For a graph \(G\) , \(\mathcal{E}\left(G\right)\) is defined as the sum of the absolute values of its eigenvalues. It is well-known that, if \(H\) and \(K\) are vertex induced subgraphs of \(G\) such that \(V\left(H\right)\) and \(V\left(K\right)\) partition \(V(G)\) , then \(\mathcal{E}\left(H\right)+\mathcal{E}\left(K\right)\le \mathcal{E}\left(G\right)\) . In this paper, we investigate the equality case. More precisely, we show that if \(\mathcal{E}\left(H\right)+\mathcal{E}\left(K\right)=\mathcal{E}\left(G\right)\) , then \(rank\left(H\right)+rank\left(K\right)\ge rank\left(G\right)\) and then we guess that \({\varvec{\rho}}\left(H\right)+{\varvec{\rho}}\left(K\right)\ge{\varvec{\rho}}\left(G\right)\) , where \({\varvec{\rho}}\left(G\right)\) denotes the spectral radius of \(G\) . Furthermore, we propose to extend the concept of graph energy to analyze the coefficient matrices of Linear Feedback Shift Registers (LFSRs). To date, there has been no research in this specific area. Coefficient matrices are generally very large. Establishing relationships between the energies of subgraphs (or submatrices) and the overall graph (or matrix) could provide valuable insights. These relationships may also simplify the computation of the energy of the main graph (matrix). This approach aims to extract various types of information from LFSR coefficient matrices by leveraging the properties of energy matrices.
