Acoustic emission source location and damage detection in a metallic structure using a graph-theory-based geodesic pproach

A geodesic-based approach using Lamb waves is proposed to locate the acoustic emission (AE) source and damage in an isotropic metallic structure. In the case of the AE (passive) technique, the elastic waves take the shortest path from the source to the sensor array distributed in the structure. The geodesics are computed on the meshed surface of the structure using graph theory based on Dijkstra's algorithm. By propagating the waves in reverse virtually from these sensors along the geodesic path and by locating the first intersection point of these waves, one can get the AE source location. The same approach is extended for detection of damage in a structure. The wave response matrix of the given sensor configuration for the healthy and the damaged structure is obtained experimentally. The healthy and damage response matrix is compared and their difference gives the information about the reflection of waves from the damage. These waves are backpropagated from the sensors and the above method is used to locate the damage by finding the point where intersection of geodesics occurs. In this work, the geodesic approach is shown to be suitable to obtain a practicable source location solution in a more general set-up on any arbitrary surface containing finite discontinuities. Experiments were conducted on aluminum specimens of simple and complex geometry to validate this new method.

Computerized synthesis of the structure of geared kinematic chains

The paper presents for the first time a fully computerized method for structural synthesis of geared kinematic chains which can be used to derive epicyclic gear drives. The method has been formulated on the basis of representing these chains by their graphs, the graphs being in turn represented algebraically by their vertex-vertex incidence matrices. It has thus been possible to make advantageous use of concepts and results from graph theory to develop a method amenable for implementation on a digital computer. The computerized method has been applied to the structural synthesis of single-freedom geared kinematic chains with up to four gear pairs, and the results obtained thereform are presented and discussed.

Extrapolation and series acceleration procedures: Hard spheres and discs as illustrations

Three new procedures - in the context of estimation of virial coefficients and summation of the partial virial series for hard discs and hard spheres - are proposed. They are based on the parametrised Euler transformation, a novel resummation, identity and the ε-convergence methods respectively. A comparison with other estimates (molecular dynamics, graph theory and empirical methods) reveals satisfactory agreement.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

Acyclic Edge Coloring of Graphs with Maximum Degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a'(G) <= Delta+2, where Delta=Delta(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Delta(G)<= 4, with the additional restriction that m <= 2n-1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m <= 2n, when Delta(G)<= 4. It follows that for any graph G if Delta(G)<= 4, then a'(G) <= 7.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

Computer-aided analysis of the structure of kinematic chains

Using the link-link incidence matrix to represent a simple-jointed kinematic chain algebraic procedures have been developed to determine its structural characteristics such as the type of freedom of the chain, the number of distinct mechanisms and driving mechanisms that can be derived from the chain. A computer program incorporating these graph theory based procedures has been applied successfully for the structural analysis of several typical chains.

Acoustic emission source location in composite structure by Voronoi construction using geodesic curve evolution

Conventional analytical/numerical methods employing triangulation technique are suitable for locating acoustic emission (AE) source in a planar structure without structural discontinuities. But these methods cannot be extended to structures with complicated geometry, and, also, the problem gets compounded if the material of the structure is anisotropic warranting complex analytical velocity models. A geodesic approach using Voronoi construction is proposed in this work to locate the AE source in a composite structure. The approach is based on the fact that the wave takes minimum energy path to travel from the source to any other point in the connected domain. The geodesics are computed on the meshed surface of the structure using graph theory based on Dijkstra's algorithm. By propagating the waves in reverse virtually from these sensors along the geodesic path and by locating the first intersection point of these waves, one can get the AE source location. In this work, the geodesic approach is shown more suitable for a practicable source location solution in a composite structure with arbitrary surface containing finite discontinuities. Experiments have been conducted on composite plate specimens of simple and complex geometry to validate this method.

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η(G) denote the largest clique minor of a graph G, and let χ(G) denote its chromatic number. Hadwiger's conjecture states that η(G)greater-or-equal, slantedχ(G) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η(G) is guaranteed not to grow too fast with respect to χ(G), since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η(G)less-than-or-equals, slant2χ(G)−1, and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.

