Narrow Results

We analyze electrically pumped atomic cavity coupled to a two-mode vacuum reservoirs via a single-port mirror whose open cavity contains N nondegenerate three-level cascade atoms. We carry out our analysis by putting the noise operators associated with a vacuum reservoir in normal order. It is found that unlike the mean photon number, the quadrature squeezing and the degree of entanglement do not depend on the number of atoms. This implies that the quadrature squeezing and the degree of entanglement of the cavity light do not depend on the number of photons. We have also shown that the light generated by the three-level laser is in a squeezed and entangled state, with maximum quadrature squeezing and degree of entanglement being 50%. Moreover, the mean photon number of the system in which the laser operating at threshold and above threshold does not depend on the spontaneous decay constant.

For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P, φ), extending the poset Clop(P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,

• Reg(P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.

• If Reg(P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.

• Clop(P, φ) is a lattice if and only if every regular closed set is clopen.

The extended permutohedron R(G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R(G) and Reg S are bounded homomorphic images of free lattices.

Let G be a finite group and H be a subgroup of G. We introduce the non-normal graph of H in G, denoted by _H,G, and give some of the graph theoretical properties of _H,G.

Merging the first and third classes in a connected graph is the operation of adding edges between all vertices at distance 3 in the original graph while keeping the original edges. We determine when merging the first and third classes in a bipartite distance-regular graph produces a distance-regular graph.

The permutability graph of subgroups of a given group G, denoted by Γ(G), is a graph with vertex set consists of all the proper subgroups of G and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are one of bipartite, star graph, C₃-free, C₅-free, K₄-free, K₅-free, K_1,4-free, K_2,3-free or P_n-free (n = 2, 3, 4). We investigate the same for infinite groups also. Moreover, some results on the girth, completeness and regularity of the permutability graphs of subgroups of groups are obtained. Among the other results, we characterize groups Q₈, S₃ and A₄ by using their permutability graphs of subgroups.

A simple graph $G$ is called $N$-bounded if for every two nonadjacent vertices $x,y$ of $G$ there exists a vertex $z$ such that $N(x)\cup N(y)\subseteq N(z)\cup \{z\}$, where $N(v)$ denotes the set of neighbors of the vertex $v$ in $G$. In this paper, some properties of $N$-bounded graphs are studied. It is shown that any bipartite $N$-bounded graph is a complete bipartite graph with at most two horns; in particular, any $N$-bounded tree is either a star or a two-star graph. Also, we prove that any non-end vertex of every $N$-bounded graph is contained in either a triangle or a rectangle. Among other results, it is shown that all regular $N$-bounded graphs are strongly regular graphs. Finally, we determine that how many edges can an $N$-bounded graph have?

An edge labeling of a connected graph $G=(V,E)$ is said to be local antimagic if it is a bijection $f:E\to \{1,\dots ,|E|\}$ such that for any pair of adjacent vertices x and y, $f^{+}(x)\ne f^{+}(y)$, where the induced vertex label $f^{+}(x)=\sum f(e)$, with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by ${\chi }_{\mathrm{\text{la}}}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with ${\chi }_{la}=2,3$.

A Kähler graph consists of two graphs having the same set of vertices. When these two graphs are regular and their adjacency operators are commutative, the probabilistic adjacency operators of derived graphs are selfadjoint. In this paper we study maximum eigenvalues of derived graphs of such a regular Kähler graph whose constituent graphs are connected, and investigate whether derived graphs are connected or not.

A Kähler graph consists of two kinds of graphs having the same set of vertices. As candidates of model Kähler graphs which correspond to homogeneous spaces admitting magnetic fields, vertex transitive normal Kähler graphs are proposed in [5]. To give many examples of such Kähler graphs, we define some product operations. In this note, we restrict ourselves to the product Kähler graphs whose first kind graphs are of tensor product type, and study whether they are connected, are bipartite, are vertex-transitive and are normal.