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

Optimal networks measured by global mean first return time