2025/04/29 by Sun, Wensheng, Yang, Yujun, Xu, Shou-Jun
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.20448
Let G be a connected graph with n vertices. The resistance distance ΩG(i,j) between any two vertices i and j of G is defined as the effective resistance between them in the electrical network constructed from G by replacing each edge with a unit resistor. The resistance matrix of G, denoted by RG, is an n × n matrix whose (i,j)-entry is equal to ΩG(i,j). The resistance curvature κi in the vertex i is defined as the i-th component of the vector (RG)-11, where 1 denotes the all-one vector. If all the curvatures in the vertices of G are equal, then we say that G has constant resistance curvature. Recently, Devriendt, Ottolini and Steinerberger \citekde conjectured that the cycle Cn is extremal in the sense that its curvature is minimum among graphs with constant resistance curvature. In this paper, we confirm the conjecture. As a byproduct, we also solve an open problem proposed by Xu, Liu, Yang and Das \citekxu in 2016. Our proof mainly relies on the characterization of maximum value of the sum of resistance distances from a given vertex to all the other vertices in 2-connected graphs.