Inertia Sets of Semicliqued Graphs

In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $\widetilde{K}G$. We establish the minimum rank and inertia sets of some $\widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $\widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.

On equitable chromatic topological indices of some Mycielski graphs

In recent years, the notion of chromatic Zagreb indices has been introduced and studied for certain basic graph classes, as a coloring parallel of different Zagreb indices. A proper coloring C of a graph G, which assigns colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one, is called an equitable coloring of G. In this paper, we introduce the equitable chromatic Zagreb indices and equitable chromatic irregularity indices of some special classes of graphs called Mycielski graphs of paths and cycles.

ELEMENTS OF PEDAGOGICAL CONTENT KNOWLEDGE (PCK) SPECIAL CLASSES OF READING AND MEMORIZING THE QURAN (KKQ) TEACHERS IN TEACHING QURANIC TAJWEED

Quranic knowledge is one of the essential knowledge in Islam. This knowledge should be taught to Muslims to find happiness in the world and the hereafter. Among the Quranic sciences that are compulsory to master are the skills of the Quran. This skill is essential in ensuring that the Quranic verse is read in the correct reading method. To achieve this desire, Quranic teachers, including Special Classes of Reading and Memorizing the Quran (KKQ) teachers, need to master effective Quranic teaching methods by mastering the Pedagogical Content Knowledge (PCK). Previous studies have found that KKQ teachers have less knowledge of PCK and that this application of Quranic teaching is low. So, this concept paper discusses the elements of PPK that KKQ teachers need to master in teaching the Quranic tajwid. Guided by theory and recent studies, this study proposes five elements of the PCK that KKQ teachers need to master to produce effective teaching in the teaching of Quranic tajwid, content knowledge, pedagogical knowledge, student knowledge, and curriculum knowledge, and teacher Quran skills.

A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.

Representing Graphs as the Intersection of Cographs and Threshold Graphs

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\dim_{COG}(G)$ (resp. $\dim_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $\dim_{COG}(G)\leqslant\mathrm{tw}(G)+2$, (b) $\dim_{TH}(G)\leqslant\mathrm{pw}(G)+1$, and (c) $\dim_{TH}(G)\leqslant\chi(G)\cdot\mathrm{box}(G)$, where $\mathrm{tw}(G)$, $\mathrm{pw}(G)$, $\chi(G)$ and $\mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $\dim_{COG}(G)$ and $\dim_{TH}(G)$ when $G$ belongs to some special graph classes.

Investigating the Impact of EFL Students’ Misuse of Prosodic Features on their Face-to-face Interactions from University Instructors’ Perspectives

The present study aims at investigating the use of prosodic features by Kurdish EFL undergraduates in their face-to-face interactions inside/outside the classroom from the university instructors’ perspectives. The study hypothesizes that the majority of Kurdish EFL undergraduates are not fully aware of the fact that any misuse of prosodic features would probably affect the emotions, feelings, and attitudes that the face-to-face interaction is intended to convey. Building on an analysis of a questionnaire given to 54 university instructors at 10 Iraqi Kurdistan Region different universities, the study concludes that the majority of problems the students face can be related to the misuse of stress, intonation, and other prosodic features. Therefore, EFL instructors should pay more attention to make students learn how to use prosodic features and enable them to send messages adequately while engaging in face-to-face interactions. This would require special classes about prosodic features so that EFL students can overcome the misuse they have in face-to-face communication. This is inevitable because accuracy and fluency in communication require EFL students to master both features: segmental and suprasegmental. The reason behind this necessity could be attributed to the fact that broken and/or incorrect pronunciation can be considered as one of the most prominent factors behind misunderstandings in communication.