Arithmetical rank of the cyclic and bicyclic graphs

TL;DR

This paper investigates the algebraic properties of edge ideals of certain cyclic and bicyclic graphs, establishing that their arithmetical rank equals the projective dimension for these classes.

Contribution

It proves that for graphs with one or two cycles connected via a vertex or path, the arithmetical rank matches the projective dimension, extending known results to new graph classes.

Findings

Arithmetical rank equals projective dimension for these graphs.

Results apply to graphs with cycles connected through vertices or paths.

Provides algebraic invariants for specific cyclic and bicyclic graphs.

Abstract

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension.