New descriptions of the Lovasz number, and the weak sandwich theorem

In the seminal work [8] L. Lovász introduced the concept of an orthonormal representation of a graph, and also a related value, now popularly known as the Lovász number of the graph. One of the remarkable properties of the Lovász number is that it lies sandwiched between the stability number and the complementer chromatic number. This fact is called the sandwich theorem. In this paper, using new descriptions of the Lovász number and linear algebraic lemmas we give three proofs for a weaker version of the sandwich theorem

On the projection onto a finitely generated cone

In this paper we study the properties of the projection onto a finitely generated cone. We show for example that this map is made up of finitely many linear parts with a structure resembling the facial structure of the finitely generated cone. An economical algorithm is also presented for calculating the projection of a fixed vector, based on Lemke’s algorithm to solve a linear complementarity problem. Some remarks on the conical inverse (a generalization of the Moore-Penrose generalized inverse) conclude the paper

On Abrams' theorem

Abrams’ theorem describes a necessary and sufficient condition for the closedness of a linear image of an arbitrary set. Closedness conditions of this type play an important role in the theory of duality in convex programming. In this paper we present generalizations of Abrams’ theorem, as well as Abrams-type theorems characterizing other properties (such as relatively openness or polyhedrality) of linear images of convex sets

Applications of the inverse theta number in stable set problems

In the paper we introduce a semidefinite upper bound on the square of the stability number of a graph, the inverse theta number, which is proved to be multiplicative with respect to the strong graph product, hence to be an upper bound for the square of the Shannon capacity of the graph. We also describe a heuristic algorithm for the stable set problem based on semidefinite programming, Cholesky factorization, and eigenvector computation

On the Projection onto a Finitely Generated Cone

In the paper we study the properties of the projection onto a finitely generated cone. We show that this map is made up of finitely many linear parts with a structure resembling the facial structure of the finitely generated cone. An economical (regarding storage) algorithm is also presented for calculating the projection of a fixed vector, based on Lemke's algorithm to solve a linear complementarity problem. Some remarks on the conical inverse (a generalization of the Moore-Penrose generalized inverse) conclude the paper

Bounds on the Stability Number of a Graph via the Inverse Theta Function

In the paper we consider degree, spectral, and semidefinite bounds on the stability number of a graph. The bounds are obtained via reformulations and variants of the inverse theta function, a notion recently introduced by the author in a previous work

Szén alapú nano-kompozitok előállítása és komplex szerkezeti jellemzése = Preparation and complex characterisation of carbon based nano-composites

A gyorsuló műszaki-technológiai fejlődés növekvő igényeket támaszt az új, különleges tulajdonságú anyagok iránt. Ezek között kitüntetett helyet foglalnak el a szénalapú rendszerek, és különösen azok vékonyrétegei. A szén kémiai kötésviszonyaitól függően igen változatos szerkezetű, mely a gyémánt, és gyémántszerű (DLC), grafit és fullerén-szerű módosulatokig terjedhet, melyek kialakulása befolyásolható az előállítás körülményeivel és adalékanyagokkal. Az elvégzett kutatások eredményként megvalósítottuk a szén-alapú nanokompozitok vékonyrétegeinek plazmás-ionsugaras aktivációval, magnetronos porlasztással és plazmaimmerziós ionimplantációval való előállítását. Feltártuk a rétegnövesztés egyes paramétereinek hatását a rétegek összetételére, kémiai kötés- és elektronszerkezetére. Modellrendszereken kimutattuk, hogy a DLC rétegek különböző kémiai kötésszerkezetű nanométeres klaszterekből épülnek fel és így szerkezeti nanokompozitoknak tekinthetők. A különböző prekurzorokból előállított Si adalékot tartalmazó DLC rétegek amorf szerkezetűek. A szilícium befolyásolja a szénklaszterek kötésszerkezetét, a Si eloszlása atomi szinten nem egyenletes, ugyanakkor nem képez többatomos elemi Si-Si és Si-C klasztereket. A rétegek alkalmazási szempontból előnyös nanomechanikai tulajdonságokkal rendelkeznek. Az átmeneti fém (W, Cr, Ni) tartalmú szénrétegek közül a Si tartalmúak 1-5 nm-es klasszterekből álló amorf, míg a csak Ni tartalmúak részben kristályos szerkezetűek. | Novel materials with exceptional properties are required by the accelerating technical development. In this project carbon based systems, especially their thin layers, have outstanding importance. The chemical bonding structure of the carbon may have a great variety, ranging from the diamond and diamond like systems to the graphitic and fullerene like systems, development of which can be influenced by the preparation conditions and additives. The preparation facilities and conditions for deposition of carbon based thin layers were developed by plasma and ion beam activation, by magnetron sputtering, and also by plasma immersion ion implantation. It was established that the DLC layers are built of nanometre size clusters of different chemical bonding environments. The Si containing DLC layers prepared from various precursors are amorphous. The Si additive influences the chemical bond structure of the carbon clusters; the distribution of Si is not homogeneous, however, it does not form multi-atomic Si-Si or SiC clusters. The layers posses beneficial nanomechanical properties. The carbon layers doped with transition metals (W, Cr, Ni) are also amorphous with 1-5 nm sized clusters, while the Ni containing layers are partially crystalline

