Research On The Extremal Problems Of Eccentric Resistance-distance Sum In Graphs

The research on the parameters and the structures of graphs based on the resistance-distance originated from the study of electric network,and eventually de-veloped into an important direction of modern graph theory.For a simple connected graph G=(VG,EG),the eccentric resistance-distance sum of G,ξR(G),is defined as(?)Ruv,where εG(w)is the eccentricity of vertex w and Ruv is the resistance-distance between u and v in G.The main content of this paper is to study graphs whose eccentric resistance-distance sum can reach the extreme values in the two types of graphs.More details are as follows:·In Chapter 1,the background and significance of the research,and the current status of this aspect are introduced.·In Chapter 2,we explain some necessary notations and terminologies involved in this paper,then introduce somelemmas used in the proof below.·In Chapter 3,among the bipartite graphs of diameter 2,the graphs having the smallest and the largest eccentric resistance-distance sums are characterized,respectively.Among the bipartite graphs of diameter 3,the graphs having the smallest and second smallest eccentric resistance-distance sums are character-ized,respectively.·In Chapter 4,the n-vertex unicyclic graphs with girth k having the smallest and second smallest eccentric resistance-distance sums are identified,respectively.Furthermore,n-vertex unicyclic graphs having the smallest and second smallest eccentric resistance-distance sums are characterized,respectively.·In Chapter 5.we summarize the full paper and make prospects for further research.