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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.20448 |
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| _version_ | 1866911545508757504 |
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| author | Sun, Wensheng Yang, Yujun Xu, Shou-Jun |
| author_facet | Sun, Wensheng Yang, Yujun Xu, Shou-Jun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20448 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the minimum constant resistance curvature conjecture of graphs Sun, Wensheng Yang, Yujun Xu, Shou-Jun Combinatorics 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. |
| title | On the minimum constant resistance curvature conjecture of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.20448 |