Characterization of graphs having extremal Randi? indices

The higher Randi? index R_t(G) of a simple graph G is defined as

     

Which generalized Randi? indices are suitable measures of molecular branching?

Molecular branching is a very important notion, because it influences many physicochemical properties of chemical compounds. However, there is no consensus on how to measure branching. Nevertheless two requirements seem to be obvious: star is the most branched graph and path is the least branched graph. Every measure of branching should have these two graphs as extremal graphs. In this paper we restrict our attention to chemical trees (i.e. simple connected graphs with maximal degree at most 4), hence we have only one requirement that the path be an extremal graph. Here, we show that the generalized Randi? index R_p(G)=∑_uv∈E(G)(d_ud_v)^p is a suitable measure for branching if and only if p∈[λ,0)∪(0,λ^′) where λ is the solution of the equation in the interval (−0.793,−0.792) and λ^′ is the positive solution of the equation 3⋅3^x−2⋅2^x−4^x=0. These results include the solution of the problem proposed by Clark and Gutman.     

SDP Relaxation of Arbitrage Pricing Bounds Based on?Option Prices and Moments

This paper develops a semidefinite programming approach to computing bounds on the range of allowable absence of arbitrage prices for a European call option when option prices at other strikes and expirations are available and when moment related information on the underlying is known. The moment related information is incorporated in the problem through the fictitious prices of polynomial valued securities. The optimization then comes from relaxing a risk neutral pricing optimization problem in terms of moments of measures from a decomposition of the risk neutral pricing measure. We demonstrate this optimization formulation with computations using moment data from the standard Black-Scholes option pricing model and Merton’s jump diffusion model.     

Conjugated trees with minimum general Randi? index

The general Randi? index R_α(G) is the sum of the weights (d_G(u)d_G(v))^α over all edges uv of a (molecular) graph G, where α is a real number and d_G(u) is the degree of the vertex u of G. In this paper, for any real number α≤−1, the minimum general Randi? index R_α(T) among all the conjugated trees (trees with a Kekulé structure) is determined and the corresponding extremal conjugated trees are characterized. These trees are also extremal over all the conjugated chemical trees.     

On a conjecture of the Randi? index

The Randi? index of a graph G is defined as , where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs.     

Comment on “Complete solution to a conjecture on Randić index”

In this paper we will show that the proof of Theorem 2.1 from “Complete solution to a conjecture on Randi? index”, by Xueliang Li, Bolian Liu and Jianxi Liu, European Journal of Operational Research 200, Issue 1, (2010), 9–13, is not correct. They tried to prove the conjecture given by M. Aouchiche, P. Hansen in “On a conjecture about the Randi? index” (Discrete Mathematics, 307 (2), 2007, 262–265), but they failed in it. The mathematical model given by them is a problem of quadratic programming which they tried to solve by wrong use of linear programming. This error invalidates all further work.     

The Randi? index and the diameter of graphs

The Randi? indexR(G) of a graph G is defined as the sum of over all edges uv of G, where d_u and d_v are the degrees of vertices u and v, respectively. Let D(G) be the diameter of G when G is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs G on n vertices the path P_n achieves the minimum values for both R(G)/D(G) and R(G)−D(G). We prove this conjecture completely. In fact, we prove a stronger theorem: If G is a connected graph, then , with equality if and only if G is a path with at least three vertices.     

Bounds on the Non-real Spectrum of a Singular Indefinite Sturm-Liouville Operator on ℝ

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat Semigroup Bounds on BV(Ω) and Related Estimates

Let \(\overline{\Omega}\) be a compact Riemannian manifold with nonempty boundary. We note that if \(f\in C^\infty(\overline{\Omega})\) does not vanish identically on the boundary, then the heat semigroup \(e^{t\Delta_D}\) (with the Dirichlet boundary condition) acting on f produces a family bounded in \(H^{1,p}(\overline{\Omega})\) if and only if p?=?1. This observation motivates the main result of this paper, which is that the heat semigroup is uniformly bounded on BV(Ω), the space of functions on Ω with bounded variation.     

Bounds on the vertex–edge domination number of a tree

A vertex–edge dominating set of a graph G is a set D of vertices of G such that every edge of G is incident with a vertex of D or a vertex adjacent to a vertex of D. The vertex–edge domination number of a graph G  , denoted by γ_ve(T)${\gamma }_{\mathrm{ve}}(T)$, is the minimum cardinality of a vertex–edge dominating set of G. We prove that for every tree T   of order n?3$n?3$ with l leaves and s   support vertices, we have (n−l−s+3)/4?γ_ve(T)?n/3$(n-l-s+3)/4?{\gamma }_{\mathrm{ve}}(T)?n/3$, and we characterize the trees attaining each of the bounds.     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

Estimates for the Upper Bounds of the First Eigenvalue on Submanifolds

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Bounds on the value of information in uncertain decision problems II†

In most stochastic decision problems one has the opportunity to collect information that would partially or totally eliminate the inherent uncertainty. One wishes to compare the cost and value of such information in terms of the decision maker's preferences to determine an optimal information gathering plan. The calculation of the value of information generally involves oneor more stochastic recourse problems as well as one or more expected value distribution problems. The difficulty and costs of obtaining solutions to these problems has led to a focus on the development of upper and lower bounds on the various subproblems that yield bounds on the value of information. In this paper we discuss published and new bounds for static problems with linear and concave preference functions for partial and perfect information. We also provide numerical examples utilizing simple production and investment problems that illustrate the calculations involved in the computation of the various bounds and provide a setting for a comparison of the bounds that yields some tentative guidelines for their use. The bounds compared are the Jensen's Inequality bound,the Conditional Jensen's Inequality bound and the Generalized Jensen and Edmundson-Madansky bounds.