abolfazl tehranian

Graph isomorphism is a central topic in graph theory, yet students often struggle to grasp its underlying principles. This study aimed to explore students' perceptions of graph isomorphism and identify the conceptual metaphors they employ when reasoning about this concept.

Using a two-stage qualitative research design, the study combined surveys and task-based interviews to collect data. The variation theory framework (Marton & Booth, 2013) was applied to analyze students' understanding and identify critical and superficial features in their reasoning.

The findings revealed that students frequently focus on superficial features, such as the equality of the number of vertices and edges, while overlooking critical features like bijective attributes. Additionally, the conceptual metaphors used by students were categorized into three main themes: Sameness, Sameness/Mapping, and Formal Definition. These metaphors reflect varying levels of conceptual understanding, with some students relying on intuitive notions of similarity and others employing more formal, mathematical definitions.

This study highlights the challenges students face in understanding graph isomorphism and underscores the importance of addressing both superficial and critical features in teaching. By identifying the conceptual metaphors students use, educators can develop targeted instructional strategies to bridge gaps in understanding and promote deeper conceptual learning.

‎Let $r\geq 2$. A subset $S$ of vertices of a graph $G$ is a $r$-hop independent dominating set if every vertex outside $S$ is at distance $r$ from a vertex of $S$, and for any pair $v, w\in S$, $d(v, w)\neq r$. A $r$-hop Roman dominating function ($r$HRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v \in V$ with $f(v) = 0$ there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$-step Roman dominating function ($r$SRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v$ with $f(v)=0$ or $2$, there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$HRDF $f$ is a $r$-hop Roman independent dominating function if for any pair $v, w$ with non-zero labels under $f$, $d(v, w)\neq r$. We show that the decision problem associated with each of $r$-hop independent domination, $r$-hop Roman domination, $r$-hop Roman independent domination and $r$-step Roman domination is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.

 This study aims to leverage psychological aspects within the Flipped Learning method by presenting a novel technique and examining the mathematical performance of undergraduate students.

Methods and Materials: 

This descriptive-analytical study was conducted in the 2022-2023 academic year with 89 undergraduate students from the Computer Science field at the Islamic Azad University, Science and Research Branch. Students were divided into three General Mathematics II classes and grouped based on the DISC Personality Test. A sample size of 75 was selected using stratified random sampling. The Flipped Learning method involved ten 120-minute sessions, with instructional scaffolding provided through the iGap Iranian application. Pretest and posttest scores, along with various performance metrics, were analyzed using SPSS version 27. Statistical tests, including Paired-Samples T-Test and Independent-Samples T-Test, were conducted to evaluate the hypotheses.

 The results indicated significant improvement in students' mathematical performance post-intervention, confirming the positive effect of the Flipped Learning method. No significant differences were found between female and male students' academic progress, suggesting equal benefits across genders. Extroverted students showed greater academic progress than introverted students, with extroverted females and males outperforming their introverted counterparts. Lecturer scaffolding significantly enhanced students' Mathematical Mind Processing (MMP), with introverted students showing more substantial MMP improvements compared to extroverts. No significant differences were found in companionship between introverts and extroverts or between female and male students.

 The study concludes that the Flipped Learning method, when combined with the DISC Personality Test and lecturer scaffolding, significantly enhances undergraduate students' mathematical performance. This approach benefits both genders equally and supports greater academic progress for extroverted students. Scaffolding plays a crucial role in improving cognitive processing, particularly for introverted students. These findings highlight the importance of personalized and interactive teaching methodologies in higher education.

This paper explores generalized hyper S-acts (GHS-acts) over a hypermonoid S as generalizations of monoid acts within the context of algebraic hyperstructures. Specifically, we extend the definition of C-injectivity to GHS-acts and investigate their internal and homological properties. It is established that for being GHS-injectivity of GHS-acts with a fixed element, it suffices to consider allinclusions from cyclic GHS-subacts into indecomposable ones. Then we introducenew concepts known as semi-injectivity and semi-C-injectivity. By providing examples, we demonstrate that injectivity and semi-injectivity (C-injectivity and semiC-injectivity) are different concepts for GHS-acts, whereas they are the same in the context of acts over monoids. It is also shown that all pure GHS-acts are injective if and only if all pure cyclic GHS-acts are C-injective. Furthermore, we establish an equivalent condition on a hypermonoid S such that all quotients of SS exhibit semi-injectivity. Finally, we derive an equivalent condition for a hypermonoid to beclassified as semi-injective.

