26,347 research outputs found
Factor-Critical Property in 3-Dominating-Critical Graphs
A vertex subset of a graph is a dominating set if every vertex of
either belongs to or is adjacent to a vertex of . The cardinality of a
smallest dominating set is called the dominating number of and is denoted
by . A graph is said to be - vertex-critical if
, for every vertex in . Let be a 2-connected
-free 3-vertex-critical graph. For any vertex , we show
that has a perfect matching (except two graphs), which is a conjecture
posed by Ananchuen and Plummer.Comment: 8 page
On Murty-Simon Conjecture II
A graph is diameter two edge-critical if its diameter is two and the deletion
of any edge increases the diameter. Murty and Simon conjectured that the number
of edges in a diameter two edge-critical graph on vertices is at most
and the extremal graph is the complete
bipartite graph .
In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al.
is not the original conjecture, indeed, it is only for the diameter two
edge-critical graphs of even order. In this paper, we completely prove the
Murty-Simon Conjecture for the graphs whose complements have vertex
connectivity , where ; and for the graphs whose
complements have an independent vertex cut of cardinality at least three.Comment: 9 pages, submitted for publication on May 10, 201
Quantum phase transitions in coupled two-level atoms in a single-mode cavity
The dipole-coupled two-level atoms(qubits) in a single-mode resonant cavity
is studied by extended bosonic coherent states. The numerically exact solution
is presented. For finite systems, the first-order quantum phase transitions
occur at the strong interatomic interaction. Similar to the original Dicke
model, this system exhibits a second-order quantum phase transition from the
normal to the superradiant phases. Finite-size scaling for several observables,
such as the average fidelity susceptibility, the order parameter, and
concurrence are performed for different interatomic interactions. The obtained
scaling exponents suggest that interatomic interactions do not change the
universality class.Comment: 13 pages, 5 figure
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