Computing closeness for some graphs

The analysis of networks involves several crucial parameters. In this paper, we consider the closeness parameter, which is based on the total distance between every pair of vertices. Initially, we delve into a discussion about the applicability of the closeness parameter to Mycielski graphs. Our findings are categorized based on the underlying graph’s diameter. The formula for calculating the closeness of a Mycielski graph is derived for graphs with a diameter of less than or equal to 4. Furthermore, we establish a sharp lower bound for the closeness of a Mycielski graph when the diameter of the underlying graph is greater than 4. To achieve this, the closeness of the Mycielski transformation of a path graph plays an important role. Additionally, leveraging the obtained results, we examine the closeness of a special planar construction composed of path and cycle graphs, as well as its Mycielski transformation.