Tapas Kumar Roy | Searcharxiv

arXiv2025

On robust toric ideals of weighted oriented graphs

In this work, we study the equivalence of various robustness properties of toric ideals of weighted oriented graphs. For any weighted oriented graph $D$, if its toric ideal $I_D$ is generalized robust (or weakly robust), then we show that $D$ does not have forbidden subgraphs $D_1,D_2$ of certain structures. We give a significant class of weighted oriented graphs $D$ whose toric ideals $I_D$ have the following equivalence. (i) $I_{D}$ is strongly robust (equivalently, $I_{D}$ is robust); (ii) $I_{D}$ is generalized robust (equivalently, $I_{D}$ is weakly robust); (iii) $D$ does not have subgraphs equal to $D_{1}$ and $D_{2}$.

math.AC↗

arXiv2024

On circuit binomials of toric ideals of weighted oriented graphs

In this work, we classify the circuit binomials of any weighted oriented graph $D$ and we explicitly compute the circuit binomials of $D$ in terms of the minors of the incidence matrix of $D$. We show that the circuit binomials of any weighted oriented graph $D$ are the primitive binomials corresponding to one of the classes: (i) a balanced cycle, (ii) two unbalanced cycles sharing a vertex, (iii) two unbalanced cycles connected by a path, (iv) two unbalanced cycles sharing a path. We explicitly prove a formula for the primitive binomial generator of the toric ideal $I_D$ in terms of the minors of the incidence matrix of $D$, where $D$ is as in (i), (ii), (iii) and (iv). Thus we explicitly compute all the circuit binomials $\C_D$ of any weighted oriented graph $D$. If $D$ is a weighted oriented graph which has at most two unbalanced cycles such that no two balanced cycles share a path in $D$ and no balanced cycle in $D$ shares an edge with the path which connects the two unbalanced cycles in $D$ if it exists, then we show that $I_D$ is a strongly robust circuit ideal and it has complete intersection initial ideal. For this class of ideals, we explicitly compute the Betti numbers.

math.AC↗

arXiv2024

Strongly robustness of toric ideals of weighted oriented graphs

In this article, we investigate the strongly robust property of toric ideals associated with weighted oriented graphs. We establish that the toric ideals of a broad class of monomial ideals are strongly robust; this class encompasses the edge ideals of weighted oriented graphs in which every edge is incident to a vertex of degree $2$.

math.AC↗