Diffusion profile embedding as a basis for graph vertex similarity

We describe here a notion of diffusion similarity, a method for defining similarity between vertices in a given graph using the properties of random walks on the graph to model the relationships between vertices. Using the approach of graph vertex embedding, we characterize a vertex vi by considering two types of diffusion patterns: the ways in which random walks emanate from the vertex vi to the remaining graph and how they converge to the vertex vi from the graph. We define the similarity of two vertices vi and vj as the average of the cosine similarity of the vectors characterizing vi and vj. We obtain these vectors by modifying the solution to a differential equation describing a type of continuous time random walk.This method can be applied to any dataset that can be assigned a graph structure that is weighted or unweighted, directed or undirected. It can be used to represent similarity of vertices within community structures of a network while at the same time representing similarity of vertices within layered substructures (e.g., bipartite subgraphs) of the network. To validate the performance of our method, we apply it to synthetic data as well as the neural connectome of the C. elegans worm and a connectome of neurons in the mouse retina. A tool developed to characterize the accuracy of the similarity values in detecting community structures, the uncertainty index, is introduced in this paper as a measure of the quality of similarity methods.

Coordination sequences of 2-uniform graphs

Explicit formulas for coordination sequences of all 20 plane 2-uniform graphs are proved. The proof is based on the concept of layer-by-layer growth and on the canonical representation of geodesic chains in terms of special chains called as rays. The method works for a wide class of plane periodic graphs satisfying the following condition: for each sector of layer-by-layer growth there exists a graph vertex that is initial for two rays determining the sector. This generaizes the previous results where it is required that all vertices are initial for all rays.

Aplikasi Pewarnaan Graf dengan Algoritma Recursive Largest First pada Penjadwalan Mata Kuliah

Penelitian ini merupakan penelitian terapan yang menggunakan algoritma Recursive Largest First (RLF) pada penjadwalan mata kuliah di salah satu program studi pada salah satu universitas di Makassar. Penjadwalan mata kuliah dilakukan dengan merepresentasikan masalah ke dalam bentuk graf berdasarkan data yang diperoleh dengan menganggap mata kuliah sebagai simpul dan dosen sebagai sisi. Dibentuk matriks ketetanggaan dari graf yang terbentuk, kemudian simpulya diwarnai menggunakan algoritma RLF, sehingga diperoleh jadwal kuliah. Penjadwalan mata kuliah menggunakan pewarnaan simpul graf dengan algoritma RLF digunakan VBA for Microsoft Excel 2016 sebagai alat bantu dalam pewarnaan simpul graf. Diperoleh 102 simpul dengan 25 warna yang menunjukkan bahwa dibutuhkan 25 waktu perkuliahan. Jumlah simpul yang diwarnai dengan warna yang sama 6 simpul artinya dalam setiap slot dibutuhkan maksimal 6 ruangan perkuliahan. Mata kuliah yang terjadwal dijamin bahwa tidak ada jadwal dosen yang bersamaan untuk setiap mata kuliah yang diajarkan.Kata Kunci: Graf, Recursive Largest First, Jadwal Kuliah This research discusses the colorization of graph vertex using algorithm of Recursive Largest First (RLF) in scheduling courses in one of the study programs at one of the universities in Makassar. Representation of problems in scheduling courses was done by making it in the shape of the graph assuming subjects as vertex and the presence of the same lecturers teach different courses as a side. The steps in solving a problem that was mapping the lectures data, representing the lectures data as a graph, forming adjacency matrix, coloring the graph vertex using algorithm of RLF, so that the schedule was retrieved. Scheduling of courses using the colorization of graph vertex with algorithm of RLF as well as the help of VBA for Microsoft Excel 2016 at Faculty of Animal Husbandry obtained 102 vertices with 25 colors indicates that it takes 25 of lecture schedules. The number of vertices that were colored with the same color ≤ 6 vertices meaning in every slot needed 6 lecture rooms in maximum. Scheduled courses are guaranteed that no schedule conflicts for each lecturer of courses taught.Keywords: Graph, Recursive Largest First, Lecture Schedules.

Adjacency and Laplacian spectra of variants of neighborhood corona of graphs constrained by vertex subsets

In this paper, we define some variants of corona of graphs namely, subdivision (respectively, [Formula: see text]-graph, [Formula: see text]-graph, total) neighborhood corona, [Formula: see text]-graph (respectively, [Formula: see text]-graph, total) semi-edge neighborhood corona, [Formula: see text]-graph (respectively, total) semi-vertex neighborhood corona of graphs constrained by vertex subsets. These corona operations generalize some existing corona operations such as subdivision ([Formula: see text]-graph, [Formula: see text]-graph, total) double neighborhood corona, subdivision vertex (respectively, edge) neighborhood corona, [Formula: see text]-graph vertex (respectively, edge) neighborhood corona of graphs. First, we consider a matrix in specific form and determine its spectrum. Then by using this, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the new graphs when the base graph is regular. Also, we deduce the characteristic polynomials of the adjacency and Laplacian matrices of the above mentioned particular cases from our results.

A Backward Single Source Shortest Path Algorithm For Directed Graph

A new directed backward variant of the Single Source Shortest Path algorithm was described in this paper. This algorithm accept that approaching adjacency list of the given graph vertex loads showed up in expanding request. The running time of forward, based strategy algorithm is the best aftereffect of O (n), which are the most ideal forward-backward SSSP consequences of Wilson et al. Moreover, the likelihood of the new algorithm additionally requires O (n) time.This is an equally improved version of exponentially and polynomial small probability derived by Wilson et al.

Distance Degree Index of Some Derived Graphs

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (DD) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a DD index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the DD index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph.