The effect on the spectral radius by attaching a pendant starlike tree.

A tree T with exactly one node u of degree larger than two is called a starlike tree and u is called the root node of T. Let S T n be a family of graphs consists of all starlike trees and the path with n + 1 nodes. For a fixed connected graph H with at least one edge, we construct a family of graphs F H (v) , n = { H (v) • T | T ∈ S T n } , where H (v) • T denotes the graph obtained by identifying some node, say v , of H with the root node of T. In this paper, we give an ordering of graphs in F H (v) , n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths. • The spectral radius of the graph powerfully characterizes dynamic processes on networks, such as virus spread and synchronization. • In this paper, we give an ordering of graphs in F H (v) , n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths. • The result generalizes the main result of Oliveira et al. in [10]. [ABSTRACT FROM AUTHOR]