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.

Brain and Spinal Cavernomas : Helsinki Experience

Cavernomas are rare neurovascular lesions, encountered in up to 10% of patients harboring vascular abnormalities of the CNS. Cavernomas consist of dilated thin-walled sinusoids or caverns covered by a single layer of endothelium. Due to advancements in neuroradiology, the number of cavernoma patients coming to be evaluated in neurosurgical practice is increasing. In the present work, we summarized our results on the treatment of cavernomas. Particular attention was paid to uncommon locations or insufficiently investigated cavernomas, including 1. Intraventricular cavernomas; 2. Multiple cavernomas; 3. Spinal cavernomas; and 4. Temporal lobe cavernomas. After analyzing the patient series with these lesions, we concluded that: 1. IVCs are characterized by a high tendency to cause repetitive hemorrhages in a short period of time after the first event. In most patients, hemorrhages were not life-threatening. Surgery is indicated when re-bleedings are frequent and the mass-effect causes progressive neurological deterioration. Modern microsurgical techniques allow safe removal of the IVC, but surgery on fourth ventricle cavernomas carries increased risk of postoperative cranial nerve deficits. 2. In MC cases, when the cavernoma bleeds or generates drug-resistant epilepsy, microsurgical removal of the symptomatic lesion is beneficial to patients. In our series, surgical removal of the most active cavernoma usually the biggest lesion with signs of recent hemorrhage - was safe and prevented further bleedings. Epilepsy outcome showed the effectiveness of active treatment of MCs. However, due to the remaining cavernomas, epileptogenic activity can persist postoperatively, frequently necessitating long-term use of antiepileptic drugs. 3. Spinal cavernomas can cause severe neurological deterioration due to low tolerance of the spinal cord to mass-effect with progressive myelopathy. When aggravated by extralesional massive hemorrhage, neurological decline is usually acute and requires immediate treatment. Microsurgical removal of a cavernoma is effective and safe, improving neurological deficits. Sensorimotor deficits and pain improved postoperatively at a high rate, whereas bladder dysfunction remained essentially unchanged, causing social discomfort to patients. 4. Microsurgical removal of temporal lobe cavernomas is beneficial for patents suffering from drug-resistant epilepsy. In our series, 69% of patients with this condition became seizure-free postoperatively. Duration of epilepsy did not correlate with seizure prognosis. The most frequent disabling symptom at follow-up was memory disorder, considered to be the result of a complex interplay between chronic epilepsy and possible damage to the temporal lobe during surgery.

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number eta(G) of a graph G is the largest integer n for which the complete graph K-n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, eta(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G square H of graphs. As the main result of this paper, we prove that eta(G(1) square G(2)) >= h root 1 (1 - o(1)) for any two graphs G(1) and G(2) with eta(G(1)) = h and eta(G(2)) = l. We show that the above lower bound is asymptotically best possible when h >= l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let G = G(1) square G(2) square ... square G(k) be the ( unique) prime factorization of G. Then G satisfies Hadwiger's conjecture if k >= 2 log log chi(G) + c', where c' is a constant. This improves the 2 log chi(G) + 3 bound in [2] 2. Let G(1) and G(2) be two graphs such that chi(G1) >= chi(G2) >= clog(1.5)(chi(G(1))), where c is a constant. Then G1 square G2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G(d) (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan [2]. ( They had shown that the Hadiwger's conjecture is true for G(d) if d >= 3).