On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

A unit cube in k dimensions (k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), a(i) + 1] on the real line. A graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of C can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes. An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We give an O(bw . n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(Delta) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and cocomparability graphs which have O(Delta) bandwidth. Thus we have: 1. cub(G) <= 3 Delta - 1, if G is an AT-free graph. 2. cub(G) <= 2 Delta + 1, if G is a circular-arc graph. 3. cub(G) <= 2 Delta, if G is a cocomparability graph. Also for these graph classes, there axe constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(Delta) width. We can thus generate the cube representation of such graphs in O(Delta) dimensions in polynomial time.

d-Regular Graphs of Acyclic Chromatic Index at Least d+2

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Suclakov and Zaks (and earlier by Fiamcik) that a'(G) <= Delta+2, where Delta = Delta(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Delta+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d>n, requires at least d+2 colors. We also show that a'(K-n,K-n) >= n+2, when n is odd using a more non-trivial argument. (Here K-n,K-n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d >= 5, n >= 2d+3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226-230, 2010.

Cubicity of Interval Graphs and the Claw Number

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b-dimensional cube is a Cartesian product l(1) x l(2) x ... x l(b), where each l(i) is a closed interval of unit length on the real line. The cub/city of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line-i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number psi(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least log(2) psi(G)]. In this article, we show that for an interval graph G log(2) psi(G)-]<= cub(G)<=log(2) psi(G)]+2. It is not clear whether the upper bound of log(2) psi(G)]+2 is tight: till now we are unable to find any interval graph with cub(G)> (log(2)psi(G)]. We also show that for an interval graph G, cub(G) <= log(2) alpha], where alpha is the independence number of G. Therefore, in the special case of psi(G)=alpha, cub(G) is exactly log(2) alpha(2)]. The concept of cubicity can be generalized by considering boxes instead of cubes. A b-dimensional box is a Cartesian product l(1) x l(2) x ... x l(b), where each I is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k-dimensional boxes. It is clear that box(G)<= cub(G). From the above result, it follows that for any graph G, cub(G) <= box(G)log(2) alpha]. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323-333, 2010

Metabolome based reaction graphs of M.tuberculosis and M.leprae: A comparative network analysis

Background. Several types of networks, such as transcriptional, metabolic or protein-protein interaction networks of various organisms have been constructed, that have provided a variety of insights into metabolism and regulation. Here, we seek to exploit the reaction-based networks of three organisms for comparative genomics. We use concepts from spectral graph theory to systematically determine how differences in basic metabolism of organisms are reflected at the systems level and in the overall topological structures of their metabolic networks. Methodology/Principal Findings. Metabolome-based reaction networks of Mycobacterium tuberculosis, Mycobacterium leprae and Escherichia coli have been constructed based on the KEGG LIGAND database, followed by graph spectral analysis of the network to identify hubs as well as the sub-clustering of reactions. The shortest and alternate paths in the reaction networks have also been examined. Sub-cluster profiling demonstrates that reactions of the mycolic acid pathway in mycobacteria form a tightly connected sub-cluster. Identification of hubs reveals reactions involving glutamate to be central to mycobacterial metabolism, and pyruvate to be at the centre of the E. coli metabolome. The analysis of shortest paths between reactions has revealed several paths that are shorter than well established pathways. Conclusions. We conclude that severe downsizing of the leprae genome has not significantly altered the global structure of its reaction network but has reduced the total number of alternate paths between its reactions while keeping the shortest paths between them intact. The hubs in the mycobacterial networks that are absent in the human metabolome can be explored as potential drug targets. This work demonstrates the usefulness of constructing metabolome based networks of organisms and the feasibility of their analyses through graph spectral methods. The insights obtained from such studies provide a broad overview of the similarities and differences between organisms, taking comparative genomics studies to a higher dimension.

Network approach for capturing ligand-induced subtle global changes in protein structures

Ligand-induced conformational changes in proteins are of immense functional relevance. It is a major challenge to elucidate the network of amino acids that are responsible for the percolation of ligand-induced conformational changes to distal regions in the protein from a global perspective. Functionally important subtle conformational changes (at the level of side-chain noncovalent interactions) upon ligand binding or as a result of environmental variations are also elusive in conventional studies such as those using root-mean-square deviations (r.m.s.d.s). In this article, the network representation of protein structures and their analyses provides an efficient tool to capture these variations (both drastic and subtle) in atomistic detail in a global milieu. A generalized graph theoretical metric, using network parameters such as cliques and/or communities, is used to determine similarities or differences between structures in a rigorous manner. The ligand-induced global rewiring in the protein structures is also quantified in terms of network parameters. Thus, a judicious use of graph theory in the context of protein structures can provide meaningful insights into global structural reorganizations upon perturbation and can also be helpful for rigorous structural comparison. Data sets for the present study include high-resolution crystal structures of serine proteases from the S1A family and are probed to quantify the ligand-induced subtle structural variations.