## 3 The g-Extra Connectivity of the BCCC Network  
In this section, we prove the the upper and lower bounds on the g-extra connectivity separately, and finally, obtain the g-extra connectivity of the \(BC_{n,k}\). For any two vertices x and y in \(BC_{n,k}\), the two numbers differ only by one digit. Let \(diffp\) denote a different position from left to right. For example, \(diffp(x,y)=0\) denotes the different \(a_{0}\) number of \(x\) and \(y\). Figure 3: Several transparent \(BC_{n,k}\)s with small parameters n and k.  
Figure 2: Structure of BCCC(3,2). **Lemma 4** ([1]). : _For any integers \(n\geq 3\) and \(k\geq 2\), let \(x\) and \(y\) be any two distinct vertices in \(BC_{n,k}\); the two have only one digit number. Then_  
\[|N_{BC_{n,k}}(x)\cap N_{BC_{n,k}}(y)|\left\{ \begin{aligned} &=k-1,&& if \quad diffp(x,y)=0\\ &=n-2,&& if\quad 1\leq diffp(x,y)\leq k+1\\ &\leq 1,&& if\quad(x,y)\
otin E(BC_{n,k})\end{aligned} \right. \tag{1}\]  
**Lemma 5**. : _For any integers \(n\geq 3\), \(k\geq 2\), \(n\geq k+1\) and \(0\leq g\leq n-1\), if \(D\subseteq V(BC_{n,k})\) with \(|D|=g+1\), we have \(|N_{BC_{n,k}}(D)|\geq(g+1)(k-1)+n\)._  
Proof. : According to Lemma 3, when \(|N_{BC_{n,k}}(D)|=(g+1)(k-1)+n\), \(BC_{n,k}\) is divided into two subgraphs, each of them has least \(g+1\) vertices. The lemma holds. Next, we consider that vertices are located in different connected components. According to Lemma 1, for any two vertices \(u\), \(v\) and \((u,v)\
ot\in E(BC_{n,k})\), they have one common neighbor at most. Similarly, for two different connected components A and B, \(N_{BC_{n,k}}(A)\cup N_{BC_{n,k}}(B)\leq 1\). Thus, \(g+1\) vertices can be divided into \(\varepsilon\) disjoint vertex sets, say \(D_{1}\),\(D_{2}\),...,\(D_{\varepsilon}\). Suppose that \(|D_{i}|=g_{i}\), where \(2\leq i\leq\varepsilon\), we have \(\sum_{i=1}^{\varepsilon}|D_{i}|=|D|=g+1\) and \(|N_{BC_{n,k}}(D_{i})|\geq g_{i}(k-1)+n-1\). Thus, we have  
\[N_{BC_{n,k}}(D) \geq|N_{BC_{n,k}}(D_{1})|+|N_{BC_{n,k}}(D_{2})|+...+|N_{BC_{n,k}} (D_{\varepsilon})|\] \[\geq[g_{1}(k-1)+n-1]+[g_{2}(k-1)+n-1]+...+[g_{\varepsilon}(k-1)+ n-1]\] \[=\sum_{1\leq i\leq\varepsilon}[g_{i}(k-1)+n-\varepsilon] \tag{2}\] \[=(g+1)(k-1)+\varepsilon n-\varepsilon\]  
Let \(f(\varepsilon)=(g+1)(k-1)+\varepsilon n-\varepsilon\) be a function on \(\varepsilon\), where \(2\leq\varepsilon\leq g+1\), it is easy to verify that \(f^{\prime}(\varepsilon)=n-1\). Since \(n\geq 3\), we have \(f^{\prime}(\varepsilon)>0\). Therefore, \(f^{\prime}(\varepsilon)\) is an increasing function, \(f(\varepsilon)\geq f(2)\). We have  
\[N_{BC_{n,k}}(D) \geq(g+1)(k-1)+\varepsilon n-\varepsilon\] \[\geq(g+1)(k-1)+2n-2 \tag{3}\] \[>(g+1)(k-1)+n\]  
In summary, if \(D\subseteq V(BC_{n,k})\) with \(|D|=g+1\), we have \(|N_{BC_{n,k}}(D)|\geq(g+1)(k-1)+n\). The lemma holds. **Lemma 6**. : _For any integers \(n\geq 3\), \(k\geq 2\), \(n\geq k+1\) and \(0\leq g\leq n-1\), if \(F\subseteq V(BC_{n,k})\) with \(|F|\leq(g+1)(k-1)+n-1\), then \(BC_{n,k}-F\) has one large component, and the remaining small components have at most \(g\) vertices in total._  
Proof. : Let \(C_{1}\), \(C_{2}\),..., \(C_{7}\), \(C_{7+1}\) be the components of \(BC_{n,k}-F\), where \(C_{7+1}\) is the largest component. Let \(\sum_{i=1}^{7}|C_{i}|=|C|\). Assume that \(|C|\geq g+1\). According to Lemma 5, \(|N_{BC_{n,k}}(C)|\geq(g+1)(k-1)+n\), this is in contradiction to \(|F|\leq(g+1)(k-1)+n-1\). Hence, \(|C|\leq g\). The lemma holds. **Lemma 7**. : _For any integers \(n\geq 3\), \(k\geq 2\), \(n\geq k+1\) and \(0\leq g\leq n-1\), if \(F\subseteq V(BC_{n,k})\) with \(|F|\leq(g+1)(k-1)+n-1\), then \(BC_{n,k}-F\) has one large component at least \(N-|F|-g\) vertices._  
Proof. : According to Lemma 6, \(|C|\leq g\) with \(|F|\leq(g+1)(k-1)+n-1\). We have  
\[|C_{7+1}| =N-|F|-|C| \tag{4}\] \[\geq N-|F|-g\]Thus, the lemma holds. Lemma 8: _For any integers \(n\geq 3\), \(k\geq 2\), \(n\geq k+1\) and \(0\leq g\leq n-1\), the upper bound of \(g\)-extra connectivity is \(\kappa_{g}(BC_{n,k})\leq(g+1)(k-1)+n\)._  
Proof.