Median Sets and Median Number of a Graph

A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances to the vertices of the profile is the median of the profile. Any subset of the vertex set such that it is the median of some profile is called a median set. The number of median sets of a graph is defined to be the median number of the graph. In this paper, we identify the median sets of various classes of graphs such as Kp−e, Kp,q for P>2, and wheel graph and so forth. The median numbers of these graphs and hypercubes are found out, and an upper bound for the median number of even cycles is established. We also express the median number of a product graph in terms of the median number of their factors.