On the minimum constant resistance curvature conjecture of graphs

TL;DR

The paper addresses the problem of minimizing the resistance curvature among graphs with constant resistance curvature, confirming the conjecture that the cycle $C_n$ is extremal. It develops a framework based on the resistance matrix $R_G$ and the system $R_G\kappa=\mathbf{1}$, employing elimination principles and Rayleigh monotonicity to relate resistive distances to graph structure. The main results show that graphs with constant resistance curvature are necessarily 2-connected, and among 2-connected graphs, $C_n$ uniquely minimizes curvature with $\mathcal{K}_G=\frac{6}{n^2-1}$, while the complete graph $K_n$ maximizes curvature with $\mathcal{K}_G=\frac{n}{2n-2}$. Additionally, the Kirchhoff index satisfies $Kf(G) \le \frac{n^3-n}{12}$ for 2-connected graphs, with equality for $C_n$, yielding sharp extremal characterizations of resistance-regular graphs in terms of both curvature and Kirchhoff index.

Abstract

Let $G$ be a connected graph with $n$ vertices. The resistance distance $Ω_{G}(i,j)$ 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 $R_G$, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $Ω_{G}(i,j)$. The resistance curvature $κ_i$ in the vertex $i$ is defined as the $i$-th component of the vector $(R_G)^{-1}\mathbf{1}$, where $\mathbf{1}$ 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 \cite{kde} conjectured that the cycle $C_n$ 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 \cite{kxu} 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.

Figures (2)

• Figure 1: The graph $G$ in the proof of Theorem \ref{['tm3.1']}.
• Figure 2: Illustration of vertex labeling of graphs $G$ and $C_n$ in Case 2.

Theorems & Definitions (14)

• Definition 1.1
• Conjecture 1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof