Cycles and Bases of Graphs and Matroids

The objective of this dissertation is to investigate the properties of cycles and bases in matroids and in graphs. In [62], Tutte defined the circuit graph of a matroid and proved that a matroid is connected if and only if its circuit graph is connected. Motivated by Tutte\u27s result, we introduce the 2nd order circuit graph of a matroid, and prove that for any connected matroid M other than U1,1, the second order circuit graph of M has diameter at most 2 if and only if M does not have a restricted minor isomorphic to U2,6.;Another research conducted in this dissertation is related to the eulerian subgraph problem in graph theory. A graph G is eulerian if G is connected without vertices of odd degrees, and G is supereulerian if G has a spanning eulerian subgraph. In [3], Boesch, Suffey and Tindel raised a problem to determine when a graph is supereulerian, and they remarked that such a problem would be a difficult one. In [55], Pulleyblank confirmed the remark by showing that the problem to determine if a graph is supereulerian, even within planar graphs, is NP-complete. Catlin in [8] introduced a reduction method based on the theory of collapsible graphs to search for spanning eulerian subgraphs in a given graph G. In this dissertation, we introduce the supereulerian width of a graph G, which generalizes the concept of supereulerian graphs, and extends the supereulerian problem to the supereulerian width problem in graphs. Further, we also generalize the concept of collapsible graphs to s-collapsible graphs and develop the reduction method based on the theory of s-collapsible graphs. Our studies extend the collapsible graph theory of Catlin. These are applied to show for any integer n \u3e 2, the complete graph Kn is (n - 3)- collapsible, and so the supereulerian width of Kn is n - 2. We also prove a best possible degree condition for a simple graph to have supereulerian width at least 3.;The number of edge-disjoint spanning trees plays an important role in the design of networks, as it is considered as a measure of the strength of the network. As disjoint spanning trees are disjoint bases in graphic matroids, it is important to study the properties related to the number of disjoint bases in matroids. In this dissertation, we develop a decomposition theory based on the density function of a matroid, and prove a decomposition theorem that partitions the ground set of a matroid M into subsets based on their densities. As applications of the decomposition theorem, we investigate problems related to the properties of disjoint bases in a matroid. We showed that for a given integer k \u3e 0, any matroid M can be embedded into a matroid M\u27 with the same rank (that is, r(M) = r( M\u27)) such that M\u27 has k disjoint bases. Further we determine the minimum value of |E( M\u27)| -- |E(M)| in terms of invariants of M. For a matroid M with at least k disjoint bases, we characterize the set of elements in M such that removing any one of them would still result in a matroid with at least k disjoint bases

Characterization of removable elements with respect to having k disjoint bases in a matroid

AbstractThe well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with k edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with k disjoint bases. For any graph G that may not have k-edge-disjoint spanning trees, the problem of determining what edges should be added to G so that the resulting graph has k edge-disjoint spanning trees has been studied by Haas (2002) [11] and Liu et al. (2009) [17], among others. This paper aims to determine, for a matroid M that has k disjoint bases, the set Ek(M) of elements in M such that for any e∈Ek(M), M−e also has k disjoint bases. Using the matroid strength defined by Catlin et al. (1992) [4], we present a characterization of Ek(M) in terms of the strength of M. Consequently, this yields a characterization of edge sets Ek(G) in a graph G with at least k edge-disjoint spanning trees such that ∀e∈Ek(G), G−e also has k edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set Ek(M) in a matroid M, or the edge subset Ek(G) for a connected graph G

Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

Densities in graphs and matroids

Certain graphs can be described by the distribution of the edges in its subgraphs. For example, a cycle C is a graph that satisfies |E(H)| |V (H)| < |E(C)| |V (C)| = 1 for all non-trivial subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)| |V (H)|−1 ≤ |E(T)| |V (T)|−1 = 1 for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such that |E(H)| |V (H)| ≤ |E(G)| |V (G)| and a 1-balanced graph is a graph such that |E(H)| |V (H)|−1 ≤ |E(G)| |V (G)|−1 for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G that satisfy the property |E(H)| ≤ k|V (H)| − l for all non-trivial subgraphs H of G play important roles in defining rigid structures. This dissertation is a formal study of a class of density functions that extends the above mentioned ideas. For a rational number r ≤ 1, a graph G is said to be r-balanced if and only if for each non-trivial subgraph H of G, we have |E(H)| |V (H)|−r ≤ |E(G)| |V (G)|−r . For r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined and the existence of these graphs is discussed. We also define a class of vulnerability measures on graphs similar to the edge-connectivity of graphs and show how it is related to r-balanced graphs. All these definitions are matroidal and the definitions of r-balanced matroids naturally extend the definitions of r-balanced graphs. The vulnerability measures in graphs that we define are ranked and are lesser than the edge-connectivity. Due to the relationship of the r-balanced graphs with the vulnerability measures defined in the dissertation, identifying r-balanced graphs and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced matroids and their weak forms are discussed. For r ∈ {0, 1}, we give a method to construct big r-balanced graphs from small r-balanced graphs. This construction is a generalization of the construction of Cartesian product of two graphs. We present an algorithmic solution of the problem of transforming any given graph into a 1-balanced graph on the same number of vertices and edges as the given graph. This result is extended to a density function defined on the power set of any set E via a pair of matroid rank functions defined on the power set of E. Many interesting results may be derived in the future by choosing suitable pairs of matroid rank functions and applying the above result

HIPERC: a novel, high performance, economic steel concept for linepipe and general structural use

The HIPERC project has examined the effects of alloying elements and processing conditions in low carbon, < 0.09 wt %., niobium containing, 0.05 - 0.12 wt.%, steels. Laboratory-scale heats and pilot rolling trials simulating air and water-cooled plate production as well as hot-rolled strip production have been made. The effects of C, Mn, Ni, Cu, Cr, Mo, Nb, Ti and B, on transformation characteristics and temperatures of recrystallisation have been determined along with regression equations for characterisation of microstructure, tensile and impact properties, and for the weldability of these steels. The properties of products processed commercially to plate and coil-plate and made into pipe and to plate for structural use were determined and these compared well with the values predicted from the regression equations. The project has shown that excellent combinations of strength, toughness and weldability can be obtained using this steel type. Additional experiences have been gained in the processing of these steels through three commercial rolling mills and benefits were seen with this steel type due to higher production rates and lower amounts of surface dressing compared with steels currently being used to satisfy equivalent property specifications. Recommendations on the limits for niobium in Euronorms have been proposed; concerns relating to weldability have been addressed by proposing varying limits based on the carbon and manganese contents of the steel. This report makes the output of this project available to CEN working groups to support the revision of Euronorms based on the gathered data

Applied Fracture Mechanics

The book "Applied Fracture Mechanics" presents a collection of articles on application of fracture mechanics methods to materials science, medicine, and engineering. In thirteen chapters, a wide range of topics is discussed, including strength of biological tissues, safety of nuclear reactor components, fatigue effects in pipelines, environmental effects on fracture among others. In addition, the book presents mathematical and computational methods underlying the fracture mechanics applications, and also developments in statistical modeling of fatigue. The work presented in this book will be useful, effective, and beneficial to mechanical engineers, civil engineers, and material scientists from industry, research, and education

