Weighted Modularity on a k-Path Graph

Author(s):  
Ying-Hong Ma ◽  
Wen-Qian Wang
Keyword(s):  
Author(s):  
K. Rajalakshmi ◽  
M. Venkatachalam ◽  
M. Barani ◽  
D. Dafik

The packing chromatic number $\chi_\rho$ of a graph $G$ is the smallest integer $k$ for which there exists a mapping $\pi$ from $V(G)$ to $\{1,2,...,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. In this paper, the authors find the packing chromatic number of subdivision vertex join of cycle graph with path graph and subdivision edge join of cycle graph with path graph.


Electronics ◽  
2021 ◽  
Vol 10 (14) ◽  
pp. 1671
Author(s):  
Jibing Gong ◽  
Cheng Wang ◽  
Zhiyong Zhao ◽  
Xinghao Zhang

In MOOCs, generally speaking, curriculum designing, course selection, and knowledge concept recommendation are the three major steps that systematically instruct users to learn. This paper focuses on the knowledge concept recommendation in MOOCs, which recommends related topics to users to facilitate their online study. The existing approaches only consider the historical behaviors of users, but ignore various kinds of auxiliary information, which are also critical for user embedding. In addition, traditional recommendation models only consider the immediate user response to the recommended items, and do not explicitly consider the long-term interests of users. To deal with the above issues, this paper proposes AGMKRec, a novel reinforced concept recommendation model with a heterogeneous information network. We first clarify the concept recommendation in MOOCs as a reinforcement learning problem to offer a personalized and dynamic knowledge concept label list to users. To consider more auxiliary information of users, we construct a heterogeneous information network among users, courses, and concepts, and use a meta-path-based method which can automatically identify useful meta-paths and multi-hop connections to learn a new graph structure for learning effective node representations on a graph. Comprehensive experiments and analyses on a real-world dataset collected from XuetangX show that our proposed model outperforms some state-of-the-art methods.


2020 ◽  
Author(s):  
K. Summer Ware

Memory is traditionally thought of as a biological function of the brain. In recent years, however, researchers have found that some stimuli-responsive molecules exhibit memory-like behavior manifested as history-dependent hysteresis in response to external excitations. One example is lysenin, a pore-forming toxin found naturally in the coelomic fluid of the common earthworm Eisenia fetida. When reconstituted into a bilayer lipid membrane, this unassuming toxin undergoes conformational changes in response to applied voltages. However, lysenin is able to "remember" past history by adjusting its conformational state based not only on the amplitude of the stimulus but also on its previous its conformational state. The current model is a simple two-state Markov description which may not describe a system with memory. In this respect, this thesis aims to provide a more accurate description of this toxin's memory and response to external stimuli by applying a more rigorous mathematical approach. The traditional setting for investigating the conformational changes of voltage-responsive channel proteins is based on analyzing the ionic currents recorded through one or many channels in response to applied voltage stimuli. However, this approach provides only indirect evidence of the conformational state of the channel, i.e open (conducting) or closed (non-conducting). Although very useful, this setting is seriously limited by the inability of electrical measurements to discern between electrically identical yet conformational different open or closed states. The literature that inspired this thesis topic consider models of diffusion on a path-graph with one open state and an arbitrary number of closed states. The mathematics typically begins with approximations from a continuous model. In this thesis we study the analytic solution of the system of linear homogeneous differential equations which are probability vectors describing the diffusion process; this involves exponential theory of weighted Laplacian graphs. Since the Laplacian matrix of the path graph is well studied, we have access to both eigenvectors and eigenvalues in terms of roots of unity making for a succinct solution. We find that polynomial weights model the hysteresis successfully.


Author(s):  
Nurma Ariska Sutardji ◽  
Liliek Susilowati ◽  
Utami Dyah Purwati

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.


2021 ◽  
Vol 207 (1) ◽  
pp. 487-493
Author(s):  
V. L. Chernyshev ◽  
D. S. Minenkov ◽  
A. A. Tolchennikov
Keyword(s):  

2020 ◽  
Vol 172 ◽  
pp. 649-654
Author(s):  
Teresa Arockiamary S ◽  
Vijayalakshmi G
Keyword(s):  

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