The existing research in this field indicates that there has been insufficient investigation into the life experiences of sixth-grade teachers and students concerning the qualitative values of mathematics lessons. Furthermore, there is a noticeable gap in the development of a scale for assessing teaching skills in the qualitative evaluation of mathematics lessons. This highlights the need for comprehensive studies that explore the lived experiences of both teachers and students in the context of qualitative values within mathematics education, along with the design of an effective scale to assess teaching skills in the qualitative evaluation of mathematics lessons.

The purpose of this research was to investigate the lived experiences of sixth grade teachers and students of the qualitative value of math lessons.

The current research was applied in terms of purpose and qualitative in terms of method and based on phenomenological strategy. The participants of this research were sixth grade teachers and students who were selected by purposive sampling. The tool used in this research was a semi-structured interview. In order to analyze the data, the coding method was used.

the results showed that the interviewees suffered from disadvantages such as lack of belief in qualitative evaluation, teachers' lack of knowledge about the philosophy of qualitative evaluation, the spirit of evaluation not being governed, and the inability of students to measure by themselves., the growth of teachers in the evaluation system has been mentioned a little. Also, teachers suffer from harms such as the reduction of healthy competition, children's insensitivity to lessons, low motivation, and lack of proper infrastructure, carelessness, students not studying, lack of enthusiasm and attractiveness in teachers, and the inability to cover weaknesses in evaluation. Qualitatively pointed out.

The aim of the current research was to investigate the lived experiences of sixth grade teachers and students of the qualitative evaluation of mathematics lessons, to design a scale of teacher skills in the qualitative evaluation of mathematics lessons. The results of data analysis showed that from the teachers' point of view, 23 main categories and 60 sub-categories, and from the students' point of view, 21 main categories and 42 sub-categories were identified as necessary factors and components and obstacles and challenges for the implementation of qualitative evaluation.

In the theory of S-acts, there exists a plethora of compelling findings that establish connections between regularity in a monoid S and injectivity (as well as other notions derived from injectivity) of Sacts. In this paper, we introduce the notion of strongly regular elements in a hypermonoid as a generalization of regular elements in a monoid and extend several classic results to hypermonoids and GHS-acts over hypermonoids. Particularly, we show that a hypermonoid S is strongly regular and injective GHS-act if and only if all right hyperideals of S are C-injective.

Specific learning disabilities include a heterogeneous group of children who have problems in various domains. These problems have life-long effects that seem to extend far beyond reading, writing, and arithmetic, and profoundly affect areas such as health, mental health, interpersonal relationships, continuing education, employment opportunities, and the like.

The aim of this study was to compare the effectiveness of game-based therapy and game-based therapy integrated with cognitive rehabilitation techniques on the math performance and Anxiety mathematical in Students With learning disabilities (MLD).

In the quasi-experimental research of pre-test-post-test multi-group designs, among the statistical population of the study, including all students of third, fourth, fifth and sixth grades of the academic year of 2018- 19 of Khomeini Shahr city in Iran, the number of 36 students those with math learning disabilities were randomly divided into three groups of 12. (2 experimental and 1 control). Then the experimental groups were treated. During this period, the control group did not receive any psychological or pharmacological interventions. Shalve mathematics questionnaires and Kazeleskis mathematical anxiety scale were used to evaluate the subjects.

The results of data analysis showed that play therapy-based education and the play therapy-based education combined with cognitive rehabilitation methods in mathematical anxiety and mathematical achievement at the level of(p< 0.05) were associated with significant effectiveness. Also, play therapy-based education combined with cognitive rehabilitation methods was more effective in the mathematical achievement variable in students with mathematic learning disabilities compared to play therapy-based education.

