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In this paper, we study symmetries and symmetry-breaking of the arithmetic graph of a composite number m , denoted by A m . We first study some properties such as the distance between vertices, the degree of a vertex and the number of twin classes in the arithmetic graphs. We describe symmetries of A m and prove that the automorphism group of A m is isomorphic to the symmetric group S n of n elements, for m = ∏ i = 1 n p i . For symmetry-breaking, we study the concept of the fixing number of the arithmetic graphs and give exact formulae of the fixing number for the arithmetic graphs A m for m = ∏ i = 1 n p i r i under different conditions on r i ., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2023 The Author(s).)

Objective: Detecting brain tumor using the segmentation technique is a big challenge for researchers and takes a long time in medical image processing. Magnetic resonance image analysis techniques facilitate the accurate detection of tissues and abnormal tumors in the brain. The size of a brain tumor can vary with the individual and the specifics of the tumor. Radiologists face great difficulty in diagnosing and classifying brain tumors., Method: This paper proposed a hybrid model-based convolutional neural network with a stationary wavelet trans-form named "CNN-SWT" to segment brain tumors using MR brain big data. We utilized 7 layers for classification in the proposed model that include 3 convolutional and 3 ReLU. Firstly, the input MR image is divided into multiple patches, and then the central pixel value of each patch is provided to the CNN-SWT. Secondly, the pre-processing stage is per-formed using the mean filter to remove the noise. Then the convolution neural network-layer approach is utilized to segment brain tumors. After segmentation, robust feature extraction such as information-extraction methods is used for the feature extraction process. Finally, a CNN-based hybrid scheme based on the stationary wavelet transform technique is used to detect tumors using MR brain images., Materials: These experiments were obtained using 11500 MR brain images data from the hospital national of oncology., Results: It was proved that the proposed hybrid achieved a high classification accuracy of (98.7 %) as compared with existing methods., Conclusion: The advantage of the hybrid novelty of the model and the ability to detect the tumor area achieved excellent overall performance using different values., (Copyright© Bentham Science Publishers; For any queries, please email at epub@benthamscience.net.)

The process of diagnosing brain tumors is very complicated for many reasons, including the brain's synaptic structure, size, and shape. Machine learning techniques are employed to help doctors to detect brain tumor and support their decisions. In recent years, deep learning techniques have made a great achievement in medical image analysis. This paper proposed a deep wavelet autoencoder model named "DWAE model", employed to divide input data slice as a tumor (abnormal) or no tumor (normal). This article used a high pass filter to show the heterogeneity of the MRI images and their integration with the input images. A high median filter was utilized to merge slices. We improved the output slices' quality through highlight edges and smoothened input MR brain images. Then, we applied the seed growing method based on 4-connected since the thresholding cluster equal pixels with input MR data. The segmented MR image slices provide two two-layer using the proposed deep wavelet auto-encoder model. We then used 200 hidden units in the first layer and 400 hidden units in the second layer. The softmax layer testing and training are performed for the identification of the MR image normal and abnormal. The contribution of the deep wavelet auto-encoder model is in the analysis of pixel pattern of MR brain image and the ability to detect and classify the tumor with high accuracy, short time, and low loss validation. To train and test the overall performance of the proposed model, we utilized 2500 MR brain images from BRATS2012, BRATS2013, BRATS2014, BRATS2015, 2015 challenge, and ISLES, which consists of normal and abnormal images. The experiments results show that the proposed model achieved an accuracy of 99.3%, loss validation of 0.1, low FPR and FNR values. This result demonstrates that the proposed DWAE model can facilitate the automatic detection of brain tumors.

The classification of brain tumors is a difficult task in the field of medical image analysis. Improving algorithms and machine learning technology helps radiologists to easily diagnose the tumor without surgical intervention. In recent years, deep learning techniques have made excellent progress in the field of medical image processing and analysis. However, there are many difficulties in classifying brain tumors using magnetic resonance imaging; first, the difficulty of brain structure and the intertwining of tissues in it; and secondly, the difficulty of classifying brain tumors due to the high density nature of the brain. We propose a differential deep convolutional neural network model (differential deep-CNN) to classify different types of brain tumor, including abnormal and normal magnetic resonance (MR) images. Using differential operators in the differential deep-CNN architecture, we derived the additional differential feature maps in the original CNN feature maps. The derivation process led to an improvement in the performance of the proposed approach in accordance with the results of the evaluation parameters used. The advantage of the differential deep-CNN model is an analysis of a pixel directional pattern of images using contrast calculations and its high ability to classify a large database of images with high accuracy and without technical problems. Therefore, the proposed approach gives an excellent overall performance. To test and train the performance of this model, we used a dataset consisting of 25,000 brain magnetic resonance imaging (MRI) images, which includes abnormal and normal images. The experimental results showed that the proposed model achieved an accuracy of 99.25%. This study demonstrates that the proposed differential deep-CNN model can be used to facilitate the automatic classification of brain tumors.

In this paper, we establish bounds on the domination number and the metric dimension of the co-normal product graph G H of two simple graphs G and H in terms of parameters associated with G and H . We also give conditions on the graphs G and H for which the domination number of G H is 1, 2, and the domination number of G . Moreover, we give formulas for the metric dimension of the co-normal product G H of some families of graphs G and H as a function of associated parameters of G and H ., Competing Interests: The authors declare that they have no competing interests.