Mathematicians long to observe and analyze all sorts of patterns across a range of different fields. This sense of regularity is illustrated by linguists throughout the syntax of an incredibly large amount of human languages, however, becomes blurred by non-configurational languages- that is languages who do not seem to abide by a rigid word order on the surface level. Although, this is not at all a pleasing result, therefore this paper exemplifies through the use of spectral graph theory and in particular the concept of algebraic connectivity that these so called non-configurational languages do indeed display preference for a particular word order that obeys existing patterns in the syntax of human languages. This paper explores the intersection of spectral graph theory and the syntax of non-configurational languages, with a particular focus on the Nyulnyulan language, Bardi. The goal of this paper is to observe the nuances in the syntax of Bardi that are revealed by the application of spectral graph theory. The intersection of spectral graph theory and syntax offers a novel perspective on understanding the structure and properties of syntactic representations. By modeling syntactic trees and dependency structures as graphs, we can apply spectral techniques to analyze relationships between constituents and their hierarchical organization. The eigenvalues and eigenvectors of adjacency and Laplacian matrices derived from these syntactic graphs can reveal insights into grammatical patterns, such as language similarity, syntactic complexity, and the connectivity of linguistic constructs. As such, spectral graph theory serves as a powerful tool for linguists, providing quantitative methods to investigate the intricacies of language structure.

We show that subdivisions of binary trees satisfy a localized version of the Erdős–Pósa property. Our first result is that if a graph G contains no two disjoint subdivisions of a given binary tree B, then there exists a subgraph H of G isomorphic to a subdivision of B and a set X⊆V(H) such that G- X contains no subgraph isomorphic to a subdivision of B, and the size of X is bounded by an exponential function of |V(B)|. We then generalize this result to settings where G does not contain k vertex-disjoint subgraphs each isomorphic to a subdivision of B. In this case, we demonstrate the existence of a set X whose size depends on both |V(B)| and k, so that G- X is B-minor-free.