The application of bipartite and regular graphs plays a vital role in the area of engineering, mathematical sciences, design of experiments, and medical fields. This study proposes an optimization method for constriction of randomized block design and Latin square design using bipartite and regular graphs with applications of warping copper plates in specimens and comparing them to burners and engines on different days. The construction methods and analysis are performed as follows: the first method is a construction of randomized block design using bipartite and complete bipartite graphs with applications for the amount of warping copper plates and different laboratories are taken to test any significant difference that exists between the mean number of responses for the labs and copper specimens. The second method is the construction of a Latin square design using regular graphs to test whether there is any significant difference between the burners, engines, and some days in statistical analysis of interaction plots, contour plots, and 3D surface plots.

Background: The main key input variables to this optimization technique for constructing incomplete block designs are using bipartite and spanning subgraphs through numerical examples of vehicle fuel consumption and emissions. The theory of graphs plays a significant role in mathematical sciences and engineering technologies. In addition, the graph models many relations and processes in physical, biological, social, and information systems. Aims and Objectives: The construction methods using Partially Incomplete Block Designs (PBIBD) with differential equations through bipartite and spanning subgraphs that predict hot stabilized vehicle fuel consumption and emission rates for different drivers using different cars are studied in this paper. The other modelling of fuel consumption and emissions have appeared as an essential tool, which helps to develop and measure vehicle techniques and to help estimate vehicle fuel consumption and emissions. This paper aims to develop an optimization technique for the construction method for incomplete block designs LSD with PBIBD(2) through vehicle fuel consumption and emissions. Method: An incomplete block design can be constructed using LSD statistical analysis with bipartite and spanning subgraphs. First, the method for the construction of LSD using bipartite graphs. The second method for the construction of PBIBD(m) using spanning subgraphs. The two construction methods are through numerical examples of an oil company testing five mixings of petrol for fuel efficiency and emission according to the variability of five drivers and five models of cars. Result: The inference of the first model of PBIBD(2) using LSD F-value of 0.08 implies the model is not significant (P-values greater than 0.05). The second model has no significant difference between petrol fuel efficiency and emissions. In the third model, there is no significant difference in fuel efficiency between different cars of petrol bunks. Conclusion: Finally, it is concluded that the response variable is represented above the maximum quality scores from our fourth driver using a second car to the fourth petrol bunk in fuel efficiency.