The Seidel Eigenvalues Polynomial and Spectrum of the Petersen Graph

For a simple undirected graph G, let $A\left(G\right)$ be the (0, 1) adjacency matrix of G. The Seidel matrix of G, is defined as $S\left(G\right)=J？I？2A\left(G\right)$ , where J is the all-one matrix and I is the identity matrix. The Seidel eigenvalues polynomial of the graph G is $S_G\left(\lambda \right)=\det \left(\lambda I？S\left(G\right)\right)$ . If all the Seidel eigenvalues of the graph G are integers, then G is called a Seidel integer graph. In this paper, we apply methods from algebraic and matrix theory to obtain the Seidel eigenvalue polynomials of the Petersen graph. Furthermore, we determine whether the Petersen graph is a Seidel integral graph.