Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:

the generalized Bethe tree B,

a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,

a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,

a graph obtained from the cycle C_r by attaching B, by its root, to each vertex of the cycle, and

a tree obtained from the path P_r by attaching B, by its root, to each vertex of the path,

is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

     

On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

On upper bounds of Fourier-Walsh coefficients

An upper bound is established for the upper bounds of the Fourier-Walsh coefficients a_n(f) whose modulus of continuity (,f) does not exceed a given modulus of continuity (). In the case of convex majorants of (), these bounds are attained for individual ordinal numbers n.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 725–736, December, 1969.     

On upper bounds of bandwidths of trees

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New improvements on connectivity of cages

A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1)² ≤ δ + $ \sqrt \delta $ \sqrt \delta − 2 < r ² and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta . These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected.     

On upper bounds for Laplacian graph eigenvalues

In this paper, we obtain the following upper bound for the largest Laplacian graph eigenvalue λ(G):

     

On rigorous upper bounds to a global optimum

In branch and bound algorithms in constrained global optimization, a sharp upper bound on the global optimum is important for the overall efficiency of the branch and bound process. Software to find local optimizers, using floating point arithmetic, often computes an approximately feasible point close to an actual global optimizer. Not mathematically rigorous algorithms can simply evaluate the objective at such points to obtain approximate upper bounds. However, such points may actually be slightly infeasible, and the corresponding objective values may be slightly smaller than the global optimum. A consequence is that actual optimizers are occasionally missed, while the algorithm returns an approximate optimum and corresponding approximate optimizer that is occasionally far away from an actual global optimizer. In mathematically rigorous algorithms, objective values are accepted as upper bounds only if the point of evaluation is proven to be feasible. Such computational proofs of feasibility have been weak points in mathematically rigorous algorithms. This paper first reviews previously proposed automatic proofs of feasibility, then proposes an alternative technique. The alternative technique is tried on a test set that caused trouble for previous techniques, and is also employed in a mathematically rigorous branch and bound algorithm on that test set.     

On upper bounds of Chalk and Hua for exponential sums

Let be a polynomial of degree with integer coefficients, any prime, any positive integer and the exponential sum . We establish that if is nonconstant when read , then . Let , let be a zero of the congruence of multiplicity and let be the sum with restricted to values congruent to . We obtain for odd, and . If, in addition, , then we obtain the sharp upper bound .

     

On upper bounds for positive solutions of semilinear equations

Suppose that E is a bounded domain of class C^2,λ in and L is a uniformly elliptic operator in E. The set of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions u_Γ?w_Γ. We also introduced a class of solutions u_ν in 1-1 correspondence with a certain class of σ-finite measures ν on ∂E. With every we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and . We called this pair the fine boundary trace of u and we denoted in tr(u).Let u⊕v stand for the maximal solution dominated by u+v. We say that u belongs to the class if the condition tr(u)=(Γ,ν) implies that u?w_Γ⊕u_ν and we say that u belongs to if the condition tr(u)=(Γ,ν) implies that u?u_Γ⊕u_ν.It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de l'Université Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C⁴ belong to where Δ is the Laplacian and ψ(u)=u². [Mselati proved that, in his case, u_Γ=w_Γ and therefore the condition tr(u)=(Γ,ν) implies u=u_Γ⊕u_ν=w_Γ⊕u_ν.]By modifying Mselati's arguments, we extend his result to ψ(u)=u^α with 1<α?2 and all bounded domains of class C^2,λ.We start from proving a general localization theorem: under broad assumptions on L, ψ if, for every y∈∂E there exists a domain such that E′⊂E and ∂E∩∂E′ contains a neighborhood of y in ∂E.     

On upper bounds of sequential stochastic production planning problems

In this paper we consider a class of problems that determine production, inventory and work force levels for a firm in order to meet fluctuating demand requirements. A production planning problem arises because of the need to match, at the firm level, supply and demand efficiently. In practice, the two common approaches to counter demand uncertainties are (i) carrying a constant safety stock from period to period, and (ii) planning with a rolling horizon. Under the rolling horizon (or sequential) strategy the planning model is repeatedly solved, usually at the end of every time period, as new information becomes available and is used to update the model parameters. The costs associated with a rolling horizon strategy are hard to compute a priori because the solution of the model in any intermediate time period depends on the actual demands of the previous periods.In this paper we derive two a priori upper bounds on the costs for a class of production planning problems under the rolling horizon strategy. These upper bounds are derived by establishing correspondences between the rolling horizon problems and related deterministic programs. One of the upper bounds is obtained through Lagrangian relaxation of the service level constraint. We propose refinements to the non-Lagrangian bounds and present limited computational results. Extensions of the main results to the multiple item problems are also discussed. The results of this paper are intended to support production managers in estimating the production costs and value of demand information under a rolling horizon strategy.     

Lower bounds and upper bounds for chromatic polynomials

In this paper we give lower bounds and upper bounds for chromatic polynomials of simple undirected graphs on n vertices having m edges and girth exceeding g © 1993 John Wiley & Sons, Inc.     

On the estimates of the upper and lower bounds of Ramanujan primes

For \(n\ge 1\), the nth Ramanujan prime is defined as the least positive integer \(R_{n}\) such that for all \(x\ge R_{n}\), the interval \((\frac{x}{2}, x]\) has at least n primes. Let \(p_{i}\) be the ith prime and \(R_{n}=p_{s}\). Sondow, Laishram, and other scholars gave a series of upper bounds of s. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow–Nicholson–Noe and prove that if \(n>10^{300}\), then \(\pi (R_{mn})\le m\pi (R_{n})\) for \(m\ge 1\).     

Obtaining upper bounds of heat kernels from lower bounds

We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.