second Zagreb eccentricity indices are considered by [PERSON] and [PERSON] in [10]. The networks having maximum also second maximum with respect to the second Zagreb eccentricity index among entire \(n\)-node bicyclic networks figured out by [PERSON] and [PERSON] in [11]. The Zagreb eccentricity indices of generalized hierarchical product is computed by [PERSON] and [PERSON] in [12]. Studies given under [13; 14; 15; 16; 17] led us to consider the extremal problem on \(n\)-node bipartite networks with given matching number and diameter among \(\mathcal{A}_{n}^{q}\) and \(\mathcal{B}_{n}^{d}\). In order to formulate our main results, the following Lemma is helpful. Lemma 1 ([8], P:121). : _Let \(\aleph\) be any connected bipartite network with order \(n\) having bipartition \(V(\aleph)=X_{1}\cup X_{2}\), \(X_{1}\cap X_{2}=\emptyset\), \(|X_{1}|=p\) and \(|X_{2}|=q\).