Game theoretic techniques have become deliberate with social network analysis. Studies show that contemporary approach on social network analysis is unable to collectively evaluate the rationality of individuals and synergies that occur between them. Thus, game theory has been selected as an alternate approach for social network analysis to overcome such shortcomings \cite{Narahari}. A field of social network analysis is to examine the strength of ties within a social group and this is referred to as \textit{social cohesion}.

The study of social groups and their tendency to stay in unity is highly correlated to interpersonal relationships and the benefits one can gain to remain in a group \textemdash \ whether it be monetary, popularity, social influence or social needs of an individual \cite{Wei}. Building upon this foundation, we design a type of coalitional game where the social influence rating of members is affected based on the affiliated type of network structure. We first define group cohesion and then assess cohesion on special classes of graphs via the core stability of a coalition. We then study the core stability of a special class of weighted graph followed by the implementation of weighted graphs as a regular expression which can be read by a finite automaton.