A Polynomial-Time Exact Algorithm for k -Depot Capacitated Vehicle Routing Problem on a Tree

This paper proposes a polynomial-time exact algorithm for the k -depot capacitated vehicle routing problem on a tree for fixed k ( k -depot CVRPT for short), which involves dispatching a fixed number of k capacitated vehicles in depots on a tree-shaped graph to serve customers with the objective of minimizing total distance traveled. The polynomial-time exact algorithm improves the 2-approximation algorithm when k is a constant.

Structural Design Exploration for Product Architecture Design

This paper proposed a design support method based on structuralization and analysis of various design candidates of product architecture design. The product architecture is a basic scheme that assigns the function of the product to physical components. In the conventional modular design method, a concise model, i.e., a graph or a matrix, is used to express the interactions of the system’s components and aims to support the designer grasping the system behavior. The Design Structure Matrix (DSM) is a representative model of system architecture and enables quantitative evaluation of design candidates. While various design candidates are generated through mathematical operations, it is difficult to understand their relationships from simple comparisons because of discrete behavior and the size of the problem. It must be a critical issue at the stage of selecting and interpreting the design candidates. In the proposed method, the design candidates are classified and structuralized as a dendrogram by the hierarchical clustering method. The comparison of clusters of each branch of dendrogram clarifies the system leverage points. The information of the system is summarized into the hierarchical tree-shaped graph that corresponds to the dendrogram. The designer can explore the design candidates with such a graph-based based interpretation of underlying structures effectively.

Detecting learning disabilities based on neuro-physiological interface of affect (NPIoA)

Learning disability (LD) is a neurological processing disorder that causes impediment in processing and understanding information. LD is not only affecting academic performance but can also influence on relationship with family, friends and colleagues. Hence, it is important to detect the learning disabilities among children prior to the school year to avoid from anxiety, bully and other social problems. This research aims to implement the learning disabilities detection based on the emotions captured from electroencephalogram (EEG) to recognize the symptoms of Autism Spectrum Disorder (ASD), Attention Deficit Hyperactivity Disorder (ADHD) and dyslexia in order to have early diagnosis and assisting the clinician evaluation. The results show several symptoms that ASD children have low alpha power with the Alpha-Beta Test (ABT) power ratio and ASD U-shaped graph, ADHD children have high Theta-Beta Test (TBT) power ratio while Dyslexia have high Left-over-Right Theta (LRT) power ratio. This can be concluded that the learning disabilities detection methods proposed in this study is applicable for ASD, ADHD and also Dyslexia diagnosis.

THE DIRICHLET PROBLEM ON THE ORIENTED GRAPHS

Differential operators on graphs often arise in mathematics and different fields of science such as mechanics, physics, organic chemistry, nanotechnology, etc. In this paper the solutions of the Dirichlet problem for a differential operator on a star-shaped graph are deduced. And the differential operator with standard matching conditions in the internal vertices and the Dirichlet boundary conditions at boundary vertices are studied. Task is a model the oscillation of a simple system of several rods with an adjacent end. In work the formula of the Green function of the Dirichlet problem for the second order equation on directed graph is showed. Spectral analysis of differential operators on geometric graphs is the basic mathematical apparatus in solving modern problems of quantum mechanics.

Planar graphs have bounded nonrepetitive chromatic number

The following seemingly simple question with surprisingly many connections to various problems in computer science and mathematics can be traced back to the beginning of the 20th century to the work of [Axel Thue](https://en.wikipedia.org/wiki/Axel_Thue): How many colors are needed to color the positive integers in a way such that no two consecutive segments of the same length have the same color pattern? Clearly, at least three colors are needed: if there was such a coloring with two colors, then any two consecutive integers would have different colors (otherwise, we would get two consecutive segments of length one with the same color pattern) and so the colors would have to alternate, i.e., any two consecutive segments of length two would have the same color pattern. Suprisingly, three colors suffice. The coloring can be constructed as follows. We first define a sequence of 0s and 1s recursively as follows: we start with 0 only and in each step we take the already constructed sequence, flip the 0s and 1s in it and append the resulting sequence at the end. In this way, we sequentially obtain the sequences 0, 01, 0110, 01101001, etc., which are all extensions of each other. The limiting infinite sequence is known as the [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence). Another view of the sequence is that the $i$-th element is the parity of the number of 1s in the binary representation of $i-1$, i.e., it is one if the number is odd and zero if it is even. The coloring of integers is obtained by coloring an integer $i$ by the difference of the $(i+1)$-th and $i$-th entries in the Thue-Morse sequence, i.e., the sequence of colors will be 1, 0, -1, 1, -1, 0, 1, 0, etc. One of the properties of the Thue-Morse sequence is that it does not containing two overlapping squares, i.e., there is no sequence X such that 0X0X0 or 1X1X1 would be a subsequence of the Thue-Morse sequence. This implies that the coloring of integers that we have constructed has no two consecutive segments with the same color pattern. The article deals with a generalization of this notion to graphs. The _nonrepetitive chromatic number_ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ in such way that no path with an even number of vertices is comprised of two paths with the same color pattern. The construction presented above yields that the nonrepetitive chromatic number of every path with at least four vertices is three. The article answers in the positive the following question of Alon, Grytczuk, Hałuszczak and Riordan from 2002: Is the nonrepetitive chromatic number of planar graphs bounded? They show that the nonrepetitive chromatic number of every planar graph is at most 768 and provide generalizations to graphs embeddable to surfaces of higher genera and more generally to classes of graphs excluding a (topological) minor. Before their work, the best upper bound on the nonrepetitive chromatic number of planar graphs was logarithmic in their number of vertices, in addition to a universal upper bound quadratic in the maximum degree of a graph obtained using probabilistic method. The key ingredient for the argument presented in the article is the recent powerful result by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood asserting that every planar graph is a subgraph of the strong product of a path and a graph of bounded tree-width (tree-shaped graph).

Spectral analysis of the matrix Sturm–Liouville operator

The self-adjoint matrix Sturm–Liouville operator on a finite interval with a boundary condition in general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on an analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm–Liouville operators on a star-shaped graph with two different types of matching conditions.