Giving loans and issuing credit cards are two of the main concerns of banks in that they include the risks of non-payment. According to the Basel 2 guidelines, banks need to develop their own credit risk assessment systems. Some banks have such systems; nevertheless they have lost a large amount of money simply because the models they used failed to accurately predict customers' defaults. Traditionally, banks have used static models with demographic or static factors to model credit risk patterns. However, economic factors are not independent of political fluctuations, and as the political environment changes, the economic environment evolves with it. This has been especially evident in Iran after the 2008-2016 USA sanctions, as many previously reliable customers became unable to repay their debt (i.e., became bad customers). Nevertheless, a dynamic model that can accommodate fluctuating politicoeconomic factors has never been developed. In this paper, we propose a model that can accommodate factors associated with politico-economic crises. Human judgement is removed from the customer evaluation process. We used a fuzzy inference system to create a rule base using a set of uncertainty predictors. First, we train an adaptive network-based fuzzy inference system (ANFIS) using monthly data from a customer profile dataset. Then, using the newly defined factors and their underlying rules, a second round of assessment begins in a fuzzy inference system. Thus, we present a model that is both more flexible to politico-economic factors and can yield results that are max compatible with real-life situations. Comparison between the prediction made by proposed model and a real non-performing loan indicates little difference between them. Credit risk specialists also approve the results. The major innovation of this research is producing a table of bad customers on a monthly basis and creating a dynamic model based on the table. The latest created model is used for assessing customers henceforth, so the whole process of customer assessment need not be repeated. We assert that this model is a good substitute for the static models currently in use as it can outperform traditional models, especially in the face of economic crisis.

Communicated by J. Walker MSC: 05C40 05C45 16P10 16P40a b s t r a c t Let R be a commutative ring. The total graph of R, denoted by T (Γ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y ∈ R, are adjacent if and only if x + y ∈ Z (R), where Z (R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ (R)), be the induced subgraph of T (Γ (R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z (R) is not an ideal. In this paper we show that if T (Γ (R)) is a connected graph, then diam(Reg(Γ (R))) diam(T (Γ (R))). Also, we prove that if R is a finite ring, then T (Γ (R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.

Abstract. In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex ∆ is vertex decomposable if and only if I ∆ ∨ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of R/I ∆ , when ∆ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph G, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when G is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a C 5 -free vertex decomposable graph G, reg(R/I(G)) = c G , where c G is the maximum number of 3-disjoint edges in G. Moreover for this class of graphs we characterize pd(R/I(G)) and depth(R/I(G)). As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.