Enumeration of graph embeddings

AbstractFor a finite connected simple graph G, let Γ be a group of graph automorphisms of G. Two 2-cell embeddings ι: G → S and j: G → S of a graph G into a closed surface S (orientable or nonorientable) are congruent with respect to Γ if there are a surface homeomorphism h:S → S and a graph automorphism γϵΓ such that hoι=joγ. In this paper, we give an algebraic characterization of congruent 2-cell embeddings, from which we enumerate the congruence classes of 2-cell embeddings of a graph G into closed surfaces with respect to a group of automorphisms of G, not just the full automorphism group. Some applications to complete graphs are also discussed. As an orientable case, the oriented congruence of a graph G into orientable surfaces with respect to the full automorphism group of G was enumerated by Mull et al. (1988)

Zeta functions of line, middle, total graphs of a graph and their coverings

AbstractWe consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G

Frobenius maps

AbstractA graph is called Frobenius if it is a connected orbital regular graph of a Frobenius group. A Frobenius map is a regular Cayley map whose underlying graph is Frobenius. In this paper, we show that almost all low-rank Frobenius graphs admit regular embeddings and enumerate non-isomorphic Frobenius maps for a given Frobenius graph. For some Frobenius groups, we classify all Frobenius maps derived from these groups. As a result, we construct some Frobenius maps with trivial exponent groups as a partial answer of a question raised by Nedela and Škoviera (Exponents of orientable maps, Proc. London Math. Soc. 75(3) (1997) 1–31)

Cubic symmetric graphs of order a small number times a prime or a prime square

AbstractA graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K3,3 and the s-regular cyclic or elementary abelian coverings of the complete graph K4 for each s⩾1 are classified when the fibre-preserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regular graphs with girth 12 is found, in which the smallest one has order 2058. As an interesting application, a complete list of pairwise non-isomorphic s-regular cubic graphs of order 4p, 6p, 4p2 or 6p2 is given for each s⩾1 and each prime p

Review of the Screening Tool for Child Abuse by Health Care Provider

The recent report regarding the death of children due to child abuse brought on an implementation of an Act on special cases concerning the punishment, etc. of crimes of child abuse in September 2014. According to the report in 2013 by the National Child Protectin Agency, there were 10,000 suspected child abuse cases, with about 6,000 confirmed cases. While health care providers are obligated to report suspected child abuse, the rate of report was only less than 1%. Several studies suggest that the ease of screening process for child abuse in hospitals could increase the insight and reporting rate. With regard to the improvement in the perception for child abuse by health care providers in Korea, the authors reviewed the health care provider's roles for the protection of child abuse victims, various kinds of screening tools for child abuse in hospitals, and clinical processes for the evaluation of child abuse by a child protection team

After-hours care models in leading countries

After-hours care (AHC) provides medical care after the regular weekday work hours of clinics. In Korea, data from the National Emergency Department Information System showed that approximately 40% of the pediatric patients need AHC. To meet this need, many countries have different models of AHC. In this article, the authors tried to summarize and emphasize the advantages and disadvantages of AHC models in several leading countries. This article can be useful in designing AHC models in Korea because the proportions of potential patients requiring AHC are substantial, and the adoption of AHC models should be seriously considered