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In this article, we study the zero-divisor graphsΓ(Zn) of rings of integers modulo η as information systems L(Γ(Zn)) using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set P of all prime divisors of η and (ii) the set consisting of P/pi, prime powers of P/pi in the prime factorization of η and the elements of the form P/pi pj, where pj ε P/pi . Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of L(Γ(Zn)) either consist of two prime divisors of η or one prime divisor P/pi combined with prime powers of P/pi . Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function forΓ(Zn). Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets ofΓ(Zn). Based on these results, we determine information-granularity measures corresponding to the notable partitions of Γ(Zn) and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory. [ABSTRACT FROM AUTHOR]

Coronary artery anomalies are abnormalities in coronary arteries by birth (congenital) and involve their anatomy, course and function. They usually occur in association with other congenital cardiac defects. Coronary artery anomalies are usually clinically silent as they do not cause symptoms and are found incidentally but relatively few become evident by causing symptoms. Objectives: Determine the frequency of coronary artery anomalies in patients undergoing coronary angiography. Study Design: Cross sectional study. Setting: Faisalabad Institute of Cardiology, Faisalabad. Period: July to December 2017. Material & Methods: Demographic information from all the patients was recorded and all of these individuals were undergone coronary angiography. All the collected data was analyzed by SPSS. Results: The mean age of the patients was 43.74 ± 15.35 years, male and female patients were equal in number. In this study coronary anomalies were present in 28(4.06%) patients. Conclusion: Coronary artery anomalies are found quite frequently but usually are incidental finding. [ABSTRACT FROM AUTHOR]

The fixing number of a graph G is the smallest cardinality of a set of vertices F ⊆ V (G) such that only the trivial automorphism of G fixes every vertex in F. In this paper, we introduce and study three new fixing parameters: fixing share, fixing polynomial and fixing value. [ABSTRACT FROM AUTHOR]

The size of a resolving set R of a non-trivial connected graph Γ of order n ≥ 2 is the number of edges in the induced subgraph < R >. The minimum cardinality of a resolving set of size k of graph Γ is called the metric dimension of size k, denoted by β(k) (Γ). We study the existence of resolving sets of size k in some families of graphs and investigate their properties. We find bounds on the metric dimension of size k of a graph Γ. We give the necessary condition for the metric dimension of size k and size (k + 1) of a graph Γ, to satisfy the inequality β(k+1)(Γ)-β(k) (Γ) ≤ 1. We will disprove a conjecture on bounds of the metric dimension of size k. For every positive integers k, l, and n such that k + 1 ≤ l ≤ n, we give a realizable result of a graph Γ of order n and l = β(k) (Γ). [ABSTRACT FROM AUTHOR]

Objective: To determine the usefulness and safety of transthoracic echocardiography (TTE) imaging for real time guidance in transcatheter device closure of atrial septal defect (ASD) secundum. Study Design: Cross-sectional study. Setting: Department of Paediatric Cardiology, CPEIC, Multan, Pakistan. Period: January 2022 to December 2023. Material & Methods: Patients of both genders aged above 3 years with clinical and echocardiographic evidence of ASD secundum with significant left to right shunt were analysed. All cases had morphologically suitable margins of defect confirmed on TTE. Usefulness and safety of intra-procedure and immediate post-procedure transcatheter device closure of secundum ASD were noted. Results: In a total of 34 patients, 20 (58.8%) were female and 14 (41.2%) male. The mean age was 21.32±17.21 years. The TTE findings revealed that the mean ASD side was 17.47±4.32 mm. Aortic margins were deficient among 28 (82.4%) cases. The mean fluoroscopy and procedure time were 9.29±6.98 minutes and 49.91±39.76. Residual leak was noted in one (2.9%) cases while none of the cases were observed to have superior vena cava, inferior vena cava or pulmonary vein obstructions. Post-procedure, transient supraventricular tachycardia (SVT) arrhythmia was the most common type of complication reported among 2 (5.9%) cases. None of the cases needed any kind of emergency surgical exploration. No cases of pericardial effusion were found. No mortality was reported. Conclusion: Transcatheter device closure of secundum atrial septal defect was found to be effective and safe using TTE. [ABSTRACT FROM AUTHOR]

