# A Note on \"Wiener Index of a Fuzzy Graph and Application to Illegal Immigration Networks\"
## 2 Counterexamples and Revision  
At first, we recall the Theorem 3.10 of [7] and give two counterexamples to it.  
Theorem 1 (Theorem 3.10 of [7]): _Let \(G=(\sigma,\mu)\) be a fuzzy graph. For \(s,t\in\sigma^{*}\), let \(P_{s,t}\) denote the path which has the minimum sum of membership values among all shortest strong paths between \(s\) and \(t\). Let \(uv\in\mu^{*}\). If \(uv\) is an \(\alpha\) or \(\beta\)-strong edge and \(uv\) is not a part of any \(P_{s,t}\) for  
Therefore, Theorem 3. 10 of [7] can be changed as follows  
Theorem 2: _Let \(G=(\sigma,\mu)\) be a fuzzy graph with each edge being strong. For \(s,t\in\sigma^{*}\), let \(P_{s,t}\) denote the path which has the minimum sum of membership values among all shortest strong paths between \(s\) and \(t\). Suppose that \(uv\) is not a part of any \(P_{s,t}\) for \(s,t\in\sigma^{*}\) with \(\{s,t\}\
eq\{u,v\}\). Then (1) If \(d_{S}^{G-uv}(u,v)>\mu(uv)\), then \(WI(G-uv)>WI(G)\). (2) If \(d_{S}^{G-uv}(u,v)=\mu(uv)\), then \(WI(G-uv)=WI(G)\). (3) If \(d_{S}^{G-uv}(u,v)<\mu(uv)\), then \(WI(G-uv)<WI(G)\)._  
Proof.: Since each edge of \(G\) is strong, it follows that each edge in \(G-uv\) is also strong edge. Owing to \(uv\) being a strong edge in \(G\), \(d_{S}^{G}(u,v)=\mu(uv)\). Let \(\{a,b\}\
eq\{u,v\}\). Since \(uv\) is not part of any \(P_{a,b},d_{S}^{G-uv}(a,b)=d_{S}^{G}(a,b)\). Thus, \(WI(G-uv)=\sum_{x,y\in\sigma^{*}}\sigma(x)\sigma(y)d_{S}^{G-uv}(x,y)=\sum_{x,y \in\sigma^{*}}\sigma(x)\sigma(y)d_{S}^{G}(x,y)+(d_{S}^{G-uv}(u,v)-d_{S}^{G}(u, v))=WI(G)+(d_{S}^{G-uv}(u,v)-\mu(uv))\). So, if \(d_{S}^{G-uv}(u,v)>\mu(uv)\), then \(WI(G-uv)>WI(G)\). If \(d_{S}^{G-uv}(u,v)=\mu(uv)\), then \(WI(G-uv)=WI(G)\). If \(d_{S}^{G-uv}(u,v)<\mu(uv)\), then \(WI(G-uv)<WI(G)\).  
**Note:** The condition \"each edge is strong\" is necessary in Theorem 2. For example, let \(G=(\sigma,\mu)\) be the fuzzy graph shown in Figure 3 with vertex set \(\{a,b,c,u,v\}\) and \(\sigma(x)=1\) for any \(x\in\sigma^{*}\), \(\mu(uv)=0.95\), \(\mu(au)=\mu(ab)=0.3\), \(\mu(cu)=\mu(cb)=0.1\), \(\mu(vb)=0.5\), \(\mu(ub)=0.4\). Then each edge of the graph \(G\) except edge \(ub\) is strong. Edge \(ub\) is a weak edge in \(G\). So \(d_{S}(u,v)=0.95\), \(d_{S}(a,u)=d_{S}(a,b)=0.3\), \(d_{S}(c,u)=d_{S}(c,b)=0.1\), \(d_{S}(v,b)=0.5\), \(d_{S}(a,c)=0.4\), \(d_{S}(a,v)=0.8\), \(d_{S}(u,b)=0.2\) and \(d_{S}(c,v)=0.6\). Therefore, \(WI(G)=\sum_{x,y\in\sigma^{*}}\sigma(x)\sigma(y)d_{S}(x,y)=4.25\). It is obvious that \(uv\) is a strong edge and \(uv\) is not a part of any \(P_{s,t}\) for \(s,t\in\sigma^{*}\) with \(\{s,t\}\
eq\{u,v\}\). It is obvious that \(ub\) is a strong edge in \(G-uv\). Hence, \(d_{S}^{G-uv}(u,v)=0.9\), \(d_{S}^{G-uv}(a,u)=d_{S}^{G-uv}(a,b)=0.3\), \(d_{S}^{G-uv}(c,u)=d_{S}^{G-uv}(c,b)=0.1\), \(d_{S}^{G-uv}(v,b)=0.5\), \(d_{S}^{G-uv}(u,b)=0.4\), \(d_{S}^{G-uv}(a,c)=0.4\), \(d_{S}^{G-uv}(a,v)=0.8\) and \(d_{S}^{G-uv}(c,v)=0.6\). So \(WI(G-uv)=\sum_{x,y\in\sigma^{*}}\sigma(x)\sigma(y)d_{S}^{G-uv}(x,y)=4.4\). Though \(d_{S}^{G-uv}(u,v)<\mu(uv)\), \(WI(G-uv)>WI(G)\).  
Figure 3: Fuzzy graphs \(G\) and \(G-uv\).  
Figure 2: \(uv\) is an \(\beta\)-strong edge and \(WI(G)=WI(G-uv)\).