Grothendieck ring class of Banana and Flower graphs

TL;DR

This paper introduces a method to associate hypersurface varieties to connected planar graphs and computes their classes in the Grothendieck ring, providing a new algebraic perspective on graph structures.

Contribution

It defines a class of hypersurface varieties from graphs and characterizes when these are irreducible, advancing the algebraic understanding of graph-related geometric objects.

Findings

Computed Grothendieck ring classes for hypersurfaces from graphs

Characterized conditions for irreducibility of these hypersurfaces

Linked graph properties to algebraic geometric invariants

Abstract

We define a special type of hypersurface varieties inside $\mathbb{P}_k^{n-1}$ arising from connected planar graphs and then find their equivalence classes inside the Gr\"othendieck ring of projective varieties. Then we find a characterization for graphs in order to define irreducible hypersurfaces in general.