## 2 Preliminaries
Then \(\xi_{i}(\aleph)\geqslant\xi_{i}(K_{p,q}\setminus e)>\xi_{i}(K_{p,q})\), where \(i=1,2\) and \(e\in K_{p,q}\)._  
Network Contain Minimum Zagreb Eccentricity Indices among All \(n\)-Node Bipartite Networks with Given Matching Number \(q\)  
In this section, we characterize the networks among \(\mathcal{A}_{n}^{q}\) having minimum Zagreb eccentricity indices. Lemma 2.: _Assume that \(\aleph\) be any connected bipartite network having \(V_{\aleph}=(X,Y)\) with \(|X|=n_{1}\), \(|Y|=n_{2}\) and \(n_{1}\geqslant n_{2}\)._  
* _If_ \(n_{1}=1\)_, then_ \(\xi_{1}(\aleph)=2\) _and_ \(\xi_{2}(\aleph)=1\)_, in this case_ \(\aleph=K_{2}\)_._
* _If_ \(n_{1}>1\)_, and_ \(n_{2}=1\) _then_ \(\xi_{1}(\aleph)=4n_{1}+1\) _and_ \(\xi_{2}(\aleph)=2n_{1}\)_, in this case_ \(\aleph=K_{1,n_{1}}\)_._
* _If_ \(n_{2}>1\)_, then_ \(\xi_{1}(\aleph)\geqslant 4(n_{1}+n_{2})\) _and_ \(\xi_{2}(\aleph)\geqslant 4n_{1}n_{2}\)_, with equality if and only if_ \(\aleph\cong K_{n_{1},n_{2}}\)_._  
Due to Lemma 2 we characterize all the bipartite networks which are connected and having order \(n>2\). Theorem 3.1.: _Assume that \(\mathcal{A}_{n}^{q}\) is an \(n\)-node bipartite network with matching number \(q\), and \(\aleph\in\mathcal{A}_{n}^{q}\)._  
* _If_ \(q=1\)_, then_ \(\xi_{1}(\aleph)=4n-3\) _and_ \(\xi_{2}(\aleph)=2n-2\)_, where_ \(\aleph\cong K_{1,n-1}\)_._
* _If_ \(q>1\)_, then_ \(\xi_{1}(\aleph)\geqslant 4n\) _and_ \(\xi_{2}(\aleph)\geqslant 4q(n-q)\)_. The equality holds if and only if_ \(\aleph\cong K_{q,n-q}\)_._  
Proof. : By a direct calculation, one has  
\[\xi_{1}(K_{q,n-q})=4n,\ \ \xi_{2}(K_{q,n-q})=4q(n-q).\]Hence, we only need to show that among \(\mathcal{A}_{n}^{q}\) with minimum Zagreb eccentricity indices is a unique network \(K_{q,n-q}\). Choose \(\aleph\), in \(\mathcal{A}_{n}^{q}\) such that its first Zagreb eccentricity index and the second Zagreb eccentricity index are minimum. For \(q=\lfloor\frac{n}{2}\rfloor\), due to Lemma 1 an extremal network is exactly \(K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}\) as desired. Therefore, we only consider the case \(q<\lfloor\frac{n}{2}\rfloor\). Let the bipartition node set in \(\aleph\) is denoted by \((X,Y)\), such that \(|Y|\geqslant|X|\geqslant q\). Assume that \(M\) is a maximal matching in \(\aleph\), then due to Lemma 1, the addition of new link(s) decreases the first Zagreb eccentricity index as well as the second Zagreb eccentricity index of a network. In what follows, if \(|X|=q\), then the extremal network is \(\aleph=K_{q,n-q}\). Hence, we consider the case \(|X|>q\). Assume that \(M\) is a matching set and \(X_{M}\) (resp. \(Y_{M}\)) be the set of nodes of \(X\) (resp. \(Y\)) which are incident to the links of \(M\). Therefore, \(|X_{M}|=|Y_{M}|=q\). Keeping in mind that \(\aleph\) does not contain links between the nodes of \(X\backslash X_{M}\) and the nodes of \(Y\backslash Y_{M}\). Otherwise, any such link together with \(M\) producing the matching of cardinality more than as that in \(M\), which contradicts the maximality in \(M\). By adding entire potential links between the nodes of \(X_{M}\) and \(Y_{M}\), \(X_{M}\) and \(Y\backslash Y_{M}\), \(X\backslash X_{M}\) and \(Y_{M}\) we get a network \(\aleph^{\prime}\) as depicted in Figure 1, with \(\xi_{1}(\aleph^{\prime})<\xi_{1}(\aleph)\) and \(\xi_{2}(\aleph^{\prime})<\xi_{2}(\aleph)\). It can be noticed that a matching number in \(\aleph^{\prime}\) is at least \(q+1\). Thus, \(\aleph^{\prime}\
otin\mathcal{A}_{n}^{q}\) and \(\aleph\cong\aleph^{\prime}\). Due to \(\aleph^{\prime}\), one can build a new network, say \(\aleph^{\prime\prime}\), which is determine by keeping \(\aleph^{\prime}\) in such a way that first delete entire links among \(X\setminus X_{M}\) and \(Y_{M}\), and then add entire links among \(X\setminus X_{M}\) as well \(X_{M}\), see Figure 1. Thereby, it is easy to see that \(\aleph^{\prime\prime}\cong K_{q,n-q}\). Assume that \(|X\backslash X_{M}|=n_{1}\), \(|Y\backslash Y_{M}|=n_{2}\) let \(n_{2}\geqslant n_{1}\).