The Number of Spanning Trees in Kn-complements of Quasi-threshold Graphs

TL;DR

This paper derives formulas for counting spanning trees in graphs formed by the complements of quasi-threshold graphs and trees, expanding understanding of their combinatorial properties.

Contribution

It introduces new formulas for the number of spanning trees in K_n-complements of quasi-threshold graphs, generalizing previous results.

Findings

Formulas for spanning trees in K_n-complements of quasi-threshold graphs

Extension of known results to broader graph classes

Application of complement spanning-tree matrix theorem

Abstract

In this paper we examine the classes of graphs whose $K_n$-complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph $H$ of $K_n$, the $K_n$-complement of $H$ is the graph $K_n-H$ which is obtained from $K_n$ by removing the edges of $H$. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form $K_n-H$ admitting formulas for the number of their spanning trees.