On a magneto-spectral invariant on finite graphs

TL;DR

This paper introduces a new magneto-spectral invariant for finite graphs that distinguishes graph classes, relates to matrix theory, and exhibits specific behaviors under graph operations and spectral properties.

Contribution

The paper defines a novel magneto-spectral invariant for finite graphs, computes it for various graph families, and explores its bounds, relations, and applications in graph theory and matrix analysis.

Findings

Invariant vanishes on trees and is maximized by complete graphs.

Provides sharp bounds for regular bipartite graphs.

Can distinguish certain cospectral graphs.

Abstract

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-\'Emery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap.