Eccentricity Sum in Trees

The eccentricity of a vertex, eccT(v)=maxu∈TdT(v,u), was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, Ecc(T), is the sum of eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios Ecc(T)/eccT(u), Ecc(T)/eccT(v), eccT(u)/eccT(v), and eccT(u)/eccT(w) where u,w are leaves of T and v is in the center of T. In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence

Outerplanar crossing numbers, the circular arrangement problem and isoperimetric functions

We extend the lower bound in [15] for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of log n the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established

HHV-8 encoded LANA-1 alters the higher organization of the cell nucleus

The latency-associated nuclear antigen (LANA-1) of Human Herpes Virus 8 (HHV-8), alternatively called Kaposi Sarcoma Herpes Virus (KSHV) is constitutively expressed in all HHV-8 infected cells. LANA-1 accumulates in well-defined foci that co-localize with the viral episomes. We have previously shown that these foci are tightly associated with the borders of heterochromatin [1]. We have also shown that exogenously expressed LANA-1 causes an extensive re-organization of Hoechst 33248 DNA staining patterns of the nuclei in non-HHV-8 infected cells [2]. Here we show that this effect includes the release of the bulk of DNA from heterochromatic areas, in both human and mouse cells, without affecting the overall levels of heterochromatin associated histone H3 lysine 9 tri-methylation (3MK9H3). The release of DNA from the heterochromatic chromocenters in LANA-1 transfected mouse cells co-incides with the dispersion of the chromocenter associated methylcytosin binding protein 2 (MECP2). The localization of 3MK9H3 to the remnants of the chromocenters remains unaltered. Moreover, exogeneously expressed LANA-1 leads to the relocation of the chromocenters to the nuclear periphery, indicating extensive changes in the positioning of the chromosomal domains in the LANA-1 harboring interphase nucleus. Using a series of deletion mutants we have shown that the chromatin rearranging effects of LANA-1 require the presence of a short (57 amino acid) region that is located immediately upstream of the internal acidic repeats. This sequence lies within the previously mapped binding site to histone methyltransferase SUV39H1. We suggest that the highly concentrated LANA-1, anchored to the host genome in the nuclear foci of latently infected cells and replicated through each cell generation, may function as "epigenetic modifier". The induction of histone modification in adjacent host genes may lead to altered gene expression, thereby contributing to the viral oncogenesis

Inducibility of d-ary trees

CITATION: Czabarka, E. et al. 2020. Inducibility of d-ary trees. Discrete Mathematics, 343(2). doi:10.1016/j.disc.2019.111671.The original publication is available at https://www.sciencedirect.com/journal/discrete-mathematicsImitating the binary inducibility, a recently introduced invariant of binary trees (Cz- abarka et al., 2017), we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2). We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T |−1/2) compared to the general case where it is shown that the difference is O(|T |−1) which, in particular, responds positively to a conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.https://www.sciencedirect.com/science/article/pii/S0012365X19303498Publishers versio