Upper bounds on the energy of graphs in terms of matching number

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

A note on sharpening of a theorem of Ankeny and Rivlin

Let p(z) = ?n?=0 a?z? be a polynomial of degree n, M(p,R) := max|z|=R?0 |p(z)|, and M(p,1) := ||p||. Then according to a well-known result of Ankeny and Rivlin, we have for R ? 1, M(p,R) ? (Rn+1/2) ||p||. This inequality has been sharpened among others by Govil, who proved that for R ? 1, M(p,R) ? (Rn+1/2) ||p||-n/2 (||p||2-4|an|2/||p||) {(R-1)||p||/||p||+2|an|- ln (1+ (R-1)||p||/||p||+2|an|)}. In this paper, we sharpen the above inequality of Govil, which in turn sharpens inequality of Ankeny and Rivlin. We present our result in terms of the LerchPhi function ?(z,s,a), implemented in Wolfram's MATHEMATICA as LerchPhi [z,s,a], which can be evaluated to arbitrary numerical precision, and is suitable for both symbolic and numerical manipulations. Also, we present an example and by using MATLAB show that for some polynomials the improvement in bound can be considerably significant.

Commutative von Neumann regular rings are 1-Gröbner

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

Theory of discrete Muckenhoupt weights and discrete Rubio de Francia extrapolation theorems

In this paper, we will prove a discrete Rubio De Francia extrapolation theorem in the theory of discrete Ap-Muckenhoupt weights for which the discrete Hardy-Littlewood maximal operator is bounded on lpw (Z+). The results will be proved by employing the self-improving property of the discrete Ap-Muckenhoupt weights and the Marcinkiewicz Interpolation Theorem.

New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

The Roman domination number of some special classes of graphs - convex polytopes

In this paper we study the Roman domination number of some classes of planar graphs - convex polytopes: An, Rn and Tn. We establish the exact values of Roman domination number for: An, R3k, R3k+1, T8k, T8k+2, T8k+3, T8k+5 and T8k+6. For R3k+2, T8k+1, T8k+4 and T8k-1 we propose new upper and lower bounds, proving that the gap between the bounds is 1 for all cases except for the case of T8k+4, where the gap is 2.

Generating functions for generalization Simsek numbers and their applications

Our perpose in this work is the complete the study of Simsek numbers. We give answer to some open problems concerning polynomial representations and associated generating function. At the end of the study we investigate a new generalization of these numbers and obtain useful identities which connect Simsek numbers and Stirling numbers of second kind.

Weighted P-partitions enumerator

To an extended generalized permutohedron we associate the weighted integer points enumerator, whose principal specialization is the f-polynomial. In the case of poset cones it refines Gessel's P-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.

Sine and cosine types of generating functions

We introduce two sine and cosine types of generating functions in a general case and apply them to the generating functions of classical hypergeometric orthogonal polynomials as well as some widely investigated combinatorial numbers such as Bernoulli, Euler and Genocchi numbers. This approach can also be applied to other celebrated sequences.

Comparison inequalities of Bernstein-type between polynomials with restricted zeros

In this paper, we establish certain comparison inequalities of Bernstein-type for a linear operator between complex polynomials under certain constraints on their zeros. A variety of interesting results follow as special cases from our results.