Related papers: Bounds on distinct and repeated dot product trees
We analyse the friendship paradox on finite and infinite trees. In particular, we monitor the vertices for which the friendship-bias is positive, neutral and negative, respectively. For an arbitrary finite tree, we show that the number of…
We are interested in nodes with fixed outdegrees in large conditioned Galton--Watson trees. We first study the scaling limits of processes coding the evolution of the number of such nodes in different explorations of the tree…
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…
All product fixed point results in ordered metric spaces based on linear contractive conditions are but a vectorial form of the fixed point statement due to Nieto and Rodriguez-Lopez [Order, 22 (2005), 223-239], under the lines in Matkowski…
We consider a problem posed by Erd\H{o}s, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the…
Limits of graphs were initiated recently in the two extreme contexts of dense and bounded degree graphs. This led to elegant limiting structures called graphons and graphings. These approach have been unified and generalized by authors in a…
The Poisson boundary of a finite direct product of affine automorphism groups of homogeneous trees is considered. The Poisson boundary is shown to be a product of ends of trees with a hitting measure for spread-out, aperiodic measures of…
The problem of {\em efficiently} finding the best match for a query in a given set with respect to the Euclidean distance or the cosine similarity has been extensively studied in literature. However, a closely related problem of efficiently…
A widely adopted approach to solving constraint satisfaction problems combines systematic tree search with constraint propagation for pruning the search space. Constraint propagation is performed by propagators implementing a certain notion…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and…
We investigate the existence and multiplicity of solutions for higher order discrete boundary value problems via critical point theory.
Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension…
We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a…
We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional…
The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the…
We prove the Erd\H os--S\'os conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a…
We study trees where each successor set is equipped with some additional structure. We introduce a family of automaton models for such trees and prove their equivalence to certain fixed-point logics. As a consequence we obtain…
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in…