Isoperimetric Problem and Meta-fibonacci Sequences

Let G = (V,E) be a simple, finite, undirected graph. For S ⊆ V, let $\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$ and $\phi(S,G) = \{ v \in V -S: \exists u \in S$ , such that (u,v) ∈ E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i ≤ ∣ V ∣, the edge and vertex isoperimetric value at i is defined as b e (i,G) =  min S ⊆ V; |S| = i |δ(S,G)| and b v (i,G) =  min S ⊆ V; |S| = i |φ(S,G)|, respectively. The edge (vertex) isoperimetric problem is to determine the value of b e (i, G) (b v (i, G)) for each i, 1 ≤ i ≤ |V|. If we have the further restriction that the set S should induce a connected subgraph of G, then the corresponding variation of the isoperimetric problem is known as the connected isoperimetric problem. The connected edge (vertex) isoperimetric values are defined in a corresponding way. It turns out that the connected edge isoperimetric and the connected vertex isoperimetric values are equal at each i, 1 ≤ i ≤ |V|, if G is a tree. Therefore we use the notation b c (i, T) to denote the connected edge (vertex) isoperimetric value of T at i. Hofstadter had introduced the interesting concept of meta-fibonacci sequences in his famous book “Gödel, Escher, Bach. An Eternal Golden Braid”. The sequence he introduced is known as the Hofstadter sequences and most of the problems he raised regarding this sequence is still open. Since then mathematicians studied many other closely related meta-fibonacci sequences such as Tanny sequences, Conway sequences, Conolly sequences etc. Let T 2 be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T 2 with the Tanny sequences which is defined by the recurrence relation a(i) = a(i − 1 − a(i − 1)) + a(i − 2 − a(i − 2)), a(0) = a(1) = a(2) = 1. In particular, we show that b c (i, T 2) = i + 2 − 2a(i), for each i ≥ 1. We also propose efficient polynomial time algorithms to find vertex isoperimetric values at i of bounded pathwidth and bounded treewidth graphs.

On a Special Co-cycle Basis of Graphs

In this paper we consider the problems of computing a minimum co-cycle basis and a minimum weakly fundamental co-cycle basis of a directed graph G. A co-cycle in G corresponds to a vertex partition (S,V ∖ S) and a { − 1,0,1} edge incidence vector is associated with each co-cycle. The vector space over ℚ generated by these vectors is the co-cycle space of G. Alternately, the co-cycle space is the orthogonal complement of the cycle space of G. The minimum co-cycle basis problem asks for a set of co-cycles that span the co-cycle space of G and whose sum of weights is minimum. Weakly fundamental co-cycle bases are a special class of co-cycle bases, these form a natural superclass of strictly fundamental co-cycle bases and it is known that computing a minimum weight strictly fundamental co-cycle basis is NP-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory-Hu tree of the underlying undirected graph of G is a minimum co-cycle basis of G and it is also weakly fundamental.

New approximation algorithms for minimum cycle bases of graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over F-2 generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. We present two new algorithms to compute an approximate minimum cycle basis. For any integer k >= 1, we give (2k - 1)-approximation algorithms with expected running time 0(kmn(1+2/k) + mn((1+1/k)(omega-1))) and deterministic running time 0(n(3+2/k)), respectively. Here omega is the best exponent of matrix multiplication. It is presently known that omega < 2.376. Both algorithms are o(m(omega)) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Theta(m(omega)) bound. We also present a 2-approximation algorithm with O(m(omega) root n log n) expected running time, a linear time 2-approximation algorithm for planar graphs and an O(n(3)) time 2.42-approximation algorithm for the complete Euclidean graph in the plane.

The enhanced Russell-based directional distance measure with undesirable outputs: Numerical example considering CO2 emissions

Following the spirit of the enhanced Russell graph measure, this paper proposes an enhanced Russell-based directional distance measure (ERBDDM) model for dealing with desirable and undesirable outputs in data envelopment analysis (DEA) and allowing some inputs and outputs to be zero. The proposed method is analogous to the output oriented slacks-based measure (OSBM) and directional output distance function approach because it allows the expansion of desirable outputs and the contraction of undesirable outputs. The ERBDDM is superior to the OSBM model and traditional approach since it is not only able to identify all the inefficiency slacks just as the latter, but also avoids the misperception and misspecification of the former, which fails to identify null-jointness production of goods and bads. The paper also imposes a strong complementary slackness condition on the ERBDDM model to deal with the occurrence of multiple projections. Furthermore, we use the Penn Table data to help us explore our new approach in the context of environmental policy evaluations and guidance for performance improvements in 111 countries.

