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

For a graph $G$ with $n$ vertices and $m$ edges, Lin $\mathit{\text{et}}\mathit{\text{al.}}$ \cite{Lin} define the $\mathit{\text{atom–bond}}\mathit{\text{sum-connectivity}}$ ($ABS$) 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.