# The G-Convexity and the G-Centroids of Composite Graphs
## 3 Graph Compositions
### The Join of Graphs  
In this subsection, we show that the weight sequence of \(G=G_{1}\lor G_{2}\) does not depend on the weight sequence of \(G_{1}\) and \(G_{2}\), rather it depends on the incident pattern of the maximum cliques of \(G_{1}\) and \(G_{2}\). If \(G_{1}\) and \(G_{2}\) are either complete or if both are incomplete, we give the weight sequence of \(G\) explicitly. If \(G_{1}\) is complete and \(G_{2}\) is incomplete, it is shown that for each \(u\in V(G_{2})\), \(S_{u}(G)=K\cup V(G_{1})\), where \(K\) is a maximum g-convex set of \(G_{2}\) not containing \(u\). The following Proposition is immediate from the definition of join of graphs. **Proposition 12**. : _Let \(G_{1}\) and \(G_{2}\) be two graphs and \(G\)= \(G_{1}\lor G_{2}\). Then, the following holds._  
1. \(G\) _has an induced subgraph isomorphic to_ \(G_{1}\) _as well as to_ \(G_{2}\)_._
2. \(G\) _is complete if and only if both_ \(G_{1}\) _and_ \(G_{2}\) _are complete._  
If both \(G_{1}\) and \(G_{2}\) are complete, then \(G\) is a complete graph on \(n=\mid V(G_{1})\mid+\mid V(G_{2})\mid\) vertices and the weight sequence of \(G\) is \(\{(n-1)^{n}\}\) and \(C_{g}(G)=V(G)\). Now consider the case when \(G_{1}\) and \(G_{2}\) are incomplete. The following proposition determines the structure of a g-convex sets when both \(G_{1}\) and \(G_{2}\) are incomplete. **Proposition 13**. : _Let \(G_{1}\) and \(G_{2}\) be two incomplete graphs and \(G=G_{1}\lor G_{2}\). Then a proper subset \(S\) of \(V(G)\) is a g-convex set of \(G\) if and only if \(<S>\) is a complete subgraph of \(G\)._  
Proof. : Let \(S\) be a proper g-convex set of \(V\). If possible, let \(u\) and \(v\) be two non adjacent vertices of \(G\) in \(S\). From the construction of \(G\), both \(u\) and \(v\) belong to either \(G_{1}\) or \(G_{2}\). Let \(u\) and \(v\) belong to \(G_{1}\). Let \(z\) be any vertex of \(G_{2}\). Then \(u\),\(z\),\(v\) is a geodesic joining \(u\) and \(v\) in \(G\). As \(S\) is convex, \(z\in S\). Thus \(V(G_{2})\subseteq S\) as \(z\) is arbitrary. As \(G_{2}\) is incomplete, \(G_{2}\) has a pair of non-adjacent vertices. Therefore by a similar argument, \(V(G_{1})\subseteq S\). Hence \(S=V(G)\), which is a contradiction. Proof of the sufficiency part follows trivially as cliques are always g-convex sets of \(G\). **Remark 2**. : _If \(M\) and \(N\) are the maximum cliques of \(G_{1}\) and \(G_{2}\), respectively, then \(M\cup N\) is a maximum clique of \(G\)._  
We now characterize the g-convex sequences of graph joins based on the incident pattern of the maximum cliques of their factor graphs. **Proposition 14**. : _Let \(G_{1}\) and \(G_{2}\) be two incomplete graphs and \(G=G_{1}\lor G_{2}\). Let \(M_{1}\), \(M_{2}\), \(\cdots\), \(M_{r}\) and \(N_{1}\), \(N_{2}\), \(\cdots\), \(N_{s}\) be the maximum cliques of \(G_{1}\) and \(G_{2}\), respectively, of sizes \(\omega_{1}\) and \(\omega_{2}\). Let \(M=\cap_{j=1}^{s}M_{j}\), \(N=\cap_{j=1}^{s}N_{j}\), \(n=\mid V(G)\mid\), \(m_{1}=\mid M\mid\) and \(m_{2}=\mid N\mid\)._  
1. _If_ \(M\
eq\emptyset\) _and_ \(N\
eq\emptyset\)_, then the gcws of_ \(G\) _is_ \(\{(\omega_{1}+\omega_{2}-1)^{m_{1}+m_{2}}(\omega_{1}+\omega_{2})^{n-(m_{1}+m_{2 })}\}\)__
2. _If_ \(M\
eq\emptyset\) _and_ \(N=\emptyset\)_, then the gcws of_ \(G\) _is_ \(\{(\omega_{1}+\omega_{2}-1)^{m_{1}},\,(\omega_{1}+\omega_{2})^{n-m_{1}}\}\)__
3. _If_ \(M\) _and_ \(N\) _are empty, then the gcws of_ \(G\) _is_ \(\{(\omega_{1}+\omega_{2})^{n}\}\)_._Let \(u\) be a vertex of \(G\). Then, by Proposition 13, \(S_{u}\) induces a complete subgraph of \(G\). (1) Let \(M\),\(N\
eq\emptyset\). Let \(x\in V(G_{1})\) and \(x\
otin M\). Then there exists an \(i\), \(1\leq i\leq r\) such that \(x\
otin M_{i}\). Therefore \(M_{i}\cup N_{j}\) (\(1\leq j\leq s\)) are maximum g-convex sets of \(G\) not containing \(x\) (as they are complete). Thus, \(w(x)=\omega_{1}+\omega_{2}\). Similarly, \(w(x)=\omega_{1}+\omega_{2}\) also if \(x\in V(G_{2})\) and \(x\
otin N\).