Analysis of the connection between herd immunity and vaccination uses a model for spreading infectious diseases with injections vaccines in three forms. The results were analyzed using complete graph to determine the relationship of group immunity to vaccination under the three forms hypothesis of vaccinations against the spread of infectious diseases and drawing a graph from Gephi program. The results showed that herd immunity was directly related to the number of people vaccinated. If the number of people vaccinated is high herd immunity will occur. But if the number of people vaccinated is low herd immunity will be difficult.

An optimal of walking path to travel in Lopburi City was the shortest distance of path anduses least travel time. The research methodology used the survey of 10 real locations togetherwith viewing satellite maps and diagrams from Google earth.com for graphing and weighting of edges (distance time). The result analysis used the basic ideas of graph theory by finding the path that has all the vertex and used the idea of matrix by finding the adjacency matrix. The result of the research shown that an optimal of walking path to travel in Lopburi City that is to be as following Wat Puen Wichayen House Prang Khaek Somdet Phra Narai National Museum Lopburi Wat Phrasi Rattana Mahathat Wat Bandai Hin Wat Inthra Wat Nakorn Kosa Phra Kal Shrine Phra Prang Sam Yot with a total distance of 2090 meters in 25 minutes.