Abstract-Social networking sites always have a unique idea to make the connection between two people compact and fast. Famous social networking sites like Facebook, Twitter, LinkedIn, etc. show recommendations of the people and pages to follow, by comparing and connecting various other mutual connections. The recommendations are based on the number of mutual connections and mutual interests between one and the other person, more the number of mutual friends and interests, more the chances of recommendation. Each social media network has a unique set of technical elements, intricate logic, and usage analytics that make up the algorithm that delivers content to its users. Various optimization algorithms and logics are the base of this technology. Our project is based on the graphs and matrix data structures, which will be used to compare the interests of people which are one hop away from adjacent nodes and recommend them as friends suggestions based on their interests. This will be achieved by optimization and maximum weight algorithm. Keywords:Mutual Connection, Optimization Algorithms, Maximum weight graphs.