The present-day society depends heavily on digital technology where it is used in many applications such as banking and e-commerce transactions, computer passwords, etc. Therefore, it is important to protect information when storing and sharing them. Cryptography is the study of secret writing which applies complex math rules to convert the original message into an incomprehensible form. Graph theory is applied in the field of cryptography as graphs can be simply converted into matrices There are two approaches of cryptography; symmetric cryptography and asymmetric cryptography. This paper proposes a new connection between graph theory and symmetric cryptography to protect the information from the unauthorized parties. This proposed methodology uses a matrix as the secret key which adds more security to the cryptosystem. It converts the plaintext into several graphs and represents these graphs in their matrix form. Also, this generates several ciphertexts. The size of the resulting ciphertexts are larger than the plaintext size.