Combinatorial Inequalities Research Articles

Sharp Bounds for the General Randic Index of \(R\)-Graph Products

A chemical structure specifies the molecular geometry of a given molecule or solid in the form of atom arrangements. One way to analyze its properties is to simulate its formation as a product of two or more simpler graphs. In this article, we take this idea to find upper and lower bounds for the generalized Randić index \(\mathcal{R}_{\alpha}\) of four types of graph products, using combinatorial inequalities. We finish this paper by providing the bounds for \(\mathcal{R}_{\alpha}\) of a line graph and rooted product of graphs.

Log-concave poset inequalities

We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concave inequal- ities for counting certain weighted feasible words, which generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. Notably, we prove matching equality conditions for both earlier inequalities and our extensions. In contrast with much of the previous work, our proofs are combinatorial and employ nothing but linear algebra. We use the language formulation of greedoids which allows a linear algebraic setup, which in turn can be analyzed recursively. The underlying non- commutative nature of matrices associated with greedoids allows us to proceed beyond polymatroids and prove the equality conditions. As further application of our tools, we rederive both Stanley’s inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.

Global multiplicity, special closure and non-degeneracy of gradient maps

Given a polynomial map F:mathbb C^nlongrightarrow mathbb C^p with finite zero set, pgeqslant n, we introduce the notion of global multiplicity {text {m}}(F) associated to F, which is analogous to the multiplicity of ideals in Noetherian local rings. This notion allows to characterize numerically the Newton non-degeneracy at infinity of F. This fact motivates us to study a combinatorial inequality concerning the normalized volume of global Newton polyhedra and to characterize the corresponding equality using special closures. We also study the Newton non-degeneracy at infinity of gradient maps.

Some new Gaussian product inequalities

The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality E[X12X22m2X32m3]≥E[X12]E[X22m2]E[X32m3] for any centered Gaussian random vector (X1,X2,X3) and m2,m3∈N. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of m2 and arbitrary m3. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.

Combinatorial inequalities arising from the inclusion-exclusion principle

Combinatorial inequalities arising from the inclusion-exclusion principle

Strictness of the log-concavity of generating polynomials of matroids

Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.

Security region-based laminar flow coordinated optimization of grids

With the increasing penetration of renewable distributed generations in smart distribution grids, the coupling relationship between transmission and distribution grids has become increasingly complex. If power flow optimization for transmission and distribution grids is performed separately, the overall generation resources will not be fully utilized. In this paper, the power flow coordinated optimization based on layered decomposition is studied for the entire transmission and distribution grids, and the proposed method can be used on any scale grids. The core objective of this paper is the minimization of the overall supply cost. To this end, based on the steady-state security region (SSSR) in power injection spaces, the distributed local selfish optimization models of the transmission grid and each of the distribution grids is established, respectively. The coordination between transmission and distribution levels is achieved by using the Karush-Kuhn-Tucker (KKT) conditions. Due to the approximate hyperplane expressions of SSSR boundaries in power injection spaces, the operational constraints are transformed into a union of linear combinatorial inequalities of decision variables, and the variables in objective function and constraints are identical. Thus the optimization process and the determination of the KKT conditions become extremely simple, and the required number of iterations is very small. Several simulation results are presented, which show that the proposed method can accelerate the speed of solving problems by orders of magnitude.

The Asymptotic Induced Matching Number of Hypergraphs: Balanced Binary Strings

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith–Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science.Our proof relies on a new combinatorial inequality that may be of independent interest. This inequality concerns how many pairs of Boolean vectors of fixed Hamming weight can have their sum in a fixed subspace.

Sharp bounds for the general Randić index of transformation graphs

Inequalities are a useful method to investigate and compare topological indices of graphs relatively. A large collection of graph associated numerical descriptors have been used to examine the whole structure of networks. In these analysis, degree related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree related indices is one of the successful topic of research. Given a molecular graph H , the general Randić connectivity index is interpreted as R α ( H ) = ∑ ℛ ∈ E ( H ) ( deg H ( a ) deg H ( b ) ) α , with α is a real quantity. Also a graph transformation of H provides a comparatively simpler isomorphic structure with an ease to work on different chemical properties. In this article, we determine the sharp bounds of general Randić index of numerous graph transformations, such that semi-total-point, semi-total-line, total and eight individual transformations H fgh , where f, g, h ∈ {+ , -} of graphs by using combinatorial inequalities.

On covering monotonic paths with simple random walk

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_{1}$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\geq 4$. The main tool is a general combinatorial inequality, that is interesting in its own right.

Critical Points of Strichartz Functional

We study a pair of infinite dimensional dynamical systems naturally associated with the study of minimizing/maximizing functions for the Strichartz inequalities for the Schrödinger equation. One system is of gradient type and the other one is a Hamiltonian system. For both systems, the corresponding sets of critical points, their stability, and the relation between the two are investigated. By a combination of numerical and analytical methods we argue that the Gaussian is a maximizer in a class of Strichartz inequalities for dimensions one, two, and three. The argument reduces to verification of an apparently new combinatorial inequality involving binomial coefficients.

Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex

Let d∈N and let γi∈[0,∞), xi∈(0,1) be such that ∑i=1d+1γi=M∈(0,∞) and ∑i=1d+1xi=1. We prove thata↦Γ(aM+1)∏i=1d+1Γ(aγi+1)∏i=1d+1xiaγi is completely monotonic on (0,∞). This result generalizes the one found by Alzer [2] for binomial probabilities (d=1). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic formulas for quantities of interest in the context of statistical density estimation based on Bernstein polynomials on the d-dimensional simplex.

On the bounds of degree-based topological indices of the Cartesian product of F-sum of connected graphs

Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity or biological activity numerically. A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. In QSAR/QSPR study, a prediction about the bioactivity of chemical compounds is made on the basis of physico-chemical properties and topological indices such as Zagreb, Randić and multiple Zagreb indices. In this paper, we determine the lower and upper bounds of Zagreb indices, the atom-bond connectivity (ABC) index, multiple Zagreb indices, the geometric-arithmetic (GA) index, the forgotten topological index and the Narumi-Katayama index for the Cartesian product of F-sum of connected graphs by using combinatorial inequalities.

Combinatorial sums and binomial identities associated with the Beta-type polynomials

In this paper, we first provide some functional equations of the gener ating functions for beta-type polynomials. Using these equations, we derive various identities of the beta-type polynomials and the Bernstein basis functions. We then obtain some novel combinatorial identities in volving binomial coefficients and combinatorial sums. We also derive some generalizations of the combinatorics identities which are related to the Gould's identities and sum of binomial coefficients. Next, we present some remarks, comments, and formulas including the combi natorial identities, the Catalan numbers, and the harmonic numbers. Moreover, by applying the classical Young inequality, we derive a combi natorial inequality related to beta polynomials and combinatorial sums. We also give another inequality for the Catalan numbers.

A $$t_k$$ t k Inequality for Arrangements of Pseudolines

Consider arrangements of n pseudolines in the real projective plane. Let $$t_k$$tk denote the number of intersection points where exactly k pseudolines are incident. We present a new combinatorial inequality: $$\begin{aligned} t_2+1.5t_3\ge 8+\sum _{k\ge 4}(2k-7.5)t_k, \end{aligned}$$t2+1.5t3?8+?k?4(2k-7.5)tk,which holds if no more than $$n-3$$n-3 pseudolines intersect at one point. It looks similar but is unrelated to the Hirzebruch inequality for arrangements of complex lines in the complex projective plane. Based on this linear inequality, we construct lower bounds for the number of regions via n and the maximal number of (pseudo)lines passing through one point.

A new companion to Capparelli's identities

We discuss a new companion to Capparelli's identities. Capparelli's identities for m=1,2 state that the number of partitions of n into distinct parts not congruent to m, −m modulo 6 is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not congruent to 3−m modulo 3, and 3l+1 and 3l+2 do not appear together as consecutive parts for any integer l has the same number of elements as the above mentioned Capparelli's partitions of n. In this study we also extend the work of Alladi, Andrews and Gordon by providing a complete set of generating functions for the refined Capparelli partitions, and conjecture some combinatorial inequalities.

Distinct Parts Partitions without Sequences

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.

Two variable extensions of the Laguerre and disc polynomials

This work contains a detailed study of a one parameter generalization of the 2D-Hermite polynomials and a two parameter extension of Zernike's disc polynomials. We derive linear and bilinear generating functions, and explicit formulas for our generalizations and study integrals of products of some of these 2D orthogonal polynomials. We also establish a combinatorial inequality involving elementary symmetric functions and solve the connection coefficient problem for our polynomials.

A Combinatorial Benders׳ decomposition for the lock scheduling problem

The Lock Scheduling Problem (LSP) is a combinatorial optimization problem that represents a real challenge for many harbours and waterway operators. The LSP consists of three strongly interconnected subproblems: scheduling lockages, assigning ships to chambers, and positioning the ships inside the chambers. These should be interpreted respectively as a scheduling, an assignment, and a packing problem. By combining the first two problems into a master problem and using the packing problem as a subproblem, a decomposition is achieved that can be solved efficiently by a Combinatorial Benders׳ approach. The master problem is solved first, thereby sequencing the ships into a number of lockages. Next, for each lockage, a packing subproblem is checked for feasibility, possibly returning a number of combinatorial inequalities (cuts) to the master problem. The result is an exact approach to the LSP. Experiments are conducted on a set of instances that were generated in correspondence with real world data. The results indicate that the decomposition approach significantly outperforms other exact approaches presented in the literature, in terms of solution quality and computation time.

Some Properties of (Inverse)<i>N</i><sub>0</sub>-Matrices with its Applications

In this paper we study some combinatorial properties and inequalities of some classes of Z-matrices. These matrices arise in many problems in the mathematical and physical sciences. We show that all (inverse) N0-matrices are irreducible and some eigenvalue inequalities of (inverse) N0-matrices.