The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles. A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph's edge multiplicities.

Abstract. The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, Γ ≤k , are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for Γ ≤2 , and we use this forbidden subgraphs to get a classification of the graphs in Γ ≤2 . As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the end of this article we give two infinite families of forbidden subgraphs.

Recently, there have been found new relations between the zero forcing number and the minimum rank of a graph with the algebraic co-rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with at most minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank.

The detection of copy-move image tampering is of paramount importance nowadays, mainly due to its potential use for misleading the opinion forming process of the general public. In this paper, we go beyond traditional forgery detectors and aim at combining different properties of copy-move detection approaches by modeling the problem on a multiscale behavior knowledge space, which encodes the output combinations of different techniques as a priori probabilities considering multiple scales of the training data. Afterward, the conditional probabilities missing entries are properly estimated through generative models applied on the existing training data. Finally, we propose different techniques that exploit the multi-directionality of the data to generate the final outcome detection map in a machine learning decision-making fashion. Experimental results on complex data sets, comparing the proposed techniques with a gamut of copy-move detection approaches and other fusion methodologies in the literature, show the effectiveness of the proposed method and its suitability for real-world applications.

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