Image captioning can be improved if the structure of the graphical representations can be formulated with conceptual positional binding. In this work, we have introduced a novel technique for caption generation using the neural-symbolic encoding of the scene-graphs, derived from regional visual information of the images and we call it Tensor Product Scene-Graph-Triplet Representation (TP$_{sgt}$R). While, most of the previous works concentrated on identification of the object features in images, we introduce a neuro-symbolic embedding that can embed identified relationships among different regions of the image into concrete forms, instead of relying on the model to compose for any/all combinations. These neural symbolic representation helps in better definition of the neural symbolic space for neuro-symbolic attention and can be transformed to better captions. With this approach, we introduced two novel architectures (TP$_{sgt}$R-TDBU and TP$_{sgt}$R-sTDBU) for comparison and experiment result demonstrates that our approaches outperformed the other models, and generated captions are more comprehensive and natural.