arXiv subjects

Naveena Ragunathan

Publications and source records attributed to Naveena Ragunathan.

3 recordsLinked to original sources

arXiv2026

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO↗

arXiv2026

Levelable graphs

We study a family of positive weighted well-covered graphs, which we call levelable graphs, that are related to a construction of level artinian rings in commutative algebra. A graph $G$ is levelable if there exists a weight function with positive integer values on the vertices of $G$ such that $G$ is well-covered with respect to this weight function. That is, the sum of the weights in any maximal independent set of vertices of $G$ is the same. We describe some of the basic properties of levelable graphs and classify the levelable graphs for some families of graphs, e.g., trees, cubic circulants, Cameron--Walker graphs. We also explain the connection between levelable graphs and a class of level artinian rings. Applying a result of Brown and Nowakowski about weighted well-covered graphs, we show that for most graphs, their edge ideals are not Cohen--Macaulay.

math.CO↗

arXiv2026

The van der Waerden Simplicial Complex and its Lefschetz Properties

The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A({\tt vdw}(n,k)) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A({\tt vdw}(n,k))$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A({\tt vdw}(n,3))$ that have the Weak Lefschetz Property when the characteristic is zero. We conjecture that $A({\tt vdw}(n,k))$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A({\tt vdw}(n,k))$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair.

math.AC↗