A note on neighborhood first Zagreb energy and its significance as a molecular descriptor

Abstract

The concept of graph spectra can be thought of as an approach to use linear algebra including, in particular, the well developed theory of matrices for unlocking a thousand secrets about graph theory and its applications. A novel neighborhood degree sum based matrix is proposed as a modification of classical adjacency matrix. Using the spectrum of this matrix, a graph energy and its Estrada index are introduced, and their role as a molecular structural descriptor in chemical graph theory is investigated. An algorithm is designed to make the computation of the energy and its Estrada index convenient. The relationship between the recently proposed matrix and its associated graph invariant is studied using the spectral moment. Several sharp bounds for spectral radius, energy, and Estrada index are computed, and the corresponding extremal graphs are characterized. The integral representation of the energy is also reported.