## 2 Preliminaries  
In this article, connected, simple and undirected networks are considered. We denote \(P_{n}\) for the path network, \(K_{n}\) for the complete network and \(K_{p,n-p}\) for the complete bipartite network with \(n\) nodes. We follow [1; 2; 3; 4], for the notation and terminology which are not defined in this article. Assume that \(\aleph=(V_{\aleph},E_{\aleph})\) is a network having node set \(V_{\aleph}\) which are connected by the link is denoted by \(E_{\aleph}\). The cardinality of network \(W\) is denoted as \(|W|\). Let \(y\in V_{\aleph}\), then denote \(N_{\aleph}(y)\) the set of entire adjacent nodes with \(y\) in \(\aleph\). We denote the degree of \(y\in\aleph\) by \(d_{\aleph}(y)=|N_{\aleph}(y)|\). The minimum degree of \(\aleph\) is denoted by \(\delta(\aleph)\), and defined as \(\delta(\aleph)=\min\{d_{\aleph}(y)|y\in V_{\aleph}\}\). Let \(\aleph[A]\) is a subset of \(V_{\aleph}\) which is induced by \(A\). The networks \(\aleph-v\) and \(\aleph-uv\) denote as any network construct from \(\aleph\) by removing the node \(v\in V_{\aleph}\) and by removing the link \(uv\in E_{\aleph}\), respectively. In the same way, \(\aleph+uv\) can be determined from \(\aleph\) by adding a link \(uv\
ot\in E_{\aleph}\). The quantity \(\aleph_{1}\cup\aleph_{2}\) denotes the _union_ of two networks \(\aleph_{1}\) and \(\aleph_{2}\) with \(V_{\aleph_{1}\cup\aleph_{2}}=V_{\aleph_{1}}\cup V_{\aleph_{2}}\) and \(E_{\aleph_{1}\cup\aleph_{2}}=E_{\aleph_{1}}\cup E_{\aleph_{2}}\). If \(\aleph_{1}\) and \(\aleph_{2}\) are disjoint nodes, then we let \(\aleph_{1}\uplus\aleph_{2}\) denote the _joint_ of \(\aleph_{1}\) and \(\aleph_{2}\), which is the network obtained from \(\aleph_{1}\cup\aleph_{2}\) by adding all the links between the nodes \(x\in V_{\aleph_{1}}\) and \(y\in V_{\aleph_{2}}\). For disjoint networks \(\aleph_{1},\aleph_{2},\ldots,\aleph_{k}\) having \(k\geq 3\), then the joining \(\aleph_{1}\uplus\aleph_{2}\uplus\cdots\uplus\aleph_{k}\) is the network \((\aleph_{1}\uplus\aleph_{2})\cup(\aleph_{2}\uplus\aleph_{3})\cup\cdots\cup(\aleph_{ k-1}\uplus\aleph_{k})\). In short, we indicate \(k\aleph\) and \([k]\aleph\) for the union and for the joining of \(k\) time disjoint copies of \(\aleph\), simultaneously. For instance, \(kK_{1}\cong\overline{K}_{k}\) is exactly the \(k\) isolated nodes, whereas \([p]\aleph_{1}\uplus\aleph_{2}\uplus[q]\aleph_{3}\) is indicating the joining \(\underbrace{\aleph_{1}\uplus\aleph_{1}\uplus\ldots\uplus\aleph_{1}}_{p}\uplus \underbrace{\aleph_{2}\uplus\underbrace{\aleph_{3}\lor\aleph_{3}\uplus\ldots\uplus \aleph_{3}}_{q}}_{q}\). For a network, the _distance_ among \(x\) and \(y\) is denoted by \(d_{\aleph}(x,y)\) and defined by the length of a shortest \(x\)-\(y\) path. Assume that \(\varepsilon\) denotes the eccentricity of a node, then it can be defined as \(\varepsilon_{\aleph}(x):=\max_{y\in V_{\aleph}}d_{\aleph}(x,y)\) be the _eccentricity_ of \(x\). Next, is the _diameter_ of a network \(\aleph\) which is defined as \(\textit{diam}(\aleph)=\max_{x\in V_{\aleph}}\varepsilon_{\aleph}(x)\). The path \(P\) is called a _diametrical path_ of a network \(\aleph\) if it satisfies \(|E_{p}|=\mathrm{diam}(\aleph)\). Let \(\aleph\) be a simple network and \(U_{\aleph}\) is the node set of \(\aleph\). Then, \(U_{\aleph}\) can be divided into two disjoint subsets \(U_{1}\) and \(U_{2}\) in such a way that there is at least one link between these two disjoint subsets, then \(\aleph\) is called a bipartite network. On the other hand if every node of \(U_{1}\) is adjacent to every node of \(U_{2}\) such network is called a complete bipartite network. Generally, it is denoted by \(K_{n_{1},n_{2}}\), where \(n_{1}=|U_{1}|\), \(n_{2}=|U_{2}|\). A node independent set of any network \(\aleph\) is the node subset in \(V_{\aleph}\) which satisfies that any of the two distinct nodes in the set are not adjacent. The independence number is defined as the maximum cardinality in all of the independent sets of \(\aleph\) and it is denoted by \(\alpha(\aleph)\).