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Nowadays, the process of message encryption is the most important thing to secure our messages and communication between people. Message encryption methods are rapidly increasing currently due to the growth and evolution of the internet and network communications. Sharing information, personal messages, images, or data from one person to another over unsecured channels opens the door for attack, or hacking. To reduce this terminology and to provide better security, cryptographic or encryption techniques play an essential role. We have many kinds of symmetric enciphering techniques like the Caesar Cipher, Atbash Cipher, Hill Cipher, etc., In this paper, we are going to give the enciphering technique with the help of a complete graph, an adjacency matrix, and a generated self-invertible key matrix to encrypt and decrypt the given messages to produce a complicated ciphertext. Since we are using the self-invertible matrix as a key matrix, the inverse of this key matrix is always existing, and while we are decrypting the ciphertext, we do not need to compute the inverse of the key matrix. It helps us to reduce the computational complexity involved in the process of finding the inverse of a key matrix.