This research was conducted with the aim of comparing the effectiveness of play therapy-based education and play therapy-based education integrated with cognitive rehabilitation methods in math anxiety and math academic progress in students with math learning disorders, by examining 36 students with math learning disorders who were in three There were experimental and control groups, the experimental groups received training based on play therapy and training based on play therapy integrated with cognitive rehabilitation methods

The purpose of this research was to investigate the effectiveness of the intervention program of using information and communication technology in mathematics teaching. This research, in terms of purpose, is applied; In terms of how to collect information and data, experimental, semi-experimental, with a pre-test-post-test plan with a control group; And in terms of the nature of the data, it was quantitative. The statistical population of the research is mathematics teachers educated at Farhangian University; and there were also the principals of the schools where these teachers were present. The statistical sample was 30 mathematics teachers in the academic year 2021-2022. Using a non-random sampling method, 15 individuals were placed in the experimental group, while 15 individuals were placed in the control group in the same manner. The control group continued its traditional method; however, the experimental group received an intervention program for the use of information and communication technology in teaching mathematics for 8 sessions and 2 hours with each session. In addition to descriptive statistics, Shapiro-Wilk test, regression slope homogeneity test, Mbox test, Levon test, multivariate covariance analysis (MANCOVA) test and Bonferroni post hoc test were used for data analysis.The findings of the research showed that the intervention program for teaching the use of information and communication technology in mathematics teaching is effective.It is concluded, for the effectiveness of mathematics teachers' performance; the institution of education should take steps towards the implementation of the intervention program for the use of information and communication technology

‎Let $ \sum_{i=0}^{n}(-1)^il_ix^{n-i}$ and $\sum_{i=0}^{n}(-1)^iq_ix^{n-i}$ be the characteristic polynomials of the Laplacian matrix and signless Laplacian matrix of an $n$-vertex graph‎, ‎respectively‎. ‎Let $\alpha_i = |q_i‎ - ‎l_i|$‎, ‎$0\leq i \leq n$‎. ‎In this paper‎, ‎we find formulas for some of $\alpha_i$'s‎. ‎In particular‎, ‎we compute $\alpha_i$'s for some fullerene graphs.

‎In this paper‎, ‎we calculate the expected values of the first and second Zagreb indices‎, ‎denoted as $\textbf{E}\left(M_1\right)$ and $\textbf{E}\left(M_2\right)$ respectively‎, ‎as well as the expected value of the forgotten index‎, ‎$\textbf{E}\left(F\right)$‎, ‎for two models of random bipartite graphs‎. ‎To evaluate our findings‎, ‎we establish the growth rate by demonstrating that for a random bipartite graph $G$ of order $n$ in either model‎, ‎the expected value of $M_1(G)$ is $O\left( n^3 \right)$‎. ‎Furthermore‎, ‎we prove that the expected values of $M_2(G)$ and $F(G)$ are both $O\left( n^4 \right)$‎.

Let $S$ be a commutative pointed monoid. In this paper, some properties of admissible (Rees) short exact sequences of $S$-acts are investigated. In particular, it is shown that every admissible short exact sequence of $S$-acts is Rees short exact. In addition, a characterization of flat acts via preserving admissible short exact sequences is established. As a consequence, we show that for a flat $S$-act $F$, the functor $F \otimes_{S} -$ preserves admissible morphisms. Finally, it is proved that the class of flat $S$-acts is a subclass of admissibly projective ones.

The aim of this study was to compare the effectiveness of game-based therapy and game-based therapy integrated with cognitive rehabilitation techniques on the math performance and attitude of students with mathematical learning disabilities (MLD). A pre-test post-test quasi-experimental design was utilized to investigate the impact of independent variables on the dependent variable, with the use of a control group. The statistical population of the study consisted of male students with MLD in third to sixth grades from a school in Khomeynishahr, Iran, in 2020. A purposive sampling method was used to select 36 students with MLD, who were randomly assigned to three groups, including an experimental group receiving game-based therapy, an experimental group receiving game-based therapy integrated with cognitive rehabilitation techniques, and a control group receiving no intervention. Shalev arithmetic disorder scale and Aiken's mathematics attitude scale were used to collect data. MANCOVA was used to test the hypotheses. The findings revealed that game-based therapy and game-based therapy integrated with cognitive rehabilitation techniques significantly improved the math performance and attitude of the students (p < 0.05). Additionally, game-based therapy integrated with cognitive rehabilitation techniques was found to be more effective in improving math performance compared to game-based therapy alone (p < 0.05), while no significant difference was observed between the two interventions in improving attitude (p = 0.792). Therefore, it can be concluded that integrating game-based therapy with cognitive rehabilitation techniques can be more effective than other interventions in improving the math performance of students with MLD.

