Why Deep Learning needs to be performed in Graphical representations? Why do we bother about devising new Deep Learning methods, which is to be applied on graphical representation of data? Why not use the traditional Deep Learning methods in the adjacency matrix representation of graphs?

Because traditional deep learning methods do not take into account a crucial property of the adjacency matrix representation: node permutation equivariance.
One axiom of deep learning is that inductive biases tend to help.
This was the case with convolutions in computer vision, recurrence in timeseries, trigonometric bases in implicit neural representations.
So it made sense to incorporate this into the architecture of Graph Neural Networks, in the hopes that it would help with performance and/or efficiency.
Some of these operations can also be directly derived from group theory (see Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges, 10.48550/arXiv.2104.13478 and Gauge Equivariant Convolutional Networks and the Icosahedral CNN, 10.48550/arXiv.1902.04615 for examples).
