1030
The Practicality of Using the Circular Matrix to Determine the Spectrum Circle Graphs and Inquiry-Based Cubic Graphs on Mathematics Education Students of Universitas
Negeri Medan
Abil Mansyur1, Mulyono2, Baharuddin3, Dina Ampera4, Muhammad Aulia Rahman Sembiring5
Abstract
This research is motivated by the lack of understanding of students toward the use of circular matrix in finding their concepts. The purpose of this study is to find out the practicality of using the Circular Matrix to Determine the Spectrum of Circle Graphs and Inquiry-Based Cubic Graphs in Mathematics Education Students of Medan State University. This type of research is development research with the IDI Model (Instructional Development Institute) which applies the principles of a systematic approach consisting of 3 major stages, namely determination or analysis of needs, development, and evaluation. At this stage, it is discussed the practicality of the use of an inquiry-based circular matrix.
The instrument used is a written interview with students and spread practicality questionnaires. Based on the results of the interview obtained information, students more easily understand the material, because it guides fellow students in finding concepts when working on examples of problems given by lecturers. Based on the results of the analysis of student questionnaires obtained a level of 86.7% (very practical).
The graph spectrum is a collection of different eigenvalues and their multiplicity. Determining the graph spectrum can be done in several ways, including using a circular matrix. Circular matrix is widely applied in various areas of life, including analyzing vibrations in differential equations, determining the spectrum of a graph, and others. In this paper, the spectrum of circle graphs and cubic graphs is determined using a circular matrix. The results of the study were obtained:
𝑆𝑝(𝐶𝑛) = [ 2 2𝑐𝑜𝑠2𝜋𝑛 ⋯ 2𝑐𝑜𝑠(𝑛−1)𝜋
𝑛
1 2 ⋯ 2 ] (𝑛 ≥ 3, 𝑛 odd) 𝑆𝑝(𝐶𝑛) = [2 2𝑐𝑜𝑠
2𝜋
𝑛 ⋯ 2𝑐𝑜𝑠(𝑛 − 2)𝜋
𝑛 −2
1 2 ⋯ 2 1
] (𝑛 ≥ 4, 𝑛 even)
( 1) 1
3 2 cos ( 1) 2 cos ( 1) 3
( )
1 2 2 1
h n
h
Sp Q h h
, for ℎ odd
1 2
2 1
1 ) 1 ) ( 1 cos( 2 ) 1 ( cos 2 ) 3 (
1
h
n h
h Q h
Sp
, for ℎ even
Key Words: Graph spectrum, Circular matrix, eigen value, multiplicity
DOI Number: 10.14704/nq.2022.20.9.NQ440115 Neuro Quantology 2022; 20(9):1030-1042 Mathematics is one of the sciences used to facilitate
problem-solving. One of them is graph theory. In everyday life, graphs are used to represent discrete objects and the relationships between them. Some examples of graphs that are often found in everyday life include organizational structure, flow charts of course taking, maps, and electrical circuits. A graph consists of a set of non-blank points and a set of lines
connecting the dots on the graph. There are several types of graphs, including circle graphs.
Based on interviews with students, information was obtained that students found it difficult to understand the material of the circle graph. The material in the book has been equipped with a summary of the material and examples of the problem, but the example of the steps given in the answer there are still
1031 abandoned. So that students feel the confusion in
understanding the contents of the book. In addition, students also said, the exercise questions in the package book vary greatly and there are still many that are not understood. This is seen in every learning process; students only accept what the lecturer gives.
As an educator, you need to develop a learning model that can help students in thinking critically and practically in finding concepts.
One of the problems in a circle graph is determining the value of the spectrum. The graph spectrum is the set of numbers in which the numbers are the eigenvalues of the finite matrix, together with their multiplicity. The general form of a graph is determined from the determination of the value of the eigen. This form provides convenience in concentrating discrete objects. A matrix is a collection of number objects, symbols, or rectangular expressions arranged by rows and columns. The use of matrices can solve many problems and facilitate the creation of analyses that include relationships between variables such as to find solutions to linear equation systems. By bringing equations into matrix form so that a form of linear equation system will be simpler in its solution and easier to find a solution.
