On \u3cem\u3ek\u3c/em\u3e-minimum and \u3cem\u3em\u3c/em\u3e-minimum Edge-Magic Injections of Graphs

An edge-magic total labelling (EMTL) of a graph G with n vertices and e edges is an injection λ:V(G) ∪ E(G)→[n+e], where, for every edge uv ∈ E(G), we have wtλ(uv)=kλ, the magic sum of λ. An edge-magic injection (EMI) μ of G is an injection μ : V(G) ∪ E(G) → N with magic sum kμ and largest label mμ. For a graph G we define and study the two parameters κ(G): the smallest kμ amongst all EMI’s μ of G, and m(G): the smallest mμ amongst all EMI’s μ of G. We find κ(G) for G ∈ G for many classes of graphs G. We present algorithms which compute the parameters κ(G) and m(G). These algorithms use a G-sequence: a sequence of integers on the vertices of G whose sum on edges is distinct. We find these parameters for all G with up to 7 vertices. We introduce the concept of a double-witness: an EMI μ of G for which both kμ=κ(G) and mμ=m(G) ; and present an algorithm to find all double-witnesses for G. The deficiency of G, def(G), is m(G)−n−e. Two new graphs on 6 vertices with def(G)=1 are presented. A previously studied parameter of G is κEMTL(G), the magic strength of G: the smallest kλ amongst all EMTL’s λ of G. We relate κ(G) to κEMTL(G) for various G, and find a class of graphs B for which κEMTL(G)−κ(G) is a constant multiple of n−4 for G ∈B. We specialise to G=Kn, and find both κ(Kn) and m(Kn) for all n≤11. We relate κ(Kn) and m(Kn) to known functions of n, and give lower bounds for κ(Kn) and m(Kn)

Edge-injective and edge-surjective vertex labellings

For a graph G = (V,E) we consider vertex-k-labellings f : V \rightarrow {1,2,...,k} for which the induced edge weighting w : E \rightarrow {2,3,..., 2k} with w(uv) = f(u) + f(v) is injective or surjective or both. We study the relation between these labellings and the number theoretic notions of an additive basis and a Sidon set, present a new construction for a so-called restricted additive basis and derive the corresponding consequences for the labellings. We prove that a tree of order n and maximum degree \triangle has a vertex-k-labelling f for which w is bijective if and only if \triangle \leq k = n/2. Using this result we prove a recent conjecture of Ivančo and Jendrol' concerning edge-irregular total labellings for graphs that are sparse enough

Locally finite graphs with ends: A topological approach, I. Basic theory

AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Release of antimicrobial compounds from glass-ionomer dental cements

This thesis reports a study of the possibility of using conventional glass-ionomer cements (GICs) as matrices for release of antimicrobial compounds. Sodium fusidate, cetyl pyridinium chloride (CPC), benzalkonium chloride (BACH), triclosan and triclosan/zinc citrate at concentrations ranging from 1% to 5% by weight were added into Fuji IX and Chemflex cements. Disc-diffusion studies showed antimicrobial effect against Streptococcus mutans. Inhibition zones were proportional to the amount of added bactericide, CPC and BACH showed highest antibacterial activity. The release of the bactericides into water was studied for time intervals up to seven weeks. The amount of additive released varied from 0.61% to 5.00% of total bactericide added and samples containing more antimicrobial agent gave higher release into the surrounding water. The release was shown to be diffusion based for the first 2-4 weeks. Compressive strength and surface hardness of reformulated materials decreased in comparison with the control specimens. Addition of bactericides also decreased the amount of fluoride released. 27Al MAS-NMR showed that aluminium switches its coordination number from four, Al (IV), in the glass phase to six, Al (VI), in the cement matrix and addition of antimicrobial agents reduced the rate of this change. Incorporation of additives also prolonged the working time. By contrast, water loss properties were not affected by additives. The overall conclusion is that the presence of additives affects the setting and maturation reactions of GICs. These results can be interpreted as showing that the additives having an effect on the conformation of the poly (acrylic acid) (PAA) component in solution. Changes in the conformation of the PAA also influence the release of key ions from the glass (Al3+, Ca2+, F- and Na+). Alteration in the balance of these ions, especially Al3+, would result in slower cross-linking processes and lower cross-link density matrix. Additionally, adsorption properties of surfactants to GI aluminosilicate glass particles can also lead to reduction in the number of available active sites on the glass which can react with PAA. The reduction in available active sites on the glass will result in a lower bonding density and thus a weaker matrix. All above will leads to the observed changes in mechanical properties, working kinetics, F- release and kinetics of conversion of Al (IV) to Al (VI)