On the eigenvalues and spectral radius of starlike trees

Abstract

Let $$k\ge 1$$ and $$n_1,\ldots ,n_k\ge 1$$ be some integers. Let $$S(n_1,\ldots ,n_k)$$ be a tree T such that T has a vertex v of degree k and $$T{\setminus } v$$ is the disjoint union of the paths $$P_{n_1},\ldots ,P_{n_k}$$ , that is $$T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}$$ so that every neighbor of v in T has degree one or two. The tree $$S(n_1,\ldots ,n_k)$$ is called starlike tree, a tree with exactly one vertex of degree greater than two, if $$k\ge 3$$ . In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if $$k\ge 4$$ and $$n_1,\ldots ,n_k\ge 2$$ , then $$\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}$$ , where $$\lambda _1(T)$$ is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval $$(-2,2)$$ .