Many practical applications of the theory are applied in a variety of disciplines, for example in biology, computer science, economics, engineering, informatics, linguistics, mathematics, health, and the social sciences. In many ways, graphs become excellent modeling tools for explaining and solving problems.
According to Roestiyah (2008: 75) Inquiry is an English term, which means a technique or way used by teachers to teach in front of the classroom. According to Sanjaya (2006: 196), the inquiry is a series of learning activities that emphasize the process of thinking critically and analytically to find and find the answer to a problem. In line with that, Sagala (2009:
53) defines the Inquiry approach as a learning strategy that seeks to instill the basics of scientific thinking in students who act as learning subjects, so that in the learning process learners learn more on their own, developing creativity in solving problems. From some of the expert opinions above, it can be concluded that the inquiry method is a series of activities that invite students to think critically by trying to instill the basic basis of critical thinking to find the answer to a problem themselves. The steps of the inquiry method according to Kindsvatter et al in Suparno (2007) are
(1) Identification and clarification of problems. The first step is to determine the problem that you want to explore or solve by the inquiry method. Problems can be prepared or raised by the teacher. The problem itself must be clear so that it can be thought of, explored, and solved by students. (2) Making a hypothesis, the next step is that students are asked to submit temporary answers about a problem. This is called a hypothesis. The student hypothesis needs to be studied whether it is clear or not. If it is not clear, educators should try to help clarify their intentions first. Educators are expected not to correct the wrong student hypothesis, but simply clarify the meaning only. The wrong hypothesis will be seen after the data collection and analysis of the data obtained. (3) Collecting data, the next step is for learners to find and collect as much data as possible to prove whether their hypothesis is correct or not. (4) Analyzing the data, the data that has been collected must be analyzed to be able to prove the hypothesis whether it is true or not.
(5) Conclude, data that has been grouped and analyzed, then concluded with generalizations. After conclusions are drawn, it is then matched with the origin hypothesis, whether our hypothesis is accepted or rejected. Before looking at the practicality of this inquiry-based calculus 2 modules, the module has been declared valid by the validator. This means that the module has met the criteria of an inquiry-based module. To see the practicality/usefulness of the module, the module can be tested on students who take calculus 2 courses. After being tested, the author distributed interview questionnaires and practical questionnaires to students. Furthermore, the questionnaire is analyzed and compared with the practical value that exists in the theory.
Models in graph theory are often used in life, such as determining the waiting time of traffic lights at intersections (traffic lights), operations research, flow charts taking courses, organizational structures, communication networks, and others. Therefore, the development of graph theory is very rapid. In addition, the development of linear algebra is also very synergized with the development of graph theory.
These two sciences have always supported each other.
Graph theory and linear algebra are closely related, as they can be studied simultaneously.
Outcomes.
1032 Theoretical Studies
Understanding Graf
In the basic concept of graphs, it is known that the definition of a graph and its elements will be arranged using a set language. Therefore, before discussing the definition of the graph, I will be explained in advance the requirements of a set. Where a set itself is required that each element in it will only appear once. A graph is a "tool" used to represent objects in a discrete and the relationship that consists between these objects.
Graph (G) is defined by a pair of stacks consisting of a non-empty set of nodes that are usually symbolized by (V/vertices), and a set of ribs symbolized by (E/edge).
Where both V/E produce a pair of knots on G. Then, the knots on G will be notated as V, and the rib set on G will be notated with E. Thus, it will produce the following formula G= (V, E). Meanwhile, Siang (2002), stated that on a graph G will be highlighted from two finite sets.
Thus, a graph G consists of sets V and E, which will then be written with (V, E), with V which is a finite set, and E consisting of a set of ribs side by side with V.
Thus, it can be concluded that graph G is a pair of sets in which will consist of a non-empty set and a set of ribs both of which are then symbolized by (V and E).
Then, to facilitate the explanation of graph G can be done with a relationship of two points, two sides, and nodes given a certain name.
a. Terms in Graph
Several terms are usually very often used in graph G, including the following:
1. Bracelet (Loop): is a rib that can be said to be a bracelet if the end of the rib has a beginning and end on the same knot.
2. Multiple Edges: is a naming given to a G graph that has more than one rib with the same pair of knots.
3. Adjacent: are two knots found on a G-directional graph that is said to be neighbors when they are directly connected to the same rib.
