Search Results (6)

The central location problem is a key aspect of graph theory, with a significant importance in various applications and studies within the field. The center of a graph is made up of nodes that have the smallest eccentricity, where eccentricity is defined as the greatest distance between a given node and any other node in the graph. To determine the graph’s center, it is essential to compute the eccentricity of each node. In this article, we explore various characteristics of the BFS tree of trapezoid graphs. We also present new properties that relate to the radius, diameter, and center of trapezoid graphs. For the trapezoid graph G, We prove that the difference between the $diameter\left(G\right)$ and the height of the BFS trees $T_t\left(1\right),T_t\left(n\right),T_t\left(a\right),$ and $T_t\left(b\right)$ is at most one. We also establish relationship between $radius\left(G\right)$ and $diameter\left(G\right)$ of trapezoid graphs. We also show that, to find the center of a trapezoid graph, it is not necessary to find the eccentricity of all vertices. Based on our studied results, we design an optimal algorithm for finding the center, radius, and diameter of trapezoid graphs. Also, we prove theoretically that our proposed algorithm compiles within $O\left(n\right)$ time. We also find an algorithmic solution to real problems (that involves finding a center location in a district to build a private hospital that minimizes the farthest distance from it to all areas of the district) with the help of the trapezoid graph model and BFS trees within $O\left(n\right)$ time. Full article

An infinite homogeneous tree is a special type of graph that has a completely symmetrical structure in all directions. For an infinite homogeneous tree $T=(\mathcal{V},\mathcal{E})$ with the natural distance d defined on graphs and a weighted measure $\mu$ of exponential growth, the authors introduce the variable Lebesgue space $L^{p(·)}\left(\mu \right)$ over $(\mathcal{V},d,\mu )$ and investigate it under the global Hölder continuity condition for $p(·)$. As an application, the strong and weak boundedness of the maximal operator relevant to admissible trapezoids on $L^{p(·)}\left(\mu \right)$ is obtained, and an unbounded example is presented. Full article

In this article, we derive the above and below bounds for parameterized-type inequalities using the Riemann–Liouville fractional integral operators and limited second derivative mappings. These established inequalities generalized the midpoint-type, trapezoid-type, Simpson-type, and Bullen-type inequalities according to the specific choices of the parameter. Thus, a generalization of many inequalities and new results were obtained. Moreover, some examples of obtained inequalities are given for better understanding by the reader. Furthermore, the theoretical results are supported by graphs in order to illustrate the accuracy of each of the inequalities obtained according to the specific choices of the parameter. Full article

The shortest path problem is a topic of increasing interest in various scientific fields. The damage to roads and bridges caused by disasters makes traffic routes that can be accurately expressed become indeterminate. A neutrosophic set is a collection of the truth membership, indeterminacy membership, and falsity membership of the constituent elements. It has a symmetric form and indeterminacy membership is their axis of symmetry. In uncertain environments, the neutrosophic number can more effectively express the edge distance. The objectives in this study are to solve the shortest path problem of the neutrosophic graph with an edge distance expressed using trapezoidal fuzzy neutrosophic numbers (TrFNN) and resolve the edge distance according to the score and exact functions based on the TrFNN. Accordingly, the use of a circle-breaking algorithm is proposed to solve the shortest path problem and estimate the shortest distance. The feasibility of this method is verified based on two examples, and the rationality and effectiveness of the approach are evaluated by comparing it with the Dijkstra and Bellman algorithms. Full article

The assessment of novel waste-to-energy technologies has several drawbacks due to the nature of the R1 formula. The 3T method, which aims to cover this gap, combines thermodynamic parameters in a radar graph and the overall efficiency is calculated from the area of the trapezoid. The present study expands the application of the 3T method in order to make it suitable for utilization in other energy-from-waste technologies. In the framework of this study, a 3T specialized solution is developed for the case of landfilling plus landfill gas recovery, with the potential inclusion of landfill mining. Numerical applications have been performed for waste-to-energy and landfilling by using both the R1 formula and the 3T method. The model Land GEM was used for the calculation of the total landfill gas. The Combined Heat and Power (CHP) efficiency of the landfill gas CHP efficiency was 16.6%–33.1%, and for the waste-to-energy plant, the CHP efficiency was over 70%. The full range of parameters, like metal recovery and quality of CHP, were not fully reflected by the R1 formula, which returned values of 1.07 for waste-to-energy and from 0.37 to 0.63 for different landfilling scenarios. Contrary to that, the 3T method calculated values between 0.091 and 0.307 for the waste-to-energy plant and values between 0.011 and 0.121 for the various landfilling scenarios. The 3T method is able to account for the recovery of materials like metals and assess the quality of the output flows. The 3T method was able to successfully provide a solution for the case of landfilling plus landfill gas recovery, with the potential inclusion of landfill mining, and directly compares the results with the conventional case of waste-to-energy. Full article

Road scene model construction is an important aspect of intelligent transportation system research. This paper proposes an intelligent framework that can automatically construct road scene models from image sequences. The road and foreground regions are detected at superpixel level via a new kind of random walk algorithm. The seeds for different regions are initialized by trapezoids that are propagated from adjacent frames using optical flow information. The superpixel level region detection is implemented by the random walk algorithm, which is then refined by a fast two-cycle level set method. After this, scene stages can be specified according to a graph model of traffic elements. These then form the basis of 3D road scene models. Each technical component of the framework was evaluated and the results confirmed the effectiveness of the proposed approach. Full article