AI 中文总结
本文针对删除p条边的完全图Kₙ-F，定义边缺陷矩阵Q=BᵀB，推导有效电阻、基尔霍夫指数与生成树数量的公式，证明匹配删除时基尔霍夫指数最小，并将公式应用于多种删除图。
AI 中文摘要
本文研究从完全图中删除p条边构成的集合F后得到的连通图Kₙ-F的有效电阻、基尔霍夫指数和生成树的数量。设B为删除边的关联矩阵，我们将矩阵Q=BᵀB称为边缺陷矩阵，这是一个p×p矩阵，带符号地记录了删除边共享其端点的方式。首先，我们根据边缺陷矩阵的预解式推导了任意两个不同顶点之间有效电阻的公式，这将通常使用n×n拉普拉斯矩阵的计算简化为使用与删除边数量对应的p×p矩阵的计算。此外，通过使用同一矩阵的特征值，我们给出了基尔霍夫指数和生成树数量的统一公式。接下来，我们推导了一个稳定性恒等式，该恒等式精确描述了Xu、Das和Zhang型下界的超出量。由此，我们表明，在可实现匹配删除的范围内，当删除边集是匹配时，基尔霍夫指数最小。此外，通过使用优化理论中的优超关系，我们证明，对于p≥2且n≥max{4,2p-1}，在所有非匹配删除边集中，最小值仅当删除图同构于P₃∪(p-2)K₂时取得。最后，我们将得到的公式应用于几种删除图，例如匹配、星图、团、路径和环。
英文摘要
In this paper, we study the effective resistance, the Kirchhoff index, and the number of spanning trees of the connected graph $K_n-F$, which is obtained from the complete graph by deleting a set $F$ of $p$ edges. Let $B$ be the incidence matrix of the deleted edges. We call the matrix $Q=B^TB$ the edge-defect matrix. This is a $p\times p$ matrix which records, with signs, the way in which the deleted edges share their end vertices. First, we derive a formula for the effective resistance between any two distinct vertices in terms of the resolvent of the edge-defect matrix. This reduces the usual computation using the $n\times n$ Laplacian matrix to a computation using a $p\times p$ matrix corresponding to the number of deleted edges. Moreover, by using the eigenvalues of the same matrix, we give unified formulas for the Kirchhoff index and the number of spanning trees. Next, we derive a stability identity which exactly describes the excess from the Xu, Das, and Zhang type lower bound. As a consequence, we show that, in the range where a matching deletion can be realized, the Kirchhoff index is minimized when the deleted edge set is a matching. Furthermore, by using majorization, we prove that, for $p\ge 2$ and $n\ge \max\{4,2p-1\}$, among all non-matching deleted edge sets, the minimum is attained only when the deletion graph is isomorphic to $P_3\cup(p-2)K_2$. Finally, we apply the obtained formulas to several deletion graphs, such as matchings, stars, cliques, paths, and cycles.
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