Algebraic Equivalence of Signed Graphs with All Eigenvalues \(\geq -2\ )

We introduce a new concept called algebraic equivalence of sigraphs to study the family of sigraphs with all eigenvalues \(\geq -2\). First, we prove that any sigraph whose least eigenvalue is \(-2\) contains a proper subgraph such that both generate the same lattice in \({R}^n\). Next, we present a characterization of the family of sigraphs with all eigenvalues \(> -2\) and obtain Witt’s classification of root lattices and the well known theorem which classifies the first mentioned family by using root systems \(D_n, n \in {N} \) and \(E_8 \). Then, we prove that any sigraph whose least eigenvalue is less than \(-2\), contains a subgraph whose least eigenvalue is \(-2\). Using this, we characterize the families of sigraphs represented by the above root systems. Finally, we prove that a sigraph generating \(E_n\) ( \(n=7\) or 8) contains a subgraph generating \(E_{n-1}\) . In short, this new concept takes the central role in unifying and explaining various aspects of the theory of sigraphs represented by root systems and in giving simpler and shorter proofs of earlier known results including Witt’s theorem and also in proving new results.