Edge-defect matrices and stability of the Kirchhoff index for complete graphs with deleted edges

Abstract

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.