Time and Energy Complexity of Distributed Computation of a Class of Functions in Wireless Sensor Networks

We consider a scenario in which a wireless sensor network is formed by randomly deploying n sensors to measure some spatial function over a field, with the objective of computing a function of the measurements and communicating it to an operator station. We restrict ourselves to the class of type-threshold functions (as defined in the work of Giridhar and Kumar, 2005), of which max, min, and indicator functions are important examples: our discussions are couched in terms of the max function. We view the problem as one of message-passing distributed computation over a geometric random graph. The network is assumed to be synchronous, and the sensors synchronously measure values and then collaborate to compute and deliver the function computed with these values to the operator station. Computation algorithms differ in (1) the communication topology assumed and (2) the messages that the nodes need to exchange in order to carry out the computation. The focus of our paper is to establish (in probability) scaling laws for the time and energy complexity of the distributed function computation over random wireless networks, under the assumption of centralized contention-free scheduling of packet transmissions. First, without any constraint on the computation algorithm, we establish scaling laws for the computation time and energy expenditure for one-time maximum computation. We show that for an optimal algorithm, the computation time and energy expenditure scale, respectively, as Theta(radicn/log n) and Theta(n) asymptotically as the number of sensors n rarr infin. Second, we analyze the performance of three specific computation algorithms that may be used in specific practical situations, namely, the tree algorithm, multihop transmission, and the Ripple algorithm (a type of gossip algorithm), and obtain scaling laws for the computation time and energy expenditure as n rarr infin. In particular, we show that the computation time for these algorithms scales as Theta(radicn/lo- g n), Theta(n), and Theta(radicn log n), respectively, whereas the energy expended scales as , Theta(n), Theta(radicn/log n), and Theta(radicn log n), respectively. Finally, simulation results are provided to show that our analysis indeed captures the correct scaling. The simulations also yield estimates of the constant multipliers in the scaling laws. Our analyses throughout assume a centralized optimal scheduler, and hence, our results can be viewed as providing bounds for the performance with practical distributed schedulers.

Automatic Road Extraction using High Resolution Satellite Image Based on Texture Progressive Analysis and Normalized Cut Method

In this paper the approach for automatic road extraction for an urban region using structural, spectral and geometric characteristics of roads has been presented. Roads have been extracted based on two levels: Pre-processing and road extraction methods. Initially, the image is pre-processed to improve the tolerance by reducing the clutter (that mostly represents the buildings, parking lots, vegetation regions and other open spaces). The road segments are then extracted using Texture Progressive Analysis (TPA) and Normalized cut algorithm. The TPA technique uses binary segmentation based on three levels of texture statistical evaluation to extract road segments where as, Normalizedcut method for road extraction is a graph based method that generates optimal partition of road segments. The performance evaluation (quality measures) for road extraction using TPA and normalized cut method is compared. Thus the experimental result show that normalized cut method is efficient in extracting road segments in urban region from high resolution satellite image.

A network representation of protein structures: Implications for protein stability

This study views each protein structure as a network of noncovalent connections between amino acid side chains. Each amino acid in a protein structure is a node, and the strength of the noncovalent interactions between two amino acids is evaluated for edge determination. The protein structure graphs (PSGs) for 232 proteins have been constructed as a function of the cutoff of the amino acid interaction strength at a few carefully chosen values. Analysis of such PSGs constructed on the basis of edge weights has shown the following: 1), The PSGs exhibit a complex topological network behavior, which is dependent on the interaction cutoff chosen for PSG construction. 2), A transition is observed at a critical interaction cutoff, in all the proteins, as monitored by the size of the largest cluster (giant component) in the graph. Amazingly, this transition occurs within a narrow range of interaction cutoff for all the proteins, irrespective of the size or the fold topology. And 3), the amino acid preferences to be highly connected (hub frequency) have been evaluated as a function of the interaction cutoff. We observe that the aromatic residues along with arginine, histidine, and methionine act as strong hubs at high interaction cutoffs, whereas the hydrophobic leucine and isoleucine residues get added to these hubs at low interaction cutoffs, forming weak hubs. The hubs identified are found to play a role in bringing together different secondary structural elements in the tertiary structure of the proteins. They are also found to contribute to the additional stability of the thermophilic proteins when compared to their mesophilic counterparts and hence could be crucial for the folding and stability of the unique three-dimensional structure of proteins. Based on these results, we also predict a few residues in the thermophilic and mesophilic proteins that can be mutated to alter their thermal stability.

