The atom bond connectivity (ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes, which is defined as ABC = uv∈E(G) du+dv −2 dudv , we denotes du the degree of vertex u in G. In this paper, we study the general formula for ABC index of certain graphs and vertex gluing of graphs.

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Blockchain technology considers the central technology that is used within many applications used frequently with human life. And the primary core of the blockchain is the consensus algorithm which may affect the security of the chain as well as the required resource consumption which affect mainly the blockchain performance directly. In recent years many consensus algorithms have been used and proposed such as proof of work (PoW) and proof of stake (PoS) and many others. However, these algorithms still need some improvement to the security and system resource consumption which will reduce the need for a huge amount of energy and save the environment as well as let the blockchain be useable within low computation ability devices such as the internet of things devices (IoT). This paper proposes a new consensus algorithm that ensures the integrity and authorization of nodes participating in the validation of the transaction and only a predefined number of nodes chosen randomly to participate in block addition which reduces the need for high computations power for mining and voting. The proposed algorithm needs lower time and computation costs comparable to the standard POW algorithm.

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