4. Side by side (Incident): so, for any rib e = (u, v), ribs e will be said to intersect with knot u and node v.
5. Empty Graff (Null Graph or Empty Graph): is a G graph whose set in the ribs is empty.
6. Journey (Walk): is an alternating line between knots and ribs from G, which will be started with knot u and then ended with node v.
7. The path is a line that intersperses between the knots and ribs of a G graph.
8. Cycle and Circuit (Circuit): is a trajectory that will start and end at the same node.
9. Connected: two nodes in graph G, will be said to be connected if there is a trajectory, if each node in the graph is connected, then the graph is called a connected graph.
10. Weighted Graph: is a graph in which each rib is given a price or weight, where each rib will have a different weight, depending on the problem modeled with the graph.
b. Types of Graphs
According to Munir (2005), who stated that the G graph will be grouped into several types that are to the grouping point of view. Where, the grouping of graphs can be assessed through the absence of double ribs, based on the number of knots, or based on the orientation of the direction on the ribs. In general, the graph itself is classified into two types, namely simple graphs and non-simple graphs, which will be explained as follows:
1. Simple Graph
According to Harju (2012), simple graphs do not have double ribs or, bracelets. Where ribs are an unordered pair (unordered pairs). In addition, according to Munir (2005), a simple graph can also be interpreted as a graph consisting of a non-empty set of nodes and a set of different unordered pairs which are then referred to as ribs. Where, according to Siang (2002) some special graphs are often used graphs, as follows:
- Complete Graph: is a simple graph that in each of the two nodes is neighboring each other.
- Circle Graph: is a simple graph that on each node is twofold.
- Regular Graph: is a graph that at each node has the same degree.
- Bipartite Graph: is a graph in which the V1 node is connected to the node on V2 which is then expressed by G (V1, V2).
2. Unsubstantiated Graph
Harju (2012), stated that the graph containing double ribs or bracelets will be called a non-simple graph (unsampled graph). There are two types of graphs that graph is not simple, namely double graph (multigraph) and pseudo graph (pseudepigrapha).
Where a double graph is a graph that contains double
1033 ribs. Meanwhile, pseudo-graphs are graphs that
contain bracelets (loops). In addition, according to Bondy and Murty (1982), unassuming graphs can also be grouped based on the orientation of the direction of the ribs. Where, the orientation of the direction is divided into two, namely the undirected graph and the directional graph.
1. Undirected Graph: is a graph whose ribs do not have a directional orientation.
2. Directed Graph: is a graph whose ribs have a directional orientation.
A graph is a pair by is a non-empty and finite set of objects called points and is a set (possibly empty) of sequenced pairs of different points in V(G) called sides. Suppose G graph with order and size and set of points. The connected matrix of the graph, notated with is a matrix with elements in the ith row and the jth column is worth if the point is neighboring to the point and is worth if the point is not next to the point.
Mathematically, the connection matrix of the graph can be written by:
𝐴(𝐺) = { 1, jika 𝑣𝑖𝑣𝑗 ∈ 𝐸(𝐺) 0, jika 𝑣𝑖𝑣𝑗 ∉ 𝐸(𝐺)
The connected matrix of a graph is a matrix of symmetry with an element and has a value on its main diagonal. The connectedness matrix is widely used to discuss the characteristics of a graph because the connectedness matrix is square. Working with a square matrix will provide a lot of conveniences compared to a non-square matrix. The discussion of the connection matrix of a graph can be attributed to the concept of eigenvalues and eigenvectors on the topic of linear algebra which produces the concept of the spectrum of a graph.
The set of eigenvalues and their multiplicity of a square matrix is called a spectrum. Suppose 𝜆1, 𝜆2, ⋯ , 𝜆𝑝 it is the eigenvalue of any matrix of the connectedness of a graph and is the multiplicity of each, then the specter of graph G, 𝜆1, 𝜆2, ⋯ , 𝜆𝑝 notated with Sp(G) can be written in the following form (Thulasiraman: 2016)
𝑆𝑝(𝐺) = [ 𝜆1 𝜆2 ⋯ 𝜆𝑝
𝑚(𝜆1) 𝑚(𝜆2) ⋯ 𝑚(𝜆𝑝)]
with. 𝜆1> 𝜆2> ⋯ > 𝜆𝑝 In addition to using
characteristic value equations, the determination of the spectrum of a graph can also be done using a circular matrix.
