On the Characteristic polynomial of ABS Matrix and ABS-Energy of Some Graphs

Abstract

For a graph $G$ with $n$ vertices and $m$ edges, Lin \textit{et al.} \cite{Lin} define the \textit{atom--bond sum-connectivity} ($ABS$) matrix of $G$ such that the $(i,j)^{\text{th}}$ entry is \[ \sqrt{1 - \frac{2}{d_i + d_j}} \] if vertex $v_i$ is adjacent to the vertex $v_j$, and $0$ otherwise. In this article, we determine the characteristic polynomial of the $ABS$ matrix for certain specific classes of graphs. Furthermore, we compute the $ABS$ eigenvalues and the $ABS$ energy for these classes.