The smallest positive eigenvalue of graphs under perturbation

Abstract

The effect on the smallest positive eigenvalue of a bipartite graph is studied when the graph is perturbed by attaching a pendant vertex at one of its vertices. Let $${\widehat{T}}(v)$$ be the graph obtained by attaching a pendant at vertex v of T. We characterize the vertices v such that the smallest positive eigenvalue of $${\widehat{T}}(v)$$ is equal or greater than that of T. As an application, we obtain the pairs of nonisomorphic noncospectral trees having the same smallest positive eigenvalue.