Based on the value of the eigence, the phenomenon of an object of study can be analyzed more deeply.
Therefore, the role of eigenvalue is very important in various studies. Applied eigenvalues are used in wave equations, population growth predictions, wave vibration analysis, determination of traffic light waiting times at intersections, optimization theory, sound signal analysis, and observing genotype differences. Research related to the spectrum has been conducted by several researchers, including Syarifuddi Side and Sutra (2013), Herianti (2016), Khoirul Umam (2019). Based on the studies mentioned above, then in this study in study in the survey of the spectrum of circle graphs and cubic graphs.
Circular Matrix
A circular matrix is one of those specialized matrices whose rows (or columns) are circular shifts from previous rows (or columns). As with ordinary matrices, circular matrices also have eigence values and eigenvectors. Some research related to the circular matrix has been conducted by Gavalec (2010) which discusses the characteristics of the structure of the circular matrix eigen chamber in max-min algebra.
Kalman (2001) has researched polynomial equations and circular matrices. His research discussed the basic concepts of circular matrix and the use of a circular matrix to solve quadratic, cubic, and quartic polynomial equations. In addition, it is also discussed about the use of a circular matrix to analyze polynomial roots.
Ruminta (2009) describes a matrix as a collection of numbers arranged specifically in the form of rows and columns to form four rectangles or squares written between two parentheses, namely () and []. The numbers in the arrangement are called entries from the matrix. The size (size) of a matrix is expressed in the number of rows (horizontal direction) and columns (vertical direction) it has (Anton et al, 2004).
The matrix is denoted by uppercase letters, while the entry (matrix element) is denoted by a lowercase letter. The concept of a matrix is very useful in solving problems in mathematics. One of the most common problems in mathematics is solving a system of linear equations. Solving linear equation system problems
1034 can be solved using matrix inverses
The circular matrix has been known to people since the early 19th century. A circular matrix is a special form of the Toeplitz matrix. The circular matrix can be used to solve polynomial equations.
Definition 2.1
The circular matrix is rectilinear () with a row element to – to be obtained by shifting the elements on the first row to the right as many steps (Biggs, 1974).
For example, the matrix and the following are circular matrices
𝐵 = [
1 3 5 7
7 1 3 5
5 3
7 5
1 7
3 1
] and 𝐶 = [
1 3 5 7
7 1 3 5
5 3
7 5
1 7
3 1 ]
Suppose 𝐶 = [
𝑐1 𝑐2 ⋯ 𝑐𝑝 𝑐𝑝 𝑐1 ⋯ 𝑐𝑝−1
⋮ 𝑐2
⋮ 𝑐3
⋱
⋯
⋮ 𝑐1
] dan 𝐻 =
[ [
0 1 ⋯ 0
0 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0 ]
]
each is a circular matrix, then the
matrix S can be expressed as follows:
𝑆 = 𝑐1[
1 0 ⋯ 0
0 1 ⋯ 0
⋮ 0
⋮ 0
⋱ ⋮
⋯ 1
] + 𝑐2[
0 1 ⋯ 0
0 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0
] +
𝑐3[
0 0 1 0
0 0 0 0
⋮ 0
⋮ 0
⋱ 0
⋮ 0
] + ⋯
+𝑐𝑝[
0 0 ⋯ 1
1 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0 ]
= 𝑐 [
1 0 ⋯ 0
0 1 ⋯ 0
⋮ 0
⋮ 0
⋱ ⋮
⋯ 1
] + 𝑐2[
0 1 ⋯ 0
0 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0
] +
𝑐3[
0 1 ⋯ 0
0 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0
] [
0 1 ⋯ 0
0 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0 ]
+ ⋯ + 𝑐𝑝 [
[
0 1 ⋯ 0
0 0 ⋯ 0
⋮ 0
⋮ 0
⋱
⋯
⋮ 0 ]
]
𝑝−1
= 𝑐1𝐻0+ 𝑐2𝐻 + 𝑐3𝐻2+ ⋯ + 𝑐𝑝𝐻𝑝−1= ∑𝑝𝑗=1𝑐𝑗𝐻𝑗−1, dengan 𝐻0= 𝐼
Theorem 2.1
Let C be a circular matrix of order p, with [𝑐1, 𝑐2, ⋯ , 𝑐𝑝], is the first-row element in matrix C, then the eigenvalue of C is 𝜆𝑟= ∑𝑝𝑗=1𝑐𝑗𝜔(𝑗−1)𝑟, with 𝜔 = 𝑒2𝜋𝑖⁄𝑛 and, vector eigen to 𝑟 y which corresponds to 𝜆 the eigenvalue of is 𝑢𝑟= [1, 𝜔𝑟, 𝜔2𝑟, ⋯ , 𝜔(𝑝−1)𝑟]𝑇 for each 𝑟 = 0,1,2, ⋯ , 𝑝 − 1.
