hub number of incidence and power graph

In graph theory, a set h ⊆ v(g) is defined as a hub set if every pair of non-adjacent vertices outside h can be interconnected by a path that exclusively traverses through the internal vertices contained in h. the hub number of a graph g refers to the minimal cardinality of such a hub set, providing crucial insights into the structural connectivity of the graph. this paper delves into the exploration of the hub number across various graph structures, specifically focusing on incidence graphs and square graphs, both of which possess unique characteristics impacting their connectivity properties. we establish theoretical bounds for the hub numbers of these graphs, facilitating a clearer understanding of their structural complexities. furthermore, we derive explicit values for the hub numbers of several special types of graphs, including path graphs, star graphs and complete graphs. through rigorous analysis and evaluation, this study contributes to the broader field of connectivity in graphs by not only identifying the hub numbers for specific examples but also by proposing methodologies for their computation. these findings have important implications for applications in network design and graph optimization, enhancing the utility of hub sets in practical scenarios.