Maurizio Imbesi

Monomial Ideals of Graphs with Loops

Abstract We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the polynomial ring related to these graphs modulo such ideals. Moreover we are able to determine the structure of the ideals of vertex covers for the edge ideals associated to the previous classes of graphs which can have loops on any vertex. Lastly, it is shown that these ideals are of linear type.

Edge ideals and connection problems

Ideals arising from graphs are investigated via Grobner bases theory in order to introduce algebraic objects useful for applications related to the field of security. In particular, the notion of s-sequence for the generators of the edge ideal I(G) of any graph G is considered, so that a description of the Grobner basis for the relation ideal J of the symmetric algebra of I(G) can be obtained. Based on this approach, the initial ideal of J with respect to a monomial order is well-determined and defines the edge ideal of a supporting graph F, essential in transmitting.

On the graphic realization of certain monomial sequences

d-sequences and s-sequences are important monomial sequences. In this paper, graphs that have edge ideals with generators forming d-sequences or s-sequences are examined. Two equivalent conditions are given.

On the symmetric algebras associated to graphs with loops

We study the symmetric algebra of monomial ideals that arise from graphs with loops. The notion of s-sequence is investigated for such ideals in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs.

Existence of three weak solutions for a perturbed anisotropic discrete Dirichlet problem

ABSTRACT In this paper, we study the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. Some examples are presented to demonstrate the application of our main results.

Multiple solutions for partial discrete Dirichlet problems depending on a real parameter

In this paper we establish the existence of multiple solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, under appropriate assumptions on the nonlinearities, we determine exact collections of parameters such that the treated problems admit at least three solutions.

On A Class of Vertex Cover Ideals

We investigate vertex cover ideals associated to a significative class of connected graphs. These ideals are stated to be Cohen-Macaulay and, using the notion of linear quotients, standard algebraic invariants for them are computed. Mathematics Subject Classification 2010: 05C25, 05E40, 13C14, 90B10.

Integrality of the symmetric algebra of graph ideals

We consider edge ideals associated to some classes of simple graphs and study the projective dimension and the integrality of the symmetric algebra for them. We also analyze criteria for torsion freeness of the symmetric powers and determine conditions of acyclicity of the Z-complex of these graph ideals.