Proof:
Because 𝐶 it is an ordered circular matrix and is a vector of eigence to – 𝑟 which corresponds to the value
of the eigence 𝜆, and, 𝑢𝑟=
[1, 𝜔𝑟, 𝜔2𝑟, ⋯ , 𝜔(𝑝−1)𝑟]𝑇for each = 0,1,2, ⋯ , 𝑝 − 1, then 𝐶𝑢𝑟= 𝜆𝑟𝑢𝑟
[
𝑐1 𝑐2 ⋯ 𝑐𝑝 𝑐𝑛 𝑐1 ⋯ 𝑐𝑝−1
⋮ 𝑐2
⋮ 𝑐3
⋱
⋯
⋮ 𝑐1
] [ 1 𝜔𝑟
⋮ 𝜔(𝑝−1)𝑟
] = 𝜆𝑟[ 1 𝜔𝑟
⋮ 𝜔(𝑝−1)𝑟
]
[
𝑐1+ 𝑐2𝜔𝑟+ ⋯ + 𝑐𝑛𝜔(𝑝−1)𝑟 𝑐𝑛+ 𝑐1𝜔𝑟+ ⋯ + 𝑐𝑛−1𝜔(𝑝−1)𝑟
⋮
𝑐2+ 𝑐3𝜔𝑟+ ⋯ + 𝑐1𝜔(𝑝−1)𝑟 ]
=
[
𝜆𝑟 𝜆𝑟𝜔𝑟
⋮ 𝜆𝑟 𝜔(𝑝−1)𝑟
]
[
𝑐1+ 𝑐2𝜔𝑟+ ⋯ + 𝑐𝑛𝜔(𝑝−1)𝑟 𝑐1𝜔𝑟+ 𝑐2𝜔𝑟+ ⋯ + 𝑐𝑛−1𝜔(𝑝−1)𝑟+ 𝑐𝑝
⋮
𝑐1𝜔(𝑝−1)𝑟+ 𝑐2+ 𝑐3𝜔𝑟+ ⋯ + 𝑐𝑛𝜔2(𝑝−1)𝑟]
=
[
𝜆𝑟 𝜆𝑟𝜔𝑟
⋮ 𝜆𝑟 𝜔(𝑝−1)𝑟
]
[
∑𝑝𝑗=1𝑐𝑗𝜔(𝑝−1)𝑟
∑𝑝𝑗=1𝑐𝑗𝜔𝑗𝑟
⋮
∑𝑝𝑗=1𝑐𝑗𝜔(𝑗+𝑝−2)𝑟]
= [ 𝜆𝑟 𝜆𝑟
⋮ 𝜆𝑟
]
So, the eigenvalue to- 𝐶 of the matrix 𝐶 is, 𝜆𝑟=
∑𝑝𝑗=1𝑐𝑗𝜔(𝑗−1)𝑟 for each. 𝑟 = 0,1,2, ⋯ , 𝑝 − 1.