ICT Developing process will change learning type. Therefore, the curricula of using ICT in mathematics education at Farhangian University should be adapted to the content of high school math textbooks. Sample was selected by quota sampling method. Data were collected by semi-structured interviews, whose validity of questions and their reliability were confirmed by the peer member and external observer reviews. Text of interviews was coded and analyzed by Maxqda2020 software. According to instructors, teaching math ICT lessons to the student-teachers does not cover the ability of teaching content of high school math textbooks by technology and headings do not match the same content. These headings have a lot of exaggeration and emphasize theoretical dimension than practical one. To be highly flexible in each training course, instructors have selective use of software. Hardware training of systems, Information security, technological thinking, using math-computer experts, math specialized software, Headings should be updated.

Universal left congruences on semigroups were studied in “Y. Dandan, V. Gould, T. Quinn-Gregson and R. Zenab, Semigroups with finitely generated universal left congruence, Monat. Math. 190 (2019) 689−724”. We consider universal congruences on acts over monoids and extend the results from semigroups to acts. Among other things, for an S-act AS with zero over a monoid S, we prove that being finitely generated of the universal congruence ωA and being pseudofinite of AS coincide.

The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute values of all eigenvalues of $G$. A graph $G$ is called reciprocal if $ \frac{1}{\lambda} $ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $ \lambda $ and $\frac{1}{\lambda}$ have the same multiplicities, for each eigenvalue $\lambda$, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631--633), it was conjectured that for every graph $G$ with maximum degree $\Delta(G)$ and minimum degree $\delta(G)$ whose adjacency matrix is non-singular, $\mathcal{E}(G) \geq \Delta(G) + \delta(G)$ and the equality holds if and only if $G$ is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if $G$ is a strong reciprocal graph, then $\mathcal{E}(G) \geq \Delta(G) + \delta(G) - \frac{1}{2}$. Recently, it has been proved that if $G$ is a reciprocal graph of order $n$ and its spectral radius, $\rho$, is at least $4\lambda_{min}$, where $ \lambda_{min}$ is the smallest absolute value of eigenvalues of $G$, then $\mathcal{E}(G) \geq n+\frac{1}{2}$. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.

The aim of this study was to investigate the relationship between teacher skills in qualitative evaluation of mathematics course with mathematical anxiety, academic procrastination and students' thinking style. The present study was a descriptive correlational study.The statistical population consisted of students and teachers of grade 6 in the academic year of 2018-2019 in Mashhad, Iran. A total of 120 teachers and 200 students were selected as respondents by using convenience sampling and inclusion/exclusion criteria. The data were analyzed using descriptive indexes (e.g., mean and standard deviation) and inferential statistics (correlation and regression to relate the variables) in SPSS V.22. It was found that teachers’ mathematical performance assessment skills had negative relationships with mathematical anxiety and academic procrastination. Furthermore, a positive relationship was found between teacher mathematical performance assessment skills and student thinking style. The regression results demonstrated that teachers’ mathematical performance assessment skills were a significant predictor of mathematical anxiety and academic procrastination. It was suggested that teachers’ skills of assessing mathematical performance be enhanced to improve their academic performance and efficiency.

For a graph $G$, a set $L$ of vertices is called a total liar's domination if $|N_G(u)cap L|geq 2$ for any $uin V(G)$ and $|(N_G(u)cup N_G(v))cap L|geq 3$ for any distinct vertices $u,vin V(G)$. The total liar’s domination number is the cardinality of a minimum total liar’sdominating set of $G$ and is denoted by $gamma_{TLR}(G)$. In this paper we study the total liar's domination numbers of join and products of graphs.

In this paper, the concept of incidence domination number of graphs  is introduced and the incidence dominating set and  the incidence domination number  of some particular graphs such as  paths, cycles, wheels, complete graphs and stars are studied.

