The spectral radius Ο(G) of a graph G is the largest eigenvalue of its
adjacency matrix A(G). For a fixed integer eβ₯1, let Gn,nβeminβ be
a graph with minimal spectral radius among all connected graphs on n vertices
with diameter nβe. Let Pn1β,n2β,...,ntβ,pm1β,m2β,...,mtββ be a tree
obtained from a path of p vertices (0βΌ1βΌ2βΌ...βΌ(pβ1)) by
linking one pendant path Pniββ at miβ for each iβ{1,2,...,t}. For
e=1,2,3,4,5, Gn,nβeminβ were determined in the literature.
Cioab\v{a}-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed eβ₯6,
Gn,nβeminβ is in the family Pn,eβ={P2,1,...1,2,nβe+12,m2β,...,meβ4β,nβeβ2ββ£2<m2β<...<meβ4β<nβeβ2}. For e=6,7, they conjectured
Gn,nβ6minβ=P2,1,2,nβ52,β2Dβ1ββ,Dβ2β and
Gn,nβ7minβ=P2,1,1,2,nβ62,β3D+2ββ,Dββ3D+2ββ,Dβ2β. In this paper, we settle
their three conjectures positively. We also determine Gn,nβ8minβ in this
paper
We prove for a generic star vector field X that, if for every chain
recurrent class C of X all singularities in C have the same index, then
the chain recurrent set of X is singular hyperbolic. We also prove that every
Lyapunov stable chain recurrent class of X is singular hyperbolic. As a
corollary, we prove that the chain recurrent set of a generic 4-dimensional
star flow is singular hyperbolic.Comment: 29 pages, version to appear in J. Mod. Dy