图的加权基尔霍夫指数的组合解释
Combinatorial explanation of the weighted Kirchhoff index of graphs
AI总结：
本文对任意连通图的加权基尔霍夫指数给出组合解释，通过加权细分图及其子图的匹配权重和表示该指数，解决了相关文献提出的问题，特例可恢复树等图的对应指数公式。
AI中文摘要：
设G是顶点集为V(G)={v₁,v₂,…,vₙ}的连通图，ω:V(G)→ℝ⁺是满足ω(vᵢ)=xᵢ的正顶点权重函数，其中每个vᵢ∈V(G)。G的加权基尔霍夫指数定义为K(G;x₁,x₂,…,xₙ)=∑₁≤i<j≤n xᵢxⱼr_G(vᵢ,vⱼ)，这里r_G(vᵢ,vⱼ)表示vᵢ与vⱼ之间的电阻距离。本文对任意连通图的加权基尔霍夫指数给出组合解释，更确切地说，我们将K(G;x₁,x₂,…,xₙ)表示为G的适当加权细分图中匹配的权重和，以及从该加权细分图中删除对应于G的2-正则子图的细分图后得到的子图中的匹配权重和。这对Li、Li和Yan在《Discrete Math. 345 (2022) 113109》中提出的问题给出了肯定回答，该问题是关于使用加权细分图及其子图中的匹配来对一般图的加权基尔霍夫指数进行组合解释。作为特例，我们的公式恢复了树和单环图的加权基尔霍夫指数的已知公式，以及任意连通图的普通基尔霍夫指数的已知公式。
英文摘要：
Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\ldots,v_n\}$, and let $ω:V(G)\to \mathbb R^+$ be a positive vertex-weight function satisfying $ω(v_i)=x_i$ for each $v_i \in V(G)$. The weighted Kirchhoff index of $G$ is defined by $K(G;x_1,x_2,\ldots,x_n)=\sum_{1\le i<j\le n}x_i x_j r_G(v_i,v_j)$, where $r_G(v_i,v_j)$ denotes the resistance distance between $v_i$ and $v_j$. In this paper, we give a combinatorial interpretation of the weighted Kirchhoff index of an arbitrary connected graph. More precisely, we express $K(G;x_1,x_2,\ldots,x_n)$ in terms of the sums of weights of matchings in an appropriately weighted subdivision graph of $G$, and in the subgraphs obtained from this weighted subdivision graph by deleting the subdivision graphs corresponding to \(2\)-regular subgraphs of $G$. This gives an affirmative answer to a question posed by Li, Li and Yan [Discrete Math. 345 (2022) 113109] concerning a combinatorial explanation of the weighted Kirchhoff index of a general graph by using matchings in weighted subdivision graphs and their subgraphs. As special cases, our formula recovers the known formulas for the weighted Kirchhoff index of trees and unicyclic graphs, as well as the known formula for the ordinary Kirchhoff index of an arbitrary connected graph.