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, υ ∈ V ( G) there is a vertex w ∈ W such that d( u,w) ≠ d( υ,w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim( G). For a vertex u of G and a subset S of V ( G), the distance between u and S is the number min d( u, s). A k-partition Π = { S, S, ..., S} of V ( G) is called a resolving partition if for every two distinct vertices u, v ∈ V ( G) there is a set S in Π such that d( u, S) ≠ d( v, S). The minimum k for which there is a resolving k-partition of V ( G) is called the partition dimension of G, denoted by pd( G). The circulant graph is a graph with vertex set ℤ, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i − j (mod n) ∈ C, where C ∈ ℤ has the property that C = −C and 0 ∉ C. The circulant graph is denoted by X where Δ = | C|. In this paper, we study the metric dimension of a family of circulant graphs X with connection set $$C = \left\{ {1,\tfrac{n} {2},n - 1} \right\}$$ and prove that dim( X) is independent of choice of n by showing that We also study the partition dimension of a family of circulant graphs X with connection set C = {±1,±2} and prove that pd( X) is independent of choice of n and show that pd( X) = 5 and . [ABSTRACT FROM AUTHOR]

Sierpiński graphs are extensively studied graphs of fractal nature with applications in topology, mathematics of Tower of Hanoi and computer science. The generalized Sierpiński graphs are defined by replication of exactly the same graph, yielding self-similar graph. Certain graph invariants referred to as topological indices are used to determine a large number of properties like physico-chemical properties, thermodynamic properties, chemical activity and biological activity of chemical graphs. In QSAR/QSPR study, these graph invariants play a vital role.In this article, we study the topological indices of generalized Sierpiński and extended Sierpiński graphs with an arbitrary base graph. We obtain bounds for the atom-bond connectivity index, harmonic index, Zagreb indices and sum-connectivity index for the generalized Sierpiński graphs and extended Sierpiński graphs. [ABSTRACT FROM AUTHOR]

In this paper, we establish bounds on the domination number and the metric dimension of the co-normal product graph GH 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 GH is 1, 2, and the domination number of G. Moreover, we give formulas for the metric dimension of the co-normal product GH of some families of graphs G and H as a function of associated parameters of G and H. [ABSTRACT FROM AUTHOR]

A locating-dominating set of a graph G is a dominating set of G such that every vertex of G outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of G is the minimum cardinality of a locating-dominating set in G. Let G 1 and G 2 be two disjoint copies of a graph G and f : V (G 1) → V (G 2) be a function. A functigraph F G f consists of vertex set V (G 1) ∪ V (G 2) and edge set E (G 1) ∪ E (G 2) ∪ { u v : v = f (u) }. In this paper, we study the variation of location-domination number in passing from G to F G f and find its sharp lower and upper bounds. We also study location-domination number of functigraphs of complete graphs for all possible definitions of function f. We also obtain location-domination number of functigraphs of a family of spanning subgraph of the complete graphs. [ABSTRACT FROM AUTHOR]

Introduction: In recent years the trans-radial approach has been increasingly employed as an alternative approach to percutaneous coronary intervention. Loss of radial pulse may lead to ischemic symptoms like pain in hand, claudication etc. There is a lack of local data regarding complication associated with trans-radial coronary catheterization. Therefore, there is need for local study to find out the frequency of complications associated with trans-radial approach. Objectives: To determine the frequency of loss of radial pulse in patients undergoing trans-radial coronary catheterization. Design: Cross sectional study. Setting: Department of Cardiology, Jinnah Hospital, Lahore. Period: From 3rd June 2014 to 2nd December 2014. Methodology: All 325 cases fulfilling the inclusion/exclusion criteria undergoing coronary catheterization were included in the study. Results: A total of 325 patients, both male and female with positive Allens test, undergoing Trans-radial angiography for intervention were included. Loss of radial pulse was evaluated clinically at four weeks follow up. In our sampled population, mean age was 57.01 ± 6.8 years and 240 patients (73.8%) were male while 85 patients (26.2%) were female. Only 14 patients (4.3%) out of 325 had loss of radial artery pulse while remaining 311 patients (95.7%) didn't have loss of radial artery pulse. There was found no effect of gender on outcome i.e. loss of radial pulse but older age made a person prone to loss of radial pulse. Conclusion: It is concluded that the frequency (percentage) of loss of radial pulse is quite low (4.3%) in our patients undergoing trans-radial coronary catheterization. [ABSTRACT FROM AUTHOR]

