The Main Eigenvalues of the Seidel Matrix

Houqing Zhou

Let $G$ be a simple graph with vertex set $V(G)$ and $(0,1)$-adjacency matrix $A$. As usual, $A^{\ast}(G) = J-I-2A$ denotes the Seidel matrix of the graph $G$. The eigenvalue $\lambda$ of $A$ is said to be a main eigenvalue of $G$ if the eigenspace $\varepsilon(\lambda)$ is not orthogonal to the all-1 vector $\mathbf{e}$. In this paper, relations between the main eigenvalues and associated eigenvectors of adjacency matrix and Seidel matrix of a graph are investigated.