  We consider the density matrices derived from combinatorial laplacian matrix
of graphs. Specifically, the star-relevant graph, which means adding certain
edges on peripheral vertices of star graph, is the focus of this paper.
Initially, we provide the spectrum of the density matrices corresponding to
star-like graph(i.e., adding an edge on star graph) and present that the Von
Neumann entropy will increase under the graph operation(adding an edge on star
graph) and the graph operation cannot simulated by local operation and
classical communication (LOCC). Subsequently, we illustrate the spectrum of
density matrices corresponding to star-alike graph(i.e, adding one edge on
star-like graph) and exhibit that the Von Neumann entropy will increase under
the graph operation(adding an edge on star-like graph)and the graph operation
cannot simulated by LOCC. Finally, the spectrum of density matrices
corresponding to star-mlike graph(i.e.,adding $m$ nonadjacent edges on the
peripheral vertices of star graph) is demonstrated and the relation between the
graph operation and Von Neumann entropy, LOCC is revealed in this paper.
