On ABC energy and its application to anticancer drugs

Abstract

For a simple connected graph $ \Gamma $ with node set $ V(\Gamma) = \{w_{1}, w_{2}, \dots, w_{n}\} $ and degree sequence $ d_{i} $, the atom-bond connectivity ($ ABC $) matrix of $ \Gamma $ has an $ (ij) $-th entry $ \sqrt{\frac{d_{i}+d_{j}-2}{d_{i}d_{j}}} $ if $ w_{i} $ is adjacent to $ w_{j} $ and $ 0 $, otherwise. The multiset of all eigenvalues of $ ABC $ matrix is known as the $ ABC $ spectrum and their absolute sum is known as the $ ABC $ energy of $ \Gamma. $ Two graphs of same order are known as $ ABC $ equienergetic if they have the same $ ABC $ energy but share different $ ABC $ spectrum. We describe the $ ABC $ spectrum of some special graph operations and as an application, we construct the $ ABC $ equienergetic graphs. Further, we give linear regression analysis of $ ABC $ index/energy with the physical properties of anticancer drugs. We observe that they are better correlated with $ ABC $-energy.

Keywords

• adjacency matrix
• abc matrix
• atom-bond connectivity
• equienergetic graphs
• correlation