Mechanical behavior of alternative multicrystalline silicon for solar cells

The usage of more inexpensive silicon feedstock for the crystallization of multicrystalline silicon blocks promises cost reduction for the photovoltaic industry. Less expensive substrates made out of metallurgical silicon (MG-Si) are used as a mechanical support for the epitaxial solar cell. Moreover, conventional inert solar cells can be produced from up-graded metallurgical silicon (UMG-Si). This feedstock has higher content of impurities which influences cell performance and mechanical strength of the wafers. Thus, it is of importance to know these effects in order to know which impurities should be preferentially removed or prevented during the crystallization process. Solar cell processing steps can also exert a change in the values of mechanical strength of processed multicrystalline silicon wafers until the fabrication of a solar cell. Bending tests, fracture toughness and dynamic elastic modulus measurements are performed in this work in order to research the mechanical behavior of multicrystalline silicon crystallized with different qualities of silicon feedstock. Bending tests and residual stress measurements allows the quantification of the mechanical strength of the wafers after every solar cell processing step. The experimental results are compared with theoretical models found in the classical literature about the mechanical properties of ceramics. The influence of second phase particles and thermal processes on the mechanical strength of silicon wafers can be predicted and analyzed with the theoretical models. Metals like Al and Cu can decrease the mechanical strength due to micro-cracking of the silicon matrix and introduction of high values of thermal residual stress. Additionally, amorphous silicon oxide particles (SiOx) lower the mechanical strength of multicrystalline silicon due to thermal residual stresses and elastic mismatch with silicon. Silicon nitride particles (Si3N4) reduce fracture toughness and cause failure by radial cracking in its surroundings due to its thermal mismatch with silicon. Finally, silicon carbide (SiC) and crystalline silicon oxide (SiOx) introduce thermal residual stresses but can have a toughening effect on the silicon matrix and hence, increase the mechanical strength of silicon wafers if the particles are smaller than a certain size. The surface of as-cut wafers after multi-wire sawing presents sharp micro-cracks that control their mechanical behavior. Subsequent removal of these micro-cracks by texture or damage etching approximately doubles the mechanical strength of silicon wafers. The mechanical behavior of the wafers is then governed by defects like cracks and particles formed during the crystallization of multicrystalline silicon blocks. Further thermal processing steps have a minor impact on the mechanical strength of the wafers compared to as-cut wafers. Finally, the mechanical strength of final solar cells is comparable to the mechanical strength of as-cut wafers due to the high residual thermal stress introduced after the formation of the metallic contacts which makes silicon prone to crack

Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a (1+?) approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate B such that B ? B^* ? ?(1+?)B?+1 where B^* is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization

The Influence of Fibre Processing and Treatments on Hemp Fibre/Epoxy and Hemp Fibre/PLA Composites