Kombinatorikus Optimalizálás: Algoritmusok, Strukturák, Alkalmazások = Combinatorial optimization: algorithms, structures, applications

Mint azt az OTKA-pályázat munkaterve tartalmazza, a pályázatban résztvevő kutatók alkotják a témavezető irányításával működő Egerváry Jenő Kombinatorikus Optimalizálási Kutatócsoportot. A csoport a kutatási tervben szereplő több témában jelentős eredményeket ért el az elmúlt 4 évben, ezekről a pályázat résztvevőinek több mint 50 folyóiratcikke jelent meg, és számos rangos nemzetközi konferencián ismertetésre kerültek. Néhány kiemelendő eredmény: sikerült polinomiális kombinatorikus algoritmust adni irányított gráf pont-összefüggőségének növelésére; jelentős előrelépés történt a háromdimenziós térben merev gráfok jellemzésével és a molekuláris sejtéssel kapcsolatban; 2 dimenzióban sikerült bizonyítani Hendrickson sejtését; a párosításelméletben egy újdonságnak számító módszerrel számos új algoritmikus eredmény született; több, gráfok élösszefüggőségét jellemző tételt sikerült hipergráfokra általánosítani. | As the research plan indicates, the researchers participating in the project are the members of the Egerváry Research Group, led by the coordinator. The group has made important progress in the past 4 years in the research topics declared in the research plan. The results have been published in more than 50 journal papers, and have been presented at several prestigious international conferences. The most significant results are the following: a polynomial algorithm has been found for the node-connectivity augmentation problem of directed graphs; considerable progress has been made towards the characterization of 3-dimensional rigid graphs and towards the proof of the molecular conjecture; Hendrickson's conjecture has been proved in 2 dimensions; several new algorithmic results were obtained in matching theory using a novel approach; several theorems characterizing connectivity properties of graphs have been generalized to hypergraphs

New descriptions of the Lovász number, and the weak sandwich theorem

In 1979, L. Lovász introduced the concept of an orthonormal representation of a graph, and also a related value, now popularly known as the Lovász number of the graph. One of the remarkable properties of the Lovász number is that it lies sandwiched between the stability number of the graph and the chromatic number of the complementary graph. This fact is called the sandwich theorem. In this paper, using new descriptions of the Lovász number and linear algebraic lemmas we give three proofs for a weaker version of the sandwich theorem

On closedness conditions, strong separation, and convex duality

In the paper, we describe various applications of closedness and duality theorems from previous works of the author. First, the strong separability of a polyhedron and a linear image of a convex set is characterized. Then, it is shown how stability conditions (known from the generalized Fenchel-Rockafellar duality theory) can be reformulated as closedness conditions. Finally, we present a generalized Lagrangian duality theorem for Lagrangian programs described with cone-convex/cone-polyhedral mappings