  Random walks have wide application in real lives, ranging from target search,
reaction kinetics, polymer chains, to the forecast of the arrive time of
extreme events, diseases or opinions. In this paper, we consider discrete
random walks on general connected networks and focus on the analysis of the
global mean first return time (GMFRT), which is defined as the mean first
return time averaged over all the possible starting positions (vertices),
aiming at finding the structures who have the maximal (or the minimal) GMFRT
among all connected graphs with the same number of vertices and edges. Our
results show that, among all trees with the same number of vertices, trees with
linear structure are the structures with the minimal GMFRT and stars are the
structures with the maximal GMFRT. We also find that, among all connected
graphs with the same number of vertices, the graphs whose vertices have the
same degree, are the structures with the minimal GMFRT; and the graphs whose
vertex degrees have the biggest difference, are the structures with the maximal
GMFRT. We also present the methods for constructing the graphs with the maximal
GMFRT (or the minimal GMFRT), among all connected graphs with the same number
of vertices and edges.
