Graph theory is quickly becoming an important area of research due to its numerous applications in areas like cryptography, coding theory, communication networks and their security. In this paper a super-encryption method of graphs is proposed via matricial approach for maintaining confidentiality in communicating the messages with two layers of encryption by taking the first level of encryption as the incidence matrix. The message is further encrypted using Fibonacci numbers (taken in matrix form) by adding to the corresponding elements of the product of the incidence matrix of the graph and message (taken as diagonal matrix). Here the graph can be taken as private key. The matrix so formed can be treated as an adjacency matrix which in turn can be represented as a graph and will be sent to the receiver as super encrypted message. The receiver writes the adjacency matrix corresponding to the graph, then subtracts the Fibonacci matrix from adjacency matrix and then multiplies it with the inverse of incidence matrix. Finally, receiver decrypts the message using the private key shared.