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Graphs with Diameter Minimizing the Spectral Radius
The spectral radius of a graph is the largest eigenvalue of its
adjacency matrix . For a fixed integer , let be
a graph with minimal spectral radius among all connected graphs on vertices
with diameter . Let be a tree
obtained from a path of vertices () by
linking one pendant path at for each . For
, were determined in the literature.
Cioab\v{a}-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed ,
is in the family . For , they conjectured
and
. In this paper, we settle
their three conjectures positively. We also determine in this
paper
On the singular hyperbolicity of star flows
We prove for a generic star vector field that, if for every chain
recurrent class of all singularities in have the same index, then
the chain recurrent set of is singular hyperbolic. We also prove that every
Lyapunov stable chain recurrent class of 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
PROFESSIONAL DEVELOPMENT OF ESP ENGLISH TEACHERS IN CHINESE COLLEGES AND UNIVERSITIES BASED ON EDUCATIONAL PSYCHOLOGY
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