In recent years, due to growing environmental awareness, considerable attention has been given to the development and production of natural fibre reinforced polymer (both thermoset and thermoplastic) composites. The main objective of this study was to reinforce epoxy and polylactic acid (PLA) with hemp fibre to produce improved composites by optimising the fibre treatment methods, composite processing methods, and fibre/matrix interfacial bonding. An investigation was conducted to obtain a suitable fibre alkali treatment method to: (i) remove non-cellulosic fibre components such as lignin (sensitive to ultra violet (UV) radiation) and hemicelluloses (sensitive to moisture) to improve long term composites stability (ii) roughen fibre surface to obtain mechanical interlocking with matrices (iii)expose cellulose hydroxyl groups to obtain hydrogen and covalent bonding with matrices (iv) separate the fibres from their fibre bundles to make the fibre surface available for bonding with matrices (v) retain tensile strength by keeping fibre damage to a minimum level and (vi) increase crystalline cellulose by better packing of cellulose chains to enhance the thermal stability of the fibres. An empirical model was developed for fibre tensile strength (TS) obtained with different treatment conditions (different sodium hydroxide (NaOH) and sodium sulphite (Na2SO3) concentrations, treatment temperatures, and digestion times) by a partial factorial design. Upon analysis of the alkali fibre treatments by single fibre tensile testing (SFTT), scanning electron microscopy (SEM), zeta potential measurements, differential thermal analysis/thermogravimetric analysis (DTA/TGA), wide angle X-ray diffraction (WAXRD), lignin analysis and Fourier transform infrared (FTIR) spectroscopy, a treatment consisting of 5 wt% NaOH and 2 wt% Na2SO3 concentrations, with a treatment temperature of 120oC and a digestion time of 60 minutes, was found to give the best combination of the required properties. This alkali treatment produced fibres with an average TS and Young's modulus (YM) of 463 MPa and 33 GPa respectively. The fibres obtained with the optimised alkali treatment were further treated with acetic anhydride and phenyltrimethoxy silane. However, acetylated and silane treated fibres were not found to give overall performance improvement. Cure kinetics of the neat epoxy (NE) and 40 wt% untreated fibre/epoxy (UTFE) composites were studied and it was found that the addition of fibres into epoxy resin increased the reaction rate and decreased the curing time. An increase in the nucleophilic activity of the amine groups in the presence of fibres is believed to have increased the reaction rate of the fibre/epoxy resin system and hence reduced the activation energies compared to NE. The highest interfacial shear strength (IFSS) value for alkali treated fibre/epoxy (ATFE) samples was 5.2 MPa which was larger than the highest value of 2.7 MPa for UTFE samples supporting that there was a stronger interface between alkali treated fibre and epoxy resin. The best fibre/epoxy bonding was found for an epoxy to curing agent ratio of 1:1 (E1C1) followed by epoxy to curing agent ratios of 1:1.2 (E1C1.2), 1: 0.8 (E1C0.8), and finally for 1:0.6 (E1C0.6). Long and short fibre reinforced epoxy composites were produced with various processing conditions using vacuum bag and compression moulding. A 65 wt% untreated long fibre/epoxy (UTLFE) composite produced by compression moulding at 70oC with a TS of 165 MPa, YM of 17 GPa, flexural strength of 180 MPa, flexural modulus of 10.1 GPa, impact energy (IE) of 14.5 kJ/m2, and fracture toughness (KIc) of 5 MPa.m1/2 was found to be the best in contrast to the trend of increased IFSS for ATFE samples. This is considered to be due to stress concentration as a result of increased fibre/fibre contact with the increased fibre content in the ATFE composites compared to the UTFE composites. Hygrothermal ageing of 65 wt% untreated and alkali treated long and short fibre/epoxy composites (produced by curing at 70oC) showed that long fibre/epoxy composites were more resistant than short fibre/epoxy composites and ATFE composites were more resistant than UTFE composites towards hygrothermal ageing environments as revealed from diffusion coefficients and tensile, flexural, impact, fracture toughness, SEM, TGA, and WAXRD test results. Accelerated ageing of 65 wt% UTLFE and alkali treated long fibre/epoxy (ATLFE) composites (produced by curing at 70oC) showed that ATLFE composites were more resistant than UTLFE composites towards hygrothermal ageing environments as revealed from tensile, flexural, impact, KIc, SEM, TGA, WAXRD, FTIR test results. IFSS obtained with untreated fibre/PLA (UFPLA) and alkali treated fibre/PLA (ATPLA) samples showed that ATPLA samples had greater IFSS than that of UFPLA samples. The increase in the formation of hydrogen bonding and mechanical interlocking of the alkali treated fibres with PLA could be responsible for the increased IFSS for ATPLA system compared to UFPLA system. Long and short fibre reinforced PLA composites were also produced with various processing conditions using compression moulding. A 32 wt% alkali treated long fibre PLA composite produced by film stacking with a TS of 83 MPa, YM of 11 GPa, flexural strength of 143 MPa, flexural modulus of 6.5 GPa, IE of 9 kJ/m2, and KIc of 3 MPa.m1/2 was found to be the best. This could be due to the better bonding of the alkali treated fibres with PLA. The mechanical properties of this composite have been found to be the best compared to the available literature. Hygrothermal and accelerated ageing of 32 wt% untreated and alkali treated long fibre/PLA composites ATPLA composites were more resistant than UFPLA composites towards hygrothermal and accelerated ageing environments as revealed from diffusion coefficients and tensile, flexural, impact, KIc, SEM, differential scanning calorimetry (DSC), WAXRD, and FTIR results. Increased potential hydrogen bond formation and mechanical interlocking of the alkali treated fibres with PLA could be responsible for the increased resistance of the ATPLA composites. Based on the present study, it can be said that the performance of natural fibre composites largely depend on fibre properties (e.g. length and orientation), matrix properties (e.g. cure kinetics and crystallinity), fibre treatment and processing methods, and composite processing methods