The vector autoregressive moving average (VARMA) model is one of the statistical analyses frequently used in several studies of multivariate time series data in economy, finance, and business. It is used in numerous studies because of its simplicity. Moreover, the VARMA model can explain the dynamic behavior of the relationship among endogenous and exogenous variables or among endogenous variables. It can also explain the impact of a variable or a set of variables by means of the impulse response function and Granger causality. Furthermore, it can be used to predict and forecast time series data. In this study, we will discuss and develop the best model that describes the relationship between two vectors of time series data export of Coal and data export of Oil in Indonesia over the period 2002-2017. Some models will be applied to the data: VARMA (1,1), VARMA (2,1), VARMA (3,1), and VARMA (4,1). On the basis of the comparison of these models using information criteria AICC, HQC, AIC, and SBC, it was found that the best model is VARMA (2,1) with restriction on some parameters: AR2_1_2 = 0, AR2_2_1 = 0, and MA1_2_1 = 0. The dynamic behavior of the data is studied through Granger causality analysis. The forecasting of the series data is also presented for the next 12 months.

Vector error-correction model (VECM) is a method of statistical analysis frequently used in many studies in time series data of economy, business and finance, and data energy. It is applied across researches due to its simplicity and limited restrictions. VECM can explain not only the dynamic behavior of the relationship among variables of endogenous and exogenous, but also among the endogenous variables. Moreover, it also explains the impact of a variable or a set of variables on others by means of impulse response function (IRF) and granger causality analysis. It can also be used for forecasting multivariate time series data. In this research, the relationship of three share price of energy (from three Asean countries: PGAS Malaysia, AKRA Indonesia, and PTT Thailand) will be studied. The data in this study were collected from October 2005 to August 2019. Based on the comparison of some VECM models, it was found that the best model is VECM (2) with cointegration rank = 3. The dynamic behavior of the data is studied through IRF, Granger Causality analysis and forecasting for the next five periods (weeks).

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Given n vertices and m edges, m ≥ 1, and for every vertex is given a label, there are lots of graphs that can be obtained. The graphs obtained may be simple or not simple, connected or disconnected. A graph G(V,E) is called simple if G(V,E) not containing loops nor paralel edges. An edge which has the same end vertex is called a loop, and paralel edges are two or more edges which connect the same set of vertices. Let N(G7,m,t) as the number of connected vertex labeled graphs of order seven with m vertices and t (t is the number edges that connect different pair of vertices). The result shows that N(G7,m,t) = ct C (m−1) t−1, with c6=6727, c7=30160 , c8=30765, c9=21000, c10=28364, c11=26880, c12=26460, c13=20790, c14=10290, c15= 8022, c16=2940, c17=4417, c18=2835, c19=210, c20= 21, c21=1.