‎Bucket recursive trees are an interesting and natural generalization of recursive trees‎. ‎In this model the nodes are buckets that can hold up to b≥ 1 labels‎. ‎The (modified) Zagreb index of a graph is defined as the sum of‎ ‎the squares of the outdegrees of all vertices in the graph‎. ‎We give the mean and variance of this index in random bucket recursive trees‎. ‎Also‎, ‎two limiting results on this index are given‎.

The main goal of this paper is to study the modified $F$-indices (modified first Zagreb index and modified forgotten topological index) of random $m$-oriented recursive trees (RMORTs). First, through two recurrence equations, we compute the mean and the variance of these indices in our random tree model. Second, we show four convergence in probability based on these indices. Third, the asymptotic normalities, through the martingale central limit theorem, are given.

Our main objective is to study some qualitative behavior of the solutions of the difference equation xn+1 = gxn−k + (axn + bxn−k) / (cxn − dxn−k), n = 0, 1, 2, ..., where the initial conditions x−k,..., x−1, x0 are arbitrary positive real numbers and the coefficients, a, b, c and d are positive constants, while k is a positive integer number.

In this paper it has been attempted to investigate the capability of the consumption-based capital asset pricing model (CCAPM), using the general method of moment (GMM), with regard to the Epstien-zin recursive preferences model for Iran's capital market. Generally speaking, recursive utility permits disentangling of the two psychologically separate concepts of risk aversion and elasticity of intertemporal substitution which are constrained to be equal to the inverse of each other for the traditional time-additive utility functions. Rather than using the stock market as a proxy for wealth, we constructed a more comprehensive return which is the weighted average of stock index return, labor wage growth (as a proxy for human capital return), housing return and deposit return. The empirical results demonstrate that the signs of the coefficient of the relative risk aversion and the intertemporal elasticity of substitution are the same, which means that investors have homogeneous attitudes toward the risk across the states of nature and the risk over time in Iran but different ones in their values. Therefore, the assumption that the relative risk aversion is equal to the reciprocal of the elasticity of substitution is not valid in Iran's stock market.

‎For a simple graph G‎, ‎the Gordon-Scantlebury index of G is equal to the number of paths of length two in G‎, ‎and the Platt index is equal to the total sum of the degrees of all edges in G‎. ‎In this paper‎, ‎we study these indices in random plane-oriented recursive trees through a recurrence equation for the first Zagreb index‎. ‎As n ∊ ∞, ‎the asymptotic normality of these indices are given‎.

If G is a connected graph with vertex set V, then the eccentric connectivity index of G, xc (G), is defined as ånÎV (G) deg (n) ecc (n) where deg (n) is the degree of a vertex n and ecc (n) is its eccentricity. In this paper we show some convergence in probability and an asymptotic normality based on this index in two classes of random bucket recursive trees.

In this paper, a new structure for image encryption using recursive cellular automata is presented. The image encryption contains three recursive cellular automata in three steps, individually. At the first step, the image is blocked and the pixels are substituted by reversible cellular automata. In the next step, pixels are scrambled by the second cellular automata and at the last step, the blocks are attached together and the pixels are substituted by the third cellular automata. Due to reversibility of cellular automata, the decryption of the image is possible by doing the steps reversely. The experimental results show that encrypted image is not understandable visually, also this algorithm has satisfactory performance of quantitative assessment from some other schemes.