For ??[0,1], let A?(G?)=?D(G)+(1??)A(G?), where G is a simple undirected graph, D(G) the diagonal matrix of its vertex degrees and A(G?) adjacency signed graph G? whose underlying G. In this paper, basic properties A?(G?) are obtained, positive semidefiniteness studied some bounds on eigenvalues derived—in particular, lower upper largest eigenvalue obtained.

Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha automorphism , then ({\mathbf v})$ also . Thus, it rather exceptional vertex-transitive to have multiplicity one. We study cubic graphs with nontrivial simple eigenvalue, discover remarkable connections arc-transitivity, regular maps, number theory.