Narrow Results

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.

Let $G=(V,E)$ be a bipartite graph over the vertex set $V=\{1,\dots ,r\}$ and let $J=J(G)$ be the cover ideal of $G$ in the polynomial ring $R=K[x_1,\dots ,x_r]$. It is known that there are integers $b$ and $t_0$ such that $\mathrm{\text{reg}}{J}^t=d(J)t+b$ is a linear function in $t$ for all $t\ge t_0$. In this paper, we give effective bounds for $b$ and $t_0$.

In this paper, we present a method for finding the absorbing radical of a finite-dimensional evolution algebra. Such a method consists of finding the acyclic vertices of an oriented graph associated with the algebra. The set of generators associated with such vertices turn out to be the generators of the absorption radical. As an application we use the absorption radical to study the decomposability of some degenerate evolution algebras.

We study the transition graph of generic Hamiltonian surface flows, whose vertices are the topological equivalence classes of generic Hamiltonian surface flows and whose edges are the generic transitions. Using the transition graph, we can describe time evaluations of generic Hamiltonian surface flows (e.g., fluid phenomena) as walks on the graph. We propose a method for constructing the complete transition graph of all generic Hamiltonian flows. In fact, we construct two complete transition graphs of Hamiltonian surface flows having three and four genus elements. Moreover, we demonstrate that a lower bound on the transition distance between two Hamiltonian surface flows with any number of genus elements can be calculated by solving an integer programming problem using vector representations of Hamiltonian surface flows.

The IOTA Tangle, a Directed Acyclic Graph (DAG)-based distributed ledger, is popular for its scalability and suitability for IoT applications, offering fee-less transactions. A critical component of IOTA’s architecture is the Cumulative Weight Calculation (CWC), essential for its tip selection mechanism. This paper introduces an optimization of the IOTA Reference Implementation (IRI) CWC process originally implemented using Breadth-First Search (BFS) by employing Depth-First Search (DFS) and Iterative Deepening Search (IDS) algorithms. We present a comparative analysis of these methods, demonstrating that DFS and IDS provide significant improvements in computational efficiency, particularly beneficial for IoT devices with limited processing capabilities. Our findings are substantiated through a series of experiments on a Tangle snapshot, highlighting the enhanced performance and reduced resource utilization of the proposed methods. This study contributes to the ongoing development of DAG-based distributed ledgers, offering insights into more efficient algorithmic solutions for large-scale, decentralized networks.