To any graph G can be associated a toric ideal I G. In this talk some recent joint work with Enrique Reyes and Christos Tatakis will be presented on the toric ideal of a graph. A characterization in graph theoretical terms of the elements of the Graver basis and the universal Gröbner basis of the toric ideal of a graph will be given. The Graver basis is the union of the primitive binomials of t...

Spiking neural P systems with anti-spikes (ASN P systems, for short) are a variant of spiking neural P systems, which are inspired by inhibitory impulses/spikes or inhibitory synapses. The typical feature of ASN P systems is when a neuron contains both spikes and anti-spikes, the spikes and anti-spikes will immediately annihilate each other in a maximal way. In this work, we consider a restrict...

Let G be a digraph with n vertices and A(G) be its adjacency matrix. A monic polynomial f(x) of degree at most n is called an annihilating polynomial of G if . 0 )) ( ( = G A f G is said to be annihilatingly unique if it possesses a unique annihilating polynomial. We prove that two families of digraphs, i.e., the ladder digraphs and the difans, are annihilatingly unique by studying the similari...

The rings considered in this article are commutative with identity which are not fields. Let R be a ring. A. Alilou, J. Amjadi and Sheikholeslami introduced and investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (0). They called this graph as a new graph as...

For a commutative ring R, we can form the zero-divisor graph Γ(R) or the ideal-divisor graph ΓI(R) with respect to an ideal I of R. We consider the diameters of direct products of zero-divisor and ideal-divisor graphs.

We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.