## 4 Network Having Minimum \(\xi_{i}\)-Value, \(i=1,2\) w.r.t \(\mathcal{B}_{n}^{d}\)  
In the current section, networks in \(\mathcal{B}_{n}^{d}\) having minimum \(\xi_{i}\)-value is considered. Assume that every member in \(\mathcal{B}_{n}^{d}\), has a diametrical path that is to say \(P=v_{0}v_{1}\ldots v_{d}\). Then for any \(\aleph=(V_{\aleph},E_{\aleph})\) in \(\mathcal{B}_{n}^{d}\), there is a partition \(V_{0},V_{1},\ldots V_{d}\) of \(V_{\aleph}\) with \(d(v_{0},v)=i\) in every node \(v\in V_{i}(i=0,1,2,\ldots,d)\). Named \(V_{i}\) to distance layer in \(V_{\aleph}\). If \(|i-j|=1\) then the two distance layers \(V_{i}\), \(V_{j}\) in \(V_{\aleph}\) are adjacent. Assume that \(|V_{i}|=l_{i}\) throughout this section. Clearly, \(l_{0}=|V_{0}|=1\). If \(3\leq d\leq n-1\), where \(d\) is odd, then suppose \(\aleph(n,d):=[\frac{d-1}{2}]K_{1}+[\frac{n-d-1}{2}]K_{1}+[\frac{n-d+1}{2}]K_{1} +[\frac{d-1}{2}]K_{1}+[\frac{d-1}{2}]K_{1}\). Whereas, if \(4\leq d\leq n-1\), and \(d\) is even then, assume \(\mathcal{H}(n,d)=\{H(n,d)=[\frac{d}{2}-1]K_{1}+a_{1}K_{1}+[\frac{n-d+2}{2}]K_{1 }+a_{2}K_{1}+[\frac{d}{2}-1]K_{1}:a_{1}+a_{2}=[\frac{n-d+2}{2}]\}\). Lemma 3: _For any network \(\aleph\in\mathcal{B}_{n}^{d}\) with the above partition of \(V_{\aleph}\), \(\aleph[V_{i}]\) induces an empty network (i.e. containing no link) for each \(i\in\{0,\ 1,\ldots,d\}\)._  
Proof: it can be seen that \(L_{0}=\{x_{0}\}\). There must be two paths \(P\) and \(Q\) such that \(P=x_{0}\ldots u\) and \(Q=x_{0}\ldots v\), once there exists a link \(uv\) in \(\aleph[L_{i}]\) for some \(i\in\{0,1,\ldots,d\}\). Meanwhile, \(P\cup Q+uv\) is an odd cycle in a network \(\aleph\), if \(P\) and \(Q\) have no internal node in common, this gives a contradiction. Else, assume that \(u_{0}\) is the last common internal node in \(P\) as well as \(Q\). Thereby, \(P(u_{0},u)\cup Q(u_{0},v)+uv\) again an odd cycle. This contradicts the statement that \(\aleph\) is a bipartite. Lemma 4: _A bipartite network \(\aleph[L_{j-1}\cup L_{j}]\) is complete in which \(j=1,2,\ldots,d\)._  
Proof: By Lemma 3, \(\aleph[L_{i}]\) is an empty network for each \(i\in\{1,2,\ldots,d\}\). In contrary, suppose that \(\aleph[L_{j-1}\cup L_{j}]\) is not a complete bipartite network, then one can construct another network \(\aleph^{\prime}\) with adding entire potential links among \(L_{j-1}\) as well as \(L_{j}\). Due to Lemma 1, one has \(\xi_{i}(\mathbb{N}^{\prime})<\xi_{i}(\mathbb{N})\), for \(i=1,2\) a contradiction. Hence, \(\aleph[L_{j-1}\cup L_{j}]\) is a complete bipartite network. Thus, we get our desired result. Theorem 2: _Assume that \(\aleph\) be any network in \(\mathcal{B}_{n}^{d}\)._  
(i) _If \(d=2\), then \(\xi_{1}(\aleph)\geqslant 4n\). The equality holds if and only if \(\aleph\cong K_{n-t,t}\), and \(\aleph\
ot\cong S_{n}\)._  
(ii) _If \(d=2\), then \(\xi_{2}(\aleph)\geqslant 4t(n-t)\). The equality holds if and only if \(\aleph\cong K_{n-t,t}\), and \(\aleph\
ot\cong S_{n}\)._  
**Proof. ** (i) Due to Lemma 1 we have \(\aleph\cong K_{n-t,t}\), as \(t,n-t\geqslant 2\). Assume that \(|Z_{1}|=n-t\), \(|Z_{2}|=t\). Thereby, it is straightforward to see that for every \(x\)(resp. \(y\)) in \(Z_{1}\)(resp. \(Z_{2}\)), we have \(\varepsilon^{2}(x)=\varepsilon^{2}(y)=4\). This gives  
\[\xi_{1}(K_{n-t,t})= \sum_{x\in V(\aleph)}\varepsilon^{2}(x)+\sum_{y\in V(\aleph)} \varepsilon^{2}(y)\] \[= 4t+4(n-t)\] \[= 4n. \tag{3}\]  
(ii) Similarly, if \(d=2\), then due to Lemma 1 one has \(\aleph\cong K_{n-t,t}\), as \(t,n-t\geqslant 2\). Assume that \(|Z_{1}|=n-t,\ |Z_{2}|=t\). Thereby, one can check it easily that for every \(x\) (resp. \(y\)) in \(Z_{1}\) (resp. \(Z_{2}\)), we have \(\varepsilon(x)=\varepsilon(y)=2\). This gives  
\[\xi_{2}(K_{n-t,t})= \sum_{xy\in E(\aleph)}\varepsilon(x)\varepsilon(y)\] \[= 4t(n-t). \tag{4}\]  
Note that, by the addition of any link(s) between any two nodes, does not increases the node eccentricity. Thus we have \(\varepsilon_{i}(\aleph+e)\leq\varepsilon_{i}(\aleph)\).