Graphs with degrees from prescribed intervals

Necessary and sufficient conditions for the existence of simple graphs with degrees from prescribed intervals, are given.     

Graphs with prescribed Levi form trace

Abstract We prove existence and uniqueness of a viscosity solution of the Dirichlet problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uniform limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers. Keywords: Levi mean curvature, Quasilinear degenerate elliptic PDE’s, Viscosity solutions, Comparison principle, Global Lipschitz estimates     

Graphs with prescribed degree sets and girth

For a finite nonempty set of integers, each of which is at least two, and an integern 3, the numberf(;n) is defined as the least order of a graph having degree set and girthn. The numberf(;n) is evaluated for several sets and integersn. In particular, it is shown thatf({3, 4}; 5) = 13 andf({3, 4}; 6) = 18.Research of the third author was partially supported by a Faculty Research Fellowship from Western Michigan University.     

Graphs with prescribed connectivity and line graph connectivity

Chartrand and Stewart have shown that the line graph of an n-connected graph is itself n-connected. This paper shows that for every pair of integers m > n > 1 there is a graph of point connectivity n whose line graph has point connectivity m. The corresponding question for line connectivity is also resolved.     

Graphs with prescribed mean curvature on hyperbolic spaces

We study a quasilinear elliptic equation in the unit ball ofℝ ^m . Using this result we get the existence of graphs with prescribed curvature on hyperbolic spacesℍ ^m inℍ ^m ×ℝ.     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.     

Circulants and their connectivities

There is diverse literature on various properties of a class of graphs known as circulants. We present a new result which answers the previously unsolved question of characterizing the connection sequence of circulants having point connectivity equal to point degree. We also develop some theorems regarding a new generalization of connectivity known as super-connectivity. In addition, we give a survey of published results pertinent to the study of connectivity of circulants.     

On connectivities of tree graphs

Let T(G) be the tree graph of a graph G with cycle rank r. Then κ(T(G)) ? m(G) ? r, where κ(T(G)) and m(G) denote the connectivity of T(G) and the length of a minimum cycle basis for G, respectively. Moreover, the lower bound of m(G) ? r is best possible.     

Matrices with prescribed characteristic polynomial and several prescribed submatrices

Let A = [A_jk],j,k = 1,2 be a partitioned matrix. We consider the problems: find necessary and sufficient conditions for the existence of A if we prescribe (i) An and the characteristic polynomial ofA, or (ii)A₁₁,A₂₂, and the characteristic polynomial of A. We solve (i) and give a partial solution of (ii).     

The connectivities of adjacent tree graphs

The connectivity and the circuit rank of a graphG are denoted byx(G) and, respectively. It is shown that ifH is the adjacent tree graph of a simple connected graphG, thenx(H)=2.     

Functions with prescribed singularities

The distributional k-dimensional Jacobian of a map u in the Sobolev space W^1,k-1 which takes values in the the sphere S^k-1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S^k-1. In case M is polyhedral, the map we construct is smooth outside M plus an additional polyhedral set of lower dimension, and can be used in the constructive part of the proof of a -convergence result for functionals of Ginzburg-Landau type, as described in [2]. Mathematics Subject Classification (2000) 46E35 (53C65, 49Q15, 26B10, 58A25)     

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53-64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119-171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173-177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91-117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q-forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17-22, 2001, TIFR, Mumbai, 2001, pp. 469-490] of D. Morris (Witte) on a completely different topic. We will answer that question too.