Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes

TL;DR

This paper extends the concept of bipartiteness from graphs to higher-dimensional simplicial complexes by characterizing disorientability through cycle structures in dual graphs, linking topology with spectral properties.

Contribution

It introduces a cycle-based characterization of disorientable simplicial complexes, generalizing bipartiteness to higher dimensions using Hodge Laplacian eigenvalues.

Findings

Disorientable simplicial complexes have no simple odd cycles in their down dual graph.

The maximum eigenvalue of the Hodge Laplacian relates to the absence of odd cycles.

Fewer odd cycles in the dual graph imply the eigenvalue approaches its maximum.

Abstract

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the construction of complex networks. Another key identifying feature is their Laplacian spectrum as bipartite graphs achieve the maximum possible eigenvalue of graph Laplacian. However, for modeling higher-order connections in complex systems, hypergraphs and simplicial complexes are required due to the limitations of graphs in representing pairwise interactions. In this article, using simple tools from graph theory, we extend the cycle-based characterization from bipartite graphs to those simplicial complexes that achieve the maximum Hodge Laplacian eigenvalue, known as disorientable simplicial complexes. We show that a $N$-dimensional simplicial complex is disorientable if its down dual graph contains no simple odd cycle of distinct edges and no twisted even cycle of distinct edges. Furthermore, we see that in a $N$-simplicial complex without twisting cycles, the fewer the number of (non-branching) simple odd cycles in its down dual graph, the closer is its maximum eigenvalue to the possible maximum eigenvalue of Hodge Laplacian. Similar to the graph case, the absence of odd cycles plays a crucial role in solving the bi-partitioning problem of simplexes in higher dimensions.