Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Epidemiological association of <it>Campylobacter jejuni</it> groups with pathogenicity-associated genetic markers

<p>Abstract</p> <p>Background</p> <p><it>Campylobacter jejuni</it>, the most leading cause for bacterial gastroenteritis worldwide, shows a high genetic diversity among its isolates. Recently, we demonstrated the existence of six <it>C. jejuni</it>-groups by combining MLST with six genetic markers. These groups were further characterized by the detection of <it>cj1321-cj1326</it>, <it>fucP</it>, <it>cj0178</it>, <it>cj0755</it>/<it>cfrA, ceuE</it>, <it>pldA</it>, <it>cstII</it>, and <it>cstIII</it> in order (I.) to show further associations between these different genetic markers and MLST CCs. Moreover, different studies were able to associate several of these markers: a sialylated lipoologosaccharide (<it>cstII/III</it>⁺), the gamma-glytamyl-transpeptidase (<it>ggt</it>⁺), and the absence of a certain allele of the enterochelin-uptake-binding-protein (<it>ceuE</it>₁₁₁₆₈^-) with severe campylobacteriosis, bloody diarrhea and unpleasant outcome. Additionally more than half of human <it>Campylobacter-</it>isolates were assigned to a non-livestock clade associated with the absence of <it>cj1321-cj1326</it>. These isolates were considered as mere colonizers.</p> <p>From the combination of marker genes, the ratio of human isolates in a specific group, and clinical data (II.) it should be demonstrated to which of the previous defined groups these <it>Campylobacter</it>-subpopulations, associated with higher virulence, correspond.</p> <p>Results</p> <p>Besides the marker gene <it>pldA</it>, all new estimated genetic markers show significant differences in their distribution among the various MLST-based groups. Especially the genes for <it>cj1321-cj1326</it>, <it>fucP</it>, <it>cj0178</it>, <it>cj0755</it>/<it>cfrA</it> are widely associated with each other and split the study population into two major and seven intermediate groups substantiating the previous group-definition, whereas <it>cstII</it> and <it>cstIII</it> indicate at least three groups following an independent distribution pattern.</p> <p>Conclusions</p> <p>Based on these data a group of <it>C. jejuni</it>-isolates characterized by the presence of <it>ansB, dmsA</it>, <it>ggt,</it> and the absence of <it>cj1365c</it>, <it>cj1585c</it>, <it>cj1321-cj1326, fucP</it>, <it>cj0178</it>, <it>cj0755</it>/<it>cfrA,</it> and <it>cstII/III</it> was associated with a higher prevalence in human campylobacteriosis, bloody diarrhea as well as hospitalization and bears obviously a higher virulence for humans. In contrast to that better livestock-adapted groups characterized by the ability to utilize L-fucose and the presence of all of the five identified putative <it>C. jejuni</it> iron-uptake systems as well as <it>cj1321-cj1326</it>, <it>cj1365c, cj1585c</it>, and <it>cstII</it> and/or <it>cstIII</it> (sialylated lipoologosaccharide) is more prevalent in animal hosts and was secondary associated with less severe campylobacteriosis.</p