Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\Gamma$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(\Gamma)$ of $\Gamma$. We determine $S(\Gamma)$ of the generalized de Bruijn graphs $\Gamma=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $S(\mathrm{DB}(n,p))$ is isomorphic to the quotient by $\langle X\rangle$ of the centralizer of $X$ in $\mathrm{PGL}_n(p)$. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field $\mathbb{F}_{p^n}$ from spanning trees in $\mathrm{DB}(n,p)$.Comment: I+24 page

A REVIEW OF PROBABILISTIC GRAPH MODELS FOR FEATURE SELECTION WITH APPLICATIONS IN ECONOMIC AND FINANCIAL TIME SERIES FORECASTING

In every field of life, people are interested to be able to forecast future.  A number of techniques are available to predict and forecasting upto a certain level of accuracy. Many techniques involve statistical tools and techniques for forecasting, modeling and control. Use of statistical techniques is growing with time and new techniques are being developed very rapidly. Especially in the field of economics and finance, the estimation and forecasting of economic and financial indicators play a vital role in decision making. Many models are developed in the last 2 decades to get better accuracy and efficiency in time series analysis and still there is a scope of learning and getting betterment in this field is available. In this research we have reviewed probability graphs, directed acyclic graphs, Bayesian networks, feature selection algorithms and Markov blankets for time series forecasting on the economic and financial problems (like stock exchange forecasting, multi-objective business risk analysis, consumers’ analysis, portfolio optimization, credit scoring etc). This is a new dimension for adaptive modeling techniques in economics and finance modeling