On the $\mathcal{A}\mathcal{B}\mathcal{S}$ spectrum and energy of graphs

Let ${\eta }_1\ge {\eta }_2\ge \cdots \ge {\eta }_n$ be the eigenavalues of $\mathcal{A}\mathcal{B}\mathcal{S}$ matrix. In this paper, we characterize connected graphs with $\mathcal{A}\mathcal{B}\mathcal{S}$ eigenvalue ${\eta }_n>-1$. As a result, we determine all connected graphs with exactly two distinct $\mathcal{A}\mathcal{B}\mathcal{S}$ eigenvalues. We show that a connected bipartite graph has three distinct $\mathcal{A}\mathcal{B}\mathcal{S}$ eigenvalues if and only if it is a complete bipartite graph. Furthermore, we present some bounds for the $\mathcal{A}\mathcal{B}\mathcal{S}$ spectral radius (resp. $\mathcal{A}\mathcal{B}\mathcal{S}$ energy) and characterize extremal graphs. Also, we obtain a relation between $\mathcal{A}\mathcal{B}\mathcal{C}$ energy and $\mathcal{A}\mathcal{B}\mathcal{S}$ energy. Finally, the chemical importance of $\mathcal{A}\mathcal{B}\mathcal{S}$ energy is investigated and it shown that the $\mathcal{A}\mathcal{B}\mathcal{S}$ energy is useful in predicting certain properties of molecules.