Objectives: To determine the frequency of multivessel coronary artery disease (CAD) in patients suffering from inferior wall acute myocardial infarction (AMI) having ST segment depression in anterior chest leads. Study Design: Cross Sectional Survey. Setting: Department of Cardiology, Faisalabad Institute of Cardiology, Faisalabad. Period: December 2013 to June 2014. Materials and Methods: After fulfilling the inclusion criteria, 120 patients with acute inferior wall MI were studied. Patients were divided in two groups. Group I included patients with ST segments depression in leads V1-V3 and Group II with ST segment depression in leads V4-V6. Age of patients ranged from 25 to 70 years. Results: The mean age was 52.5±10.5 years. As regards sex distribution, 100 patients were male, 54 in Group I and 46 in Group II. There were 20 female patients, 11 in Group I and 9 in Group II. Thirty six (30%) patients were diabetic, 22(33.8%) in Group I and 14(25.5%) in Group II. Forty five (37.5%) patients were hypertensive. Fifty two (43.3%) patients were smokers. Forty three (35.8%) patients had family history of ischemic heart disease, 25(38.5%) in Group I and 18(32.7%) in Group II. Thirty five (29.2%) had hyperlipidemia, 20 (30.8%) in Group I and 15(29.2%) in Group II. All patients underwent coronary angiogram. Thirty eight (31.7%) patients had single vessel disease, 29(44.6%) patients in Group I and 9(16.4%) in Group II. Forty three (35.8%) patients had double vessel coronary artery disease, 23 patients (35.4%) in Group I and 20(36.4%) in Group II. Thirty nine (32.5%) had three vessel coronary artery disease, 13 (20%) in Group I and 26 (47.3%) in Group II. Eighty three (69.2%) patients had multivessel coronary artery disease 37(56.8%) in Group I and 46 (83.6%) in Group II (p value <0.002). Conclusion: The presence of precordial ST depression during acute inferior wall myocardial correlates well with multivessel CAD. Precordial ST-segment depression in acute inferior wall MI is related to anterior injury due to LAD stenosis and these patients tend to have more severe CAD. [ABSTRACT FROM AUTHOR]

A vertexvof a connected graphGdistinguishes a pairu, wof vertices ofGifd(v, u)≠d(v, w), whered(·,·) denotes the length of a shortest path between two vertices inG. Ak-partition Π={S1,S2, …,Sk} of the vertex set ofGis said to be a locatic partition if for every pair of distinct verticesvandwofG, there exists a vertexs∈Sifor all 1≤i≤kthat distinguishesvandw. The cardinality of a largest locatic partition is called the locatic number ofG. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of ordernhaving locatic numbern, n−1 andn−2. Some realizable results are also given in this paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by A m . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of A m is two or three. Conditions under which the vertices have the same degrees and neighborhoods have also been identified. Symmetric behavior of the vertices lead to the study of the metric dimension of A m which gives minimum cardinality of vertices to distinguish all vertices in the graph. We give exact formulae for the metric dimension of A m , when m has exactly two distinct prime divisors. Moreover, we give bounds on the metric dimension of A m , when m has at least three distinct prime divisors. [ABSTRACT FROM AUTHOR]

In this paper, we consider the family of generalized Petersen graphs P( n, 4). We prove that the metric dimension of P( n, 4) is 3 when n ≡ 0 (mod 4), and is 4 when n = 4 k + 3 ( k is even). For n ≡ 1,2 (mod 4) and n = 4 k + 3 ( k is odd), we prove that the metric dimension of P( n, 4) is bounded above by 4. This shows that each graph of the family of generalized Petersen graphs P( n, 4) has constant metric dimension. [ABSTRACT FROM AUTHOR]

Abstract: Let be a connected graph and be the distance between the vertices and in . A subset of vertices is called a resolving set or locating set for if for every two distinct vertices , there is a vertex such that for . A resolving set containing the minimum number of vertices is called a metric basis for and the number of vertices in a metric basis is its metric dimension, denoted by . Let be a family of connected graphs depending on as follows: the order and . If there exists a constant such that for every then we shall say that has bounded metric dimension. The metric dimension of a class of circulant graphs has been determined by Javaid and Rahim (2008) . In this paper, we extend this study to an infinite class of circulant graphs . We prove that the circulant graphs have metric dimension equal to 4 for . For only 5 vertices appropriately chosen suffice to resolve all the vertices of , thus implying that except when . [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let G 1 and G 2 be disjoint copies of a graph G and g : V (G 1) → V (G 2) be a function. A functigraph F G consists of the vertex set V (G 1) ∪ V (G 2) and the edge set E (G 1) ∪ E (G 2) ∪ { u v : g (u) = v } . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to F G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs. [ABSTRACT FROM AUTHOR]