Some remarks on generalized roundness

By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank  ≥ 2 equals 1. This answers a question of J-F. Lafont and S. Prassidis.      

Metric stability of trees and tight spans

We prove optimal extension results for roughly isometric relations between metric ( ${\mathbb{R}}$ R -)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov–Hausdorff (GH) distance, for certain metric spanning constructions: the GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.     

Metric regularity and systems of generalized equations

The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some “error bound” type estimates.     

On generalized minimum spanning trees

The generalized minimum spanning tree problem consists of designing a minimum cost tree spanning several clusters. The purpose of this note is to pinpoint several inaccuracies contained in a previous publication and to propose a valid formulation for this problem.     

Goodness of trees for generalized books

A connected graphG is said to beF-good if the Ramsey numberr(F, G) is equal to(x(F) ? 1)(p(G) ? 1) + s(F), wheres(F) is the minimum number of vertices in some color class under all vertex colorings by _χ (F) colors. It is of interest to know which graphsF have the property that all trees areF-good. It is shown that any large tree isK(1, 1,m ₁,m ₂,...,m _t )-good.     

On generalized independent subsets of trees

A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ?-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.     

Metric regularity of a positive order for generalized equations

This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of a positive order at its a given solution. The provided conditions are expressed in terms of the Fréchet coderivative/or the Mordukhovich coderivative/or the Clarke one of the corresponding multifunction formulated the generalized equation. In addition, we show that such sufficient conditions turn out to be also necessary for the metric regularity of a positive order of the generalized equation in the case where the multifunction established the generalized equation is closed and convex.     

Metric subregularity for composite-convex generalized equations in Banach spaces

In this paper we consider a convex-composite generalized constraint equation in Banach spaces. Using variational analysis technique, in terms of normal cones and coderivatives, we first establish sufficient conditions for such an equation to be metrically subregular. Under the Robinson qualification, we prove that these conditions are also necessary for the metric subregularity. In particular, some existing results on error bound and metric subregularity are extended to the composite-convexity case from the convexity case.     

Metric subregularity for proximal generalized equations in Hilbert spaces

In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson-Ursescu theorem type for prox-regular multifunctions.     

The generalized Randić index of trees

The generalized Randi?; index of a tree T is the sum over the edges of T of where is the degree of the vertex x in T. For all , we find the minimal constant such that for all trees on at least 3 vertices, , where is the number of vertices of T. For example, when . This bound is sharp up to the additive constant—for infinitely many n we give examples of trees T on n vertices with . More generally, fix and define , where is the number of leaves of T. We determine the best constant such that for all trees on at least 3 vertices, . Using these results one can determine (up to terms) the maximal Randi?; index of a tree with a specified number of vertices and leaves. Our methods also yield bounds when the maximum degree of the tree is restricted. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 270–286, 2007     

Spectra of generalized Bethe trees attached to a path

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P_m be a path of m vertices. Let {B_i:1?i?m} be a set of generalized Bethe trees. Let P_m{B_i:1?i?m} be the tree obtained from P_m and the trees B₁,B₂,…,B_m by identifying the root vertex of B_i with the i-th vertex of P_m. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P_m{B_i:1?i?m}. In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B₁=B₂=…=B_m.     

Metric regularity, tangential distances and generalized differentiation in Banach spaces

In this paper we establish new generalized differentiation rules in general Banach spaces regarding normal cones to set images under functions, coderivatives of compositions of set-valued mappings, as well as calculus results for normal compactness of sets and their images. In addition to the metric regularity of mappings, our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations, basic properties, as well as relations to similar notions. Some related results are also established.     

Maximizing and minimizing the number of generalized colorings of trees

We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.     

Applications of generalized perron trees to maximal functions and density bases

In this article we give some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate L^p-functions or give Hardy-Littlewood type maximal functions which are bounded on L^p, p>1. This is done by proving that a well-known method, the construction of a Perron Tree, can be applied to a larger collection of subsets of the unit circle than was earlier known. As applications, we prove a partial converse of a well-known result of Nagel et al. [6] regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and prove a result regarding the cardinality of subsets of arithmetic progressions in sets of the type described above. Acknowledgements and Notes. This research was partially supported by NSERC.     

Line graph of combinations of generalized Bethe trees: Eigenvalues and energy

We characterize the eigenvalues and energy of the line graph L(G) whenever G is (i) a generalized Bethe tree, (ii) a Bethe tree, (iii) a tree defined by generalized Bethe trees attached to a path, (iv) a tree defined by generalized Bethe trees having a common root, (v) a graph defined by copies of a generalized Bethe tree attached to a cycle and (vi) a graph defined by copies of a star attached to a cycle; in this case, explicit formulas for the eigenvalues and energy of L(G) are derived.