Edge-Defect Spectral Methods for Higher-Order Rankings of Spanning Tree Counts

Source abstract

In this paper, we study the higher-order ranking, by spanning-tree count, of graphs obtained from a complete graph by deleting a fixed number of edges. Using the edge-defect matrix determined by the deleted edges and the interaction number measuring the local overlap among them, we derive a stability inequality that quantitatively estimates the decrease in the number of spanning trees from the matching-deletion case. Combining this stability estimate with a classification of deletion graphs having small interaction number, we determine, up to isomorphism, the nine deletion graphs with the largest spanning-tree counts for $p\geq6$ and $n\geq2p$. We also clarify the relation between the interaction number, the local structure of the deletion graph, and the decrease in the number of spanning trees through a logarithmic expansion of the normalized spanning-tree count.