The Bonferroni mean ($BM$) operator has extended the class of interrelationship handling fusion functions by modeling homogeneous pairwise interactions among data entities. This operator has been further extended to diverse directions with the intention of capturing dissimilar association that exists within data sets of different real-world systems like, social network systems, biological systems, etc. It has been observed that some of the existing forms of the $BM$ operator pre-assumes a specific model of association among data entities during its formulation, which may not be feasible in many systems. In this study, an effort has been made to develop the framework of the $BM$ operator by generalizing the structure of relationship patterns among data entities\color{black}. Classical graphs are considered as a prominent tool of describing pairwise relations among data entities. Such consideration prompted the proposal of a systematized framework of an improved version of the $BM$ operator, denoted by the $f_G^{BM}$ operator, where the unconventional association among data entities are portrayed through different graphical patterns. The $f_G^{BM}$ operator has been formulated in a way that the knowledge of interactional information depicted through graphs is embedded into its processing system with the aim of capturing precise interconnections among entities. The generalized variation of the $f_G^{BM}$ operator has also been proposed by substituting the sub-components of the $f_G^{BM}$ operator with other precise forms of the aggregation functions to provide an illustrative alignment, which is quite expressible and interpretable, and also facilitates modeling mandatory prerequisites of the decision systems. For an applicatory aspect, the proposed operators have been utilized over the Hacker attack system and have been presented with a numerical example. A detailed comprehensive analysis has been presented to demonstrate the efficiency of the proposed operators.