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

Stavros D. Nikolopoulos·Charis Papadopoulos
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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.

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Stavros D. Nikolopoulos, Charis Papadopoulos. 2005-02-07. The Number of Spanning Trees in Kn-complements of Quasi-threshold Graphs. https://arxiv.org/abs/cs/0502038

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