# The G-Convexity and the G-Centroids of Composite Graphs
## 1 Introduction  
Graph theory is a branch of mathematics that has applications in every other field including Arts, Humanities, Sociology, Anthropology, and Engineering. Whenever there is a pairwise relation existing among any entities, graphs are then used to model the system. This paper deals with finding a central structure. The problem of finding a central structure is an active research area. They are classified under facility location problems. Using discrete mathematical models, centrality can be defined in several different ways. One of them is using the eccentricity of vertices. This concept uses the traditional Euclidean distance metric applied to the underlying graph. Centrality can also be defined through the topological properties of graphs. This paper deals with geodesic-convexity (g-convexity for short). g-convexity in graphs have been studied by several different authors. A good review is presented in our earlier paper [1]. In [1], we define the g-convex weight sequences for graphs based on g-convexity and provided characterization for certain classes of graphs including trees. A central structure called the g-centroid is defined through g-convexity. In another paper [2], the author demonstrated an application of g-convexity and g-centroids in Mobile Ad hoc Networks (MANET). They also find applications in other areas, including measuring dissimilarities in dynamical systems and dynamic search in graphs. In [3], [PERSON] constituted a first to a systematic study on the size of convex sets in graphs. Due to its application, the location of g-centroid for any arbitrary graphs as well as its characterization had gained importance. In [4], we have demonstrated that the g-centroidlocation problem is \(\mathcal{NP}\)-hard for disconnected graphs. In a later paper [2], [PERSON] closed the gap by proving \(\mathcal{NP}\)-hardness for connected graphs. As there are no tractable solutions available to locate the g-centroid for arbitrary graphs, it is essential to design an approximate solution for the location problem. A graph is defined as a composite graph if it can be obtained from two or more graphs through well-defined graph operations. Currently, there is no work available in the literature that analyze the relation between the structure of the g-centroid of a graph and its factors. In this paper, we study the structure of g-centroid and the g-convex weight sequences (gcws) of composite graphs through some popular graph operations such as graph-joins, cartesian products, and corona. Through the derived properties, we may able to design efficient approximate algorithms to locate the g-centroid for composite graphs in the future. In this paper, we make the following contributions. * We present an application of g-convexity and g-centroids in MANET. This forms a strong motivation. We then present an overview of the \(\mathcal{NP}\)-hardness proof for the g-centroid location for arbitrary graphs. * For the join of two graphs \(G_{1}\) and \(G_{2}\), the weight sequence of the composite graph does not depend on the weight sequences of its factors; rather it depends on the incident pattern of the maximum cliques of \(G_{1}\) and \(G_{2}\). We also characterize the structure of the g-centroid under various cases. * For the Cartesian product of \(G_{1}\) and \(G_{2}\) and the prism of a graph, we establish the relationship between the composite graph and its factors. * For Corona operations, we characterize the g-centroid based on its factors. The rest of the paper is organized as follows. In Section 2, we present preliminary definitions that are necessary for this paper. We also outline the \(\mathcal{NP}\)-hardness algorithm for the g-centroid location problem. In Section 3, we present our main results. We consider three important types of graph operations in this paper. They are Graph Joins, Cartesian Products, and Corona. As a special case of Cartesian products, we considered the prism of a graph. Section 4 deals with the Conclusion and Future direction. In this section, we present two important open problems.