The independence Graph Ind(G) of a Graph G is the Graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence Graph of Cartesian product of d copies of complete Graphs Kq, which is known as the Hamming Graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming Graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

Deep convolutional neural networks (CNNs) have attained remarkable success in numerous visual recognition tasks. There are two challenges when adopting CNNs in real-world applications: a) Existing CNNs are computationally expensive and memory intensive, impeding their use in edge computing; b) there is no standard methodology for designing the CNN architecture for the intended problem. Network Pruning/compression has emerged as a research direction to address the first challenge, and it has proven to moderate CNN computational load successfully. For the second challenge, various evolutionary algorithms have been proposed thus far. The algorithm proposed in this paper can be viewed as a solution to both challenges. Instead of using constant predefined criteria to evaluate the filters of CNN layers, the proposed algorithm establishes evaluation criteria in online manner during network training based on the combination of each filter’s profit in its layer and the next layer. In addition, the novel method suggested that it inserts new filters into the CNN layers. The proposed algorithm is not simply a Pruning strategy but determines the optimal number of filters. Training on multiple CNN architectures allows us to demonstrate the efficacy of our approach empirically. Compared to current Pruning algorithms, our algorithm yields a network with a remarkable prune ratio and accuracy. Despite the relatively high computational cost of an epoch in the proposed algorithm in Pruning, altogether it achieves the resultant network faster than other algorithms.

Knowledge representation in the form of a concept map can be a good idea to categorize domain terms and their relations and help to generate ontology,Supplementing detail information to and Pruning useless data from the concept map,which likes a skeleton in evolving ontology,can be semantically accomplished using the domain knowledge,In this paper,we propose a method using structural knowledge resources as well as tacit knowledge of experts to generate the ontology of eLearning domain,The concept map of eLearning is manually improved and finally verified using the group of eLearning experts,In order to enrich the ontology with merging into upcoming terms,the paper proposed an automatic method based on two external knowledge sources,Wikipedia and WordNet,The semantic similarity of concepts which is measured using the words hierarchy of WordNet combined with relations of concepts extracted from the Wikipedia Graph is applied to link the new eLearning concepts to the domain ontology,The generated ontology is a dynamic knowledge source which can improve itself gradually,This integrated knowledge of eLearning domain can be used to model educational activities and to build,organize,and update specific learning resources,

Background: Accurate semantic segmentation of kidney tumors in computed tomoGraphy (CT) images is difficult because tumors feature varied forms and occasionally, look alike. The KiTs19 challenge sets the groundwork for future advances in kidney tumor segmentation. Methods: We present weight Pruning (WP)-UNet, a deep network model that is lightweight with a small scale,it involves few parameters with a quick assumption time and a low floating-point computational complexity. Results: We trained and evaluated the model with CT images from 210 patients. The findings implied the dominance of our method on the training Dice score (0. 98) for the kidney tumor region. The proposed model only uses 1, 297, 441 parameters and 7. 2e floating-point operations, three times lower than those for other network models. Conclusions: The results confirm that the proposed architecture is smaller than that of UNet, involves less computational complexity, and yields good accuracy, indicating its potential applicability in kidney tumor imaging.

In this paper, we find the star chromatic number of central Graph of complete bipartite Graph and corona Graph of complete Graph with path and cycle.

Let R be a commutative ring and M be an R-module. The M-intersection Graph of ideals of R, denoted by GM(R) is a Graph with the vertex set I(R) ∗, , and two distinct vertices I and J are adjacent if and only if IM ∩,JM ̸, = 0. In this paper, we study GR/J (R/I), where I and J are ideals of R and I ⊆,J. We characterize all ideals I and J for which GR/J (R/I) is planar, outerplanar or ring Graph.

‎Graph coloring is the assignment of one color to each vertex of a Graph so that two adjacent vertices are not of the same color‎. ‎The Graph coloring problem (GCP) is a matter of combinatorial optimization‎, ‎and the goal of GCP is determining the chromatic number $\chi(G)$‎. ‎Since GCP is an NP-hard problem‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a Graph adjacency matrix to colorize or separate a Graph‎. ‎To prove the correctness of the proposed algorithm‎, ‎we implement it in MATLAB software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software‎, ‎and we present the results in tables of various Graphs‎. ‎Several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the Graph adjacency matrix to select the vertex that has the smallest degree for coloring‎. ‎We provide some examples to compare the performance of our algorithm to other available methods‎. ‎We make use of the Dolan-Mor\'e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.

Let $G=(V,E)$ be a Graph of order $n$ and size $m.$ The Graph $Sp(G)$ obtained from $G$ by adding a new vertex $v'$ for every vertex $v\in V$ and joining $v'$ to all neighbors of $v$ in $G$ is called the splitting Graph of $G.$ In this paper, we determine the domination number, the total domination number, connected domination number, paired domination number and independent domination number for the splitting Graph $Sp(G).$

The effect of Apple Pruning wood particles in mixed with industrial wood particles for particleboard manufacturing was studied. One layer laboratory panels with density of 0.6 gr/cm3 with two variables amount of Apple Pruning wood particles content (0/100, 15/85, 30/70, 45/55 percent weight of Apple Pruning particles/wood particles) and a press time of 5 and 7 min were produced. The physical and mechanical properties of panels were measured. The results showed that the amount of Apple Pruning wood particle had significant influence on the most physical and mechanical properties of panels such as MOR, MOE, IB, TS and WA while they were improved as the Apple Pruning wood particles increased. The results also indicated that press time had significant influence on the IB and TS of panels and they were increased as the press time increased.