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In this paper, a complete characterization of the (super) edge-magic linear forests with two components is provided. In the process of establishing this characterization, the super edge-magic, harmonious, sequential and felicitous properties of certain 2-regular graphs are investigated, and several results on super edge-magic and felicitous labelings of unions of cycles and paths are presented. These labelings resolve one conjecture on harmonious graphs as a corollary, and make headway towards the resolution of others. They also provide the basis for some new conjectures (and a weaker form of an old one) on labelings of 2-regular graphs.

Fluorescence imaging in the over-thousand nanometer (OTN-) near-infrared (NIR) wavelength region is an emerging technique for real-time bioimaging. OTN-NIR probes are made from micellar nanoparticles encapsulating IR-1061 dye in the core of poly(ethylene glycol) (PEG) phospholipid (PL), such as 1, 2-distearoyl-sn-glycero-3-phosphoethanolamine (DSPE)-N-[methoxy PEG] micelles. The property investigation revealed that the probe is less stable in albumin and PBS while remaining unchanged in water and saline. The results are critical for applying OTN-NIR probe from DSPE-PEG micelles in physiological environments.

A bipartite graph G with partite sets X and Y is called consecutively super edge-magic if there exists a bijective function f : V (G) ∪ E (G) → {1, 2,. .. , |V (G)| + |E (G)|} with the property that f (X) = {1, 2,. .. , |X|}, f (Y) = {|X| + 1, |X| + 2,. .. , |V (G)|} and f (u) + f (v) + f (uv) is constant for each uv ∈ E (G). The question studied in this paper is for which bipartite graphs it is possible to add a finite number of isolated vertices so that the resulting graph is consecutively super edge-magic. If it is possible for a bipartite graph G, then we say that the minimum such number of isolated vertices is the consecutively super edge-magic deficiency of G; otherwise, we define it to be +∞. This paper also includes a detailed discussion of other concepts that are closely related to the consecutively super edge-magic deficiency.

The beta-number, β (G), of a graph G is defined to be either the smallest positive integer n for which there exists an injective function f : V (G) → {0, 1,. .. , n} such that each uv ∈ E (G) is labeled |f (u) − f (v)| and the resulting set of edge labels is {c, c + 1,. .. , c + |E (G)| − 1} for some positive integer c or +∞ if there exists no such integer n. If c = 1, then the resulting beta-number is called the strong beta-number of G and is denoted by β s (G). In this paper, we show that if G is a bipartite graph and m is odd, then β (mG) ≤ mβ (G) + m − 1. This leads us to conclude that β (mG) = m |V (G)| − 1 if G has the additional property that G is a graceful nontrivial Full PDF DMGT Page tree. In addition to these, we examine the (strong) beta-number of forests whose components are isomorphic to either paths or stars.