Let $ mathbb {Z}_{n} $ be the ring of integers modulo $ n $. The unitary Cayley graph of $ mathbb {Z}_{n} $ is defined as the graph $ G( mathbb {Z}_{n} ) $ with the vertex set $ mathbb {Z}_{n} $ and two distinct vertices $a,b$ are adjacent if and only if  $a-bin Uleft( mathbb {Z}_{n}right)$, where $ Uleft( mathbb {Z}_{n}right) $ is the set of units of $ mathbb {Z}_{n} $. Let $Gamma ( mathbb {Z}_{n} ) $ be the complement of $ G( mathbb {Z}_{n} )  $. In this paper, we determine the independence number of $ Gamma ( mathbb {Z}_{n} ) $. Also it is proved that $ Gamma ( mathbb {Z}_{n} ) $ is well-covered.  Among other things, we provide condition under which $ Gamma ( mathbb {Z}_{n} ) $ is vertex decomposable.the formula is not displayed correctly!

Divergent thinking increases the ability of children to solve problems and provides them with a wealth of associations and possible solutions. The purpose of this study was to investigate the validity and reliability of the divergent thinking questionnaire among primary school students.

The statistical population included all 7-year-old students of public schools in Zahedan. The sample size consisted of 336 students (176 girls and 169 boys) who were selected by available sampling method and answered the divergent questionnaire (Kaun et al., 2006). To analyze the data, Cronbach's alpha coefficient was used for reliability evaluation, confirmatory factor analysis for evaluating construct validity and exploratory factor analysis for factor load calculation.

The Cronbach's alpha coefficient for Persian translation of the  item initial score was 0.74 and test retest reliability of the questionnaire based on the results of the two tests was 0.79. Validity indices derived from confirmatory factor analysis showed that the scale had proper construct validity and all factors had an appropriate load factor of more than 0.20.

According to the results of this study, as well as the importance of measuring divergent thinking in children and the lack of appropriate tools validated in Iranian culture, it can be concluded that divergent thinking questionnaire is a good tool for applying educational situations in schools.

Terms like commutativity degree, non-commuting graph and isoclinism are far well-known for much of the group theorists nowadays. There are so many papers about each of these concepts and also about their relationships in finite groups. Also, there are some recent researches about generalizing these notions in finite rings and their connexions.The concepts of commutativity degree and non-commuting graph are also extended to non-associative structures such as Moufang loops and some part of the known results in group theory in these contexts have been expanded to them.In this paper, we are going to generalize the notion of isoclinism in finite Moufang loops and then study the relationships between these three concepts. Among other results, we prove that two isoclinic finite Moufang loops have the same commutativity degree and if they have the same sizes of centers and commutants then they have isomorphic non-commuting graphs. Also, the converses of these results have been investigated.Furthermore, it has been proved that a finite simple group can be characterized by its non-commuting graph. We will prove the same is true for a finite simple Moufang loop by imposing one additional hypothesis, namely, the isoclinism of the regarding loops.

Abstract. The Laplacian characteristic polynomial of an n-vertex graph G has the form f(G,x) = x^n+∑l_ix^n-i. In this paper, the fourth and fifth coefficient of f(G,x), will be investigated, where G is a T(k,t) tree in which a rooted tree with degree sequence k,k,···,k,1,1,···,1 is denoted by T(k,t).

In this paper, we present upper bounds for the multiplicative forgotten topological index of several graph operations such as sum, Cartesian product, corona product, composition, strong product, disjunction and symmetric difference in terms of the F–index and the first Zagreb index of their components. Also, we give explicit formulas for this new graph invariant under two graph operations such as union and Tensor product. Moreover, we obtain the expressions for this new graph invariant of subdivision graphs and vertex – semitotal graphs. Finally, we compare the discriminating ability of indices.

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in $mathbb{Int}(A)$ is equivalent to the finiteness of the number of subacts of $A$. Finally, we determine the clique number of the graphs of certain classes of $S$-acts.

Let G be a finite simple graph on the vertex set V(G) and let S⊆V(G) . Adding a whisker to G at x means adding a new vertex y and edge xy to G where x∈V(G) . The graph G∪W(S) is obtained from G by adding a whisker to every vertex of S . We prove that if G∖S is either a graph with no chordless cycle of length other than 3 or 5 , chordal graph or C 5 , then G∪W(S) is a vertex decomposable graph.