Abstract
Chemical structures are transformed into real numbers employing topological indices, which enable numerical computations to substitute pricey wet lab experiments. The spectral properties of topological indices can be investigated by appropriate modification of adjacency matrix. Gutman and Trinajstic, pioneers in the field of chemical graph theory, presented the Zagreb indices, which have governed topological indices research since 1972. The modified adjacency matrix associated to the second Zagreb index (\( M_2 \)) is studied in this work. The second Zagreb energy (\( ZE_2 \)) is generated from this matrix. The application potential of \( ZE_2 \) is explored by investigating its efficiency in modelling physico-chemical properties of molecules and isomer discrimination. We aim to set up crucial bounds of spectral radius and spread with identifying extremal graphs. Moreover, the \( ZE_2 \) energy is investigated for numerous special graphs including regular, semi-regular and chain graphs. We construct a divisor type matrix to estimate the \( M_2 \)-spectrum and \( ZE_2 \) energy of the chain graph.
