# The G-Convexity and the G-Centroids of Composite Graphs
## 3 Graph Compositions
### The Corona  
In this section, we deal with the corona of two graphs. Let \(G\) be the corona of \(G_{1}\) with \(G_{2}\). Then, we prove that \(C_{g}(G)=C_{g}(G_{1})\).  
**Proposition 26**.: _Let \(G=G_{1}\circ G_{2}\). Let \(u\in V(G_{1})\) and \(v\in V(i,G_{2})\). Then,_  
1. \(S_{u}(G)=S_{u}(G_{1})\cup\{V(i,G_{2}):i\in S_{u}(G_{1})\}\) _and_
2. \(S_{v}(G)=S\cup\big{(}V(G)-V(i,G_{2})\big{)}\)_, where_ \(S\) _is a maximum_ \(g\)_-convex set in_ \(V(i,G_{2})\) _not containing_ \(v\)_._  
Proof.: (1) Let \(S^{\prime}\) be any maximum g-convex set of \(G_{1}\) not containing \(u\). i.e, \(S^{\prime}=S_{u}(G_{1})\). We now show that \(S=S^{\prime}\cup\{V(i,G_{2}):i\in S^{\prime}\}\) is a g-convex set of \(G\).  
If \(x,y\in V(G_{1})\cap S\), then \(x,y\in S^{\prime}\) and all \(x-y\) geodesics will lie in \(>\) (as \(S^{\prime}\subseteq S\)).  
Let \(x,y\in V(i,G_{2})\). In this case, if \(d_{G_{2}}(x,y)\geq 3\), then \(x,i,y\) is the only geodesic joining \(x\) and \(y\) in \(G\), and by our definition of \(S\), \(i\in S\). If \(d_{G_{2}}(x,y)=2\), then as before \(i\in S\) and if \(x,z,y\) is any other geodesic joining \(x,y\) in \(G\), then clearly \(z\in V(i,G_{2})\subseteq S\). Therefore, in all these cases, all \(x-y\) geodesics lie in \(>\).  
Let \(x\in V(i,G_{2})\) and \(y\in V(j,G_{2})\) with \(i,j\in S^{\prime}\). In this case it is easy to see that if \(P\) is a geodesic joining \(i\) and \(j\) in \(G_{1}\), then \(x\cup V(P)\cup y\) is a geodesic joining \(x\) and \(y\) in \(G\) and conversely. Since \(i,j\in S^{\prime}\), all \(i-j\) geodesics lie in \(^{\prime}>\). Thus all \(x-y\) geodesics lie in \(>\).  
Proof of the case when \(x\in V(i,G_{2})\) and \(y\in V(G_{1})\) is similar to that of the above case.  
Thus the convexity of \(S\) is established. Obviously, \(S\) does not contain \(u\). The reason that every such g-convex set \(S\) of \(G\) arises this way follows from the fact that \(S^{\prime}=S\cap V(G_{1})\) is a g-convex set of \(G_{1}\) not containing \(u\) and every \(V(i,G_{2})\) is a g-convex set of \(G\). A maximum g-convex set of \(G\) not containing \(u\) is obtained by taking for \(S^{\prime}\) a maximum g-convex set of \(G_{1}\) not containing \(u\); that is an \(S_{u}(G_{1})\). This establishes (1).  
(2) Let \(v\in V(i,G_{2})\). Let \(K^{\prime}\) be a maximum \(g_{2}\)-convex set in \(G_{2}\) not containing \(v\). Then as before by considering various cases, we can prove that \(K^{\prime}\cup(V-V(i,G_{2}))\) is a g-convex set of \(G\) not containing \(v\). If \(S\) is a maximum g-convex set of \(G\) not containing \(v\), then trivially \(S\cap V(i,G_{2})\) is a \(g_{2}\)-convex set in \(G_{2}\) not containing \(v\). Therefore \(S_{v}(G)=S\cup(V(G)-V(i,G_{2}))\).  
In the case of the corona, we prove that the g-centroid of \(G\) and \(G_{1}\) are the same.  
**Proposition 27**.: _Let \(G_{1}\), \(G_{2}\) be two connected graphs. Let \(G=G_{1}\circ G_{2}\). Then, \(C_{g}(G)=C_{g}(G_{1})\)._  
Proof.: Let \(x\in V(G_{1})\). Then \(w(x:G_{1})+n_{2}xw(x:G_{1})=w(x:G)(1+n_{2})\) (by (1) of the above proposition). If \(y\) is any vertex not in \(C_{g}(G_{1})\), then \(w(y:G)>w(x:G)\) (as \(w(y:G_{1})>w(x:G_{1})\)). If \(z\in V(i,G_{2})\), then \(V(G)-V(i,G_{2})\) is a convex set of \(G\) not containing \(z\) with cardinality \(n_{1}n_{2}+n_{1}-n_{2}\). Therefore \(w(z:G)\geq n_{1}n_{2}+n_{1}-n_{2}\). For any \(x\in C_{g}(G_{1})\), \(w(x:G)=w(x:G_{1})(1+n_{2})\leq(n_{1}-1)(1+n_{2})=n_{1}n_{2}-n_{2}+n_{1}-1<w(z)\). Thus \(w(z)>w(x)\). Therefore \(C_{g}(G)=C_{g}(G_{1})\).