# A Note on \"Wiener Index of a Fuzzy Graph and Application to Illegal Immigration Networks\"
## 1 Introduction  
In many real world problems we get only partial information about the problem, and the vagueness in the description and uncertainty has led to the growth of fuzzy graph theory. A mathematical framework to describe uncertainty in real life situation was first suggested by [PERSON] [1]. [PERSON] [2] introduced the notion of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as paths, cycles and connectedness. Wiener index of graphs has been studied in the field of mathematics, chemistry, and molecular biology [3; 4; 5]. There are many situations which are modeled by a connected fuzzy graph. Wiener index is such an accepted index used in various fields like communication networks, facility location, cryotopology, medicine, etcs. Let us start with a basic definition and concepts of fuzzy graphs; most of them can be found in [6]. Let \(S\) be a set. A _fuzzy graph_\(G=(\sigma,\mu)\) is a pair of membership functions on fuzzy sets \(\sigma:S\to[0,1]\) and \(\mu:S\times S\to[0,1]\) such that \(\mu(u,v)\leq\sigma(u)\wedge\sigma(v)\). Here \(\wedge\) represents the minimum. Throughout the paper, we assume that \(S\) is finite and nonempty, \(\mu\) is reflexive and symmetric. We denote the underlying crisp graph by \(G^{*}=(\sigma^{*},\mu^{*})\) where \(\sigma^{*}=\{u\in V:\sigma(u)>0\}\) and \(\mu^{*}=\{(u,v)\in V\times V:\mu(u,v)>0\}\). We denote an element \((x,y)\) of \(\mu^{*}\) by \(xy\) and call it an _edge_ of \(G\). Elements of \(\sigma^{*}\) are called _vertices_ of the fuzzy graph \(G\). A fuzzy graph \(H=(\tau,\
u)\) is called a _partial fuzzy subgraph_ of \(G=(\sigma,\mu)\) if \(\tau(v)=\sigma(v)\) for all \(vs.\in\tau^{*}\) and \(v(uv)=\mu(uv)\) for all \(uv\in v^{*}\). Note that \(G-uv\) denotes the fuzzy subgraph of \(G\) in which \(\mu(uv)=0\) and \(G-u\) is used for the fuzzy subgraph of \(G\) in which \(\sigma(u)=0\). A fuzzy graph \(G=(\sigma,\mu)\), a _path_\(P\) of length \(n\) is a sequence of distinct vertices \(u_{0},u_{1},\ldots,u_{n}\) such that \(\mu(u_{i-1},u_{i})>0\), \(i=1,2,\ldots,n\) and the degree of membership of a weakest edge is defined to be the _strength_ of the path \(P\). A path \(P\) is called a _cycle_ if \(u_{0}=u_{n}\). For any two vertices \(x\) and \(y\), let \(d(x,y)\) denotes the length of the shortest path between \(x\) and \(y\). The diameter of \(G\), denoted by \(diam(G)\), is the maximum distance \(d(x,y)\) for any two vertices \(x,y\) in \(G\). The _strength of connectedness_ between two vertices \(x\) and \(y\) is definedas the maximum of the strengths of all paths between \(x\) and \(y\) and is denoted by \(Conn_{G}(x,y)\). If the strength of a path \(P\) is equal to \(Conn_{G}(x,y)\), then a path \(P\) is called a _strongest_\(x-y\) path. A fuzzy graph \(G=(\sigma,\mu)\) is _connected_ if for every \(u,v\in\sigma^{*}\), \(Conn_{G}(u,v)>0\). An edge \(xy\) of a fuzzy graph \(G=(\sigma,\mu)\) is called \(\alpha\)-strong if \(\mu(xy)>Conn_{G-xy}(x,y)\). An edge \(xy\) of a fuzzy graph \(G=(\sigma,\mu)\) is called \(\beta\)-strong if \(\mu(xy)=Conn_{G-xy}(x,y)\). An edge \(xy\) of a fuzzy graph \(G=(\sigma,\mu)\) is called \(\delta\)-strong if \(\mu(xy)<Conn_{G-xy}(x,y)\). An edge is called strong if it is either \(\alpha\)-strong or \(\beta\)-strong. A path \(P\) is called a _strong path_ if all of its edges are strong. Let \(G=(\sigma,\mu)\) be a fuzzy graph and \(x,y\in\sigma^{*}\). A strong path \(P\) from \(x\) to \(y\) is called _geodesic_ if there is no shorter strong path from \(x\) to \(y\). The weight of a geodesic is the sum of membership values of all edges in the geodesic. Let \(G=(\sigma,\mu)\) be a fuzzy graph. The _Wiener index (WI)_ of \(G\) is defined by \(WI(G)=\sum_{u,v\in\sigma^{*}}\sigma(u)\sigma(v)d_{S}(u,v)\), where \(d_{S}(u,v)\) is the minimum sum of weights of geodesic from \(u\) to \(v\). In this paper, it is assumed that \(\sigma(u)=1\) for \(u\in\sigma^{*}\) in all examples of fuzzy graphs \(G=(\sigma,\mu)\), for convenience. The outline of this paper is organized as follows.