Abstract
The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal v-w entry is the weight of a minimum edge cut separating vertices v and w. Its computation is a classical topic of combinatorial optimization since at least the seminal work of Gomory and Hu. In this article, we investigate spectral properties of these matrices. In particular, we provide tight bounds on the smallest eigenvalue and the energy. Moreover, we study the eigenvector structure and show in which cases eigenvectors can be easily obtained from matrix entries. These results in turn rely on a new characterization of those nonnegative matrices that can actually occur as edge-connectivity matrices.
Highlights
IntroductionAn auxiliary result of Gomory and Hu, which is important for our investigation, is a characterization of those matrices that can occur as edge-connectivity matrices of Figure 1: A graph G for which the matrix P (G) + D(G) is not positive semidefinite weighted graphs
For an undirected graph G = (V, E) with nonnegative edge weights its edgeconnectivity matrix is the V ×V matrix C(G) whose off-diagonal v-w entry denotes the minimum weight of an edge set whose removal disconnects the vertices v and w
We have a lower bound on the smallest eigenvalue of an ultrametric distance matrix, which essentially relies on the following result of Zhan [17, Therorem 1] for general symmetric interval matrices
Summary
An auxiliary result of Gomory and Hu, which is important for our investigation, is a characterization of those matrices that can occur as edge-connectivity matrices of Figure 1: A graph G for which the matrix P (G) + D(G) is not positive semidefinite weighted graphs. In another article [13], Patekar and Shikare claim that P (G) + D(G) is positive semidefinite, where D(G) denotes the diagonal matrix of vertex degrees. This would immediately imply the conjecture about the energy. For graph theoretical terminology we refer to Diestel [3]
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