EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G, ε(G) ≤ w(n,t), where w(n, t) = (n - 1) t/2 - r(t – r - 1)/2 and r = (n - 1 ) - (t - 1) ⎣(n - 1)/(t - 1)⎦. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta and Vijayan (1991). The edge- maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on : Let G(n, c, c ) denote the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c . The problem of interest is that of determining the bounds of the number of edges of a graph G ∈ G(n, c, c ) and characterize the extremal graphs of G(n, c, c ). We discuss the class G(n, c, c ) and present some results for small c. In particular for c = 4 and c = n - 2, we provide a complete solution. Key words : extremal graph, circumferenc

The Modified CW1 Algorithm For The Degree Restricted Minimum Spanning Tree Problem

Given edge weighted graph G (all weights are non-negative), The Degree Constrained Minimum Spanning Tree Problem is concerned with finding the minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem arises naturally in communication networks where the degree of a vertex represents the number of line interfaces available at a terminal (center). The applications of the Degree Constrained Minimum Spanning Tree problems that may arise in real-life include: the design of telecommunication, transportation, and energy networks. It is also used as a subproblem in the design of networks for computer communication, transportation, sewage and plumbing. Since, apart from some trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been proposed. In this paper we will discuss the modification of CW1 Algorithm that already proposed by Wamiliana and Caccetta (2003). The results on540 random table problems will be discussed

On The Graphs and Their Complements with Prescribed Circumference

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta & Vijayan (1991). The edge-maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on the class G (n, c, c) denoting the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c. The problem of interest is that of determining the bounds of the number of edges of a graph G∈ G(n, c, c) and characterize the extremal graphs of G(n, c, c). We discuss the class G (n, c, c) and present some results for small c. In particular for c=4 and c =n-2, we provide a complete solution

Computational aspects of the optimal transit path problem

In this paper we present a novel method for short term forecast of time series based on Knot-Optimizing Spline Networks (KOSNETS). The time series is first approximated by a nonlinear recurrent system. The resulting recurrent system is then approximated by feedforward B-spline networks, yielding a nonlinear optimization problem. In this optimization problem, both the knot points and the coefficients of the B-splines are decision variables so that the solution to the problem has both optimal coefficients and partition points. To demonstrate the usefulness and accuracy of the method, numerical simulations and tests using various model and real time series are performed. The numerical simulation results are compared with those from a well-known regression method, MARS. The comparison shows that our method outperforms MARS for nonlinear problems

Teucrium polium: Potential Drug Source for Type 2 Diabetes Mellitus

The prevalence of type 2 diabetes mellitus is rising globally and this disease is proposed to be the next pandemic after COVID-19. Although the cause of type 2 diabetes mellitus is unknown, it is believed to involve a complex array of genetic defects that affect metabolic pathways which eventually lead to hyperglycaemia. This hyperglycaemia arises from an inability of the insulin-sensitive cells to sufficiently respond to the secreted insulin, which eventually results in the inadequate secretion of insulin from pancreatic β-cells. Several treatments, utilising a variety of mechanisms, are available for type 2 diabetes mellitus. However, more medications are needed to assist with the optimal management of the different stages of the disease in patients of varying ages with the diverse combinations of other medications co-administered. Throughout modern history, some lead constituents from ancient medicinal plants have been investigated extensively and helped in developing synthetic antidiabetic drugs, such as metformin. Teucrium polium L. (Tp) is a herb that has a folk reputation for its antidiabetic potential. Previous studies indicate that Tp extracts significantly decrease blood glucose levels r and induce insulin secretion from pancreatic β-cells in vitro. Nonetheless, the constituent/s responsible for this action have not yet been elucidated. The effects appear to be, at least in part, attributable to the presence of selected flavonoids (apigenin, quercetin, and rutin). This review aims to examine the reported glucose-lowering effect of the herb, with a keen focus on insulin secretion, specifically related to type 2 diabetes mellitus. An analysis of the contribution of the key constituent flavonoids of Tp extracts will also be discussed

The Modified CW1 Algorithm for the Degree Restricted Minimum Spanning Tree Problem

Given edge weighted graph G (all weights are non-negative), The Degree Constrained Minimum Spanning Tree Problem is concerned with finding the minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem arises naturally in communication networks where the degree of a vertex represents the number of line interfaces available at a terminal (center). The applications of the Degree Constrained Minimum Spanning Tree problems that may arise in real-life include: the design of telecommunication, transportation, and energy networks. It is also used as a subproblem in the design of networks for computer communication, transportation, sewage and plumbing. Since, apart from some trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been proposed. In this paper we will discuss the modification of CW1 Algorithm that already proposed by Wamiliana and Caccetta (2003). The results on540 random table problems will be discussed

Optimal design of all-pass variable fractional-delay digital filters

This paper presents a computational method for the optimal design of all-pass variable fractional-delay (VFD) filters aiming to minimize the squared error of the fractional group delay subject to a low level of squared error in the phase response. The constrained optimization problem thus formulated is converted to an unconstrained least-squares (LS) optimization problem which is highly nonlinear. However, it can be approximated by a linear LS optimization problem which in turn simply requires the solution of a linear system. The proposed method can efficiently minimize the total error energy of the fractional group delay while maintaining constraints on the level of the error energy of the phase response. To make the error distribution as flat as possible, a weighted LS (WLS) design method is also developed. An error weighting function is obtained according to the solution of the previous constrained LS design. The maximum peak error is then further reduced by an iterative updating of the error weighting function. Numerical examples are included in order to compare the performance of the filters designed using the proposed methods with those designed by several existing methods