Boxicity and cubicity of asteroidal triple free graphs

The boxicity of a graph G, denoted box(G), is the least integer d such that G is the intersection graph of a family of d-dimensional (axis-parallel) boxes. The cubicity, denoted cub(G), is the least dsuch that G is the intersection graph of a family of d-dimensional unit cubes. An independent set of three vertices is an asteroidal triple if any two are joined by a path avoiding the neighbourhood of the third. A graph is asteroidal triple free (AT-free) if it has no asteroidal triple. The claw number psi(G) is the number of edges in the largest star that is an induced subgraph of G. For an AT-free graph G with chromatic number chi(G) and claw number psi(G), we show that box(G) <= chi(C) and that this bound is sharp. We also show that cub(G) <= box(G)([log(2) psi(G)] + 2) <= chi(G)([log(2) psi(G)] + 2). If G is an AT-free graph having girth at least 5, then box(G) <= 2, and therefore cub(G) <= 2 [log(2) psi(G)] + 4. (c) 2010 Elsevier B.V. All rights reserved.

Comparative analysis of thermophilic and mesophilic proteins using Protein Energy Networks

Background: Thermophilic proteins sustain themselves and function at higher temperatures. Despite their structural and functional similarities with their mesophilic homologues, they show enhanced stability. Various comparative studies at genomic, protein sequence and structure levels, and experimental works highlight the different factors and dominant interacting forces contributing to this increased stability. Methods: In this comparative structure based study, we have used interaction energies between amino acids, to generate structure networks called as Protein Energy Networks (PENs). These PENs are used to compute network, sub-graph, and node specific parameters. These parameters are then compared between the thermophile-mesophile homologues. Results: The results show an increased number of clusters and low energy cliques in thermophiles as the main contributing factors for their enhanced stability. Further more, we see an increase in the number of hubs in thermophiles. We also observe no community of electrostatic cliques forming in PENs. Conclusion: In this study we were able to take an energy based network approach, to identify the factors responsible for enhanced stability of thermophiles, by comparative analysis. We were able to point out that the sub-graph parameters are the prominent contributing factors. The thermophiles have a better-packed hydrophobic core. We have also discussed how thermophiles, although increasing stability through higher connectivity retains conformational flexibility, from a cliques and communities perspective.

Comparative analysis of thermophilic and mesophilic proteins using Protein Energy Networks

Background: Thermophilic proteins sustain themselves and function at higher temperatures. Despite their structural and functional similarities with their mesophilic homologues, they show enhanced stability. Various comparative studies at genomic, protein sequence and structure levels, and experimental works highlight the different factors and dominant interacting forces contributing to this increased stability. Methods: In this comparative structure based study, we have used interaction energies between amino acids, to generate structure networks called as Protein Energy Networks (PENs). These PENs are used to compute network, sub-graph, and node specific parameters. These parameters are then compared between the thermophile-mesophile homologues. Results: The results show an increased number of clusters and low energy cliques in thermophiles as the main contributing factors for their enhanced stability. Further more, we see an increase in the number of hubs in thermophiles. We also observe no community of electrostatic cliques forming in PENs. Conclusion: In this study we were able to take an energy based network approach, to identify the factors responsible for enhanced stability of thermophiles, by comparative analysis. We were able to point out that the sub-graph parameters are the prominent contributing factors. The thermophiles have a better-packed hydrophobic core. We have also discussed how thermophiles, although increasing stability through higher connectivity retains conformational flexibility, from a cliques and communities perspective.

Non-disruptive and null-deflection mass flow measurement by a pressure compensation technique

Accurate mass flow measurement is very important in various monitoring and control applications. This paper proposes a novel method of fluid flow measurement by compensating the pressure drop across the ends of measuring unit using a compensating pump. The pressure drop due to the flow is balanced by a feedback control loop. This is a null-deflection type of measurement. As the insertion of such a measuring unit does not affect the functioning of the systems, this is also a non-disruptive flow measurement method. The implementation and design of such a unit are discussed. The system is modeled and simulated using the bond graph technique and it is experimentally validated. (C) 2009 Elsevier Ltd. All rights reserved.

Clustering in Concept Association Networks

We view association of concepts as a complex network and present a heuristic for clustering concepts by taking into account the underlying network structure of their associations. Clusters generated from our approach are qualitatively better than clusters generated from the conventional spectral clustering mechanism used for graph partitioning.