The graph 𝐺 is said to be a circular graph if the connection matrix of the graph is circular. Based on the fact that the connectedness matrix is a symmetry matrix with zero entries on its main diagonal, this results in that if the first row of the connection matrix of the circular graph is(𝑎1, 𝑎2, ⋯ , 𝑎𝑝), then 𝑎1 = 0 and, 𝑎𝑖= 𝑎𝑝−𝑖+2, for each. 𝑖 = 2,3, ⋯ , 𝑝
1035 Theorem 2.2
Suppose [0, 𝑐1, ⋯ , 𝑐𝑝]it is the first line of the connection matrix of the circular graph, then the eigenvalue of the graph 𝐺 is
𝜆𝑟= ∑𝑝𝑗=2𝑐𝑗𝜔(𝑗−1)𝑟
for each 𝑟 = 0,1,2, ⋯ , 𝑝 − 1 with. 𝜔 = 𝑒2𝜋𝑖⁄𝑛 Proof:
Since 𝐺, the connection matrix 𝐺 of can be written in the form of 𝐴(𝐺) =
[
0 𝑐2 𝑐3 ⋯ 𝑐𝑝 𝑐𝑝 0 𝑐2 ⋯ 𝑐𝑝−1 𝑐𝑝−1
⋮ 𝑐2
𝑐𝑝
⋮ 𝑐3
0
⋮ 𝑐4
⋯
⋱
⋯ 𝑐𝑝−2
⋮ 0 ]
According to Theorem 2.1, the obtained value of to-𝑟 the Eigen to-from the graph is 𝐺
𝜆𝑟= ∑ 𝑐𝑗𝜔(𝑗−1)𝑟
𝑝
𝑗=1
= 𝑐1𝜔0.𝑟+ 𝑐2𝜔1.𝑟+ 𝑐3𝜔2.𝑟+ ⋯ + 𝑐𝑛𝜔(𝑝−1)𝑟
Because of 𝑐1= 0, then, 𝜆𝑟= ∑𝑝𝑗=2𝑐𝑗𝜔(𝑗−1)𝑟, for every 𝑟 = 0,1,2, ⋯ , 𝑝 − 1.
Inquiry Model in Mathematics Learning
According to Djuanda (2015), the inquiry model is a series of learning activities that emphasize the process of thinking critically and analytically to find and find the answer to a questionable problem for yourself.
Inquiry is an expansion of the discovery process used more deeply. inquiry means question. or examination, investigation. inquiry as a general process carried out by humans to find or understand information. The inquiry model is a learning model whose presentation provides opportunities for students to find information with or without the help of lecturers.
Inquiry is a word that has many meanings for many people in a variety of different contexts. In science, inquiry means the art or science of asking about nature and finding answers to those questions. Inquiry is carried out through steps such as observation and measurement, hypothesis, interpretation, and theoretical preparation.
Inquiry requires experimentation, reflection, and
introduction to the strengths and weaknesses of the methods used (Kusmayono and Setiawati, 2013).
Inquiry-Based Teaching-learning restriction is a teaching approach that mandates lecturers to create situations that position students. Learners take the initiative to question a phenomenon, propose hypotheses, make observations in the field, analyze data, draw conclusions, and explain their findings to others. The expected answer to the question is not singular but plural, the important thing is that in finding answers, students work using certain clear standards so that the results can be accounted for, so that students can integrate and synergize various disciplines and/or different methods (Kusmayono and Setiawati. 2013).
According to Wallace and Husid (2017), Inquiry-Based Learning (IBL) is structured and guided yet open to allowing students to assume personal roles in their learning. there is fluidity. The more students work with the process, the less they will rely on which step they are attaining. Their information literacy skills, use of Bloom's taxonomy, and progress with IBL merge into self-actualized courses of action.
According to Wallace and Husid (2017), Inquiry- Based Learning (IBL) is structured and guided but open to allow students to take a personal role in their learning. There is fluidity. The more students work with the process, the less they depend on which steps they achieve. their information literacy skills, Bloom's taxonomic use, and progress with Inquiry-Based Learning (IBL) joining into self-actualized action programs.
IBL is an instructional practice where students explore content by posing, investigating, and answering questions. Students are at the center of the learning experience and take ownership of their learning (Wells di Caswel and LaBrie, 2017). They often work independently and in small collaborative groups. As Mahaveer et al. state, in an IBL classroom, "the instructor plays the role of coach, mentor, collaborator, guide, and occasional cheerleader" More specifically, the teacher's role in IBL is to guide students and promote thinking and curiosity. This takes purposeful planning to manage multiple student investigations simultaneously. Teachers monitor the progress of each student and provide immediate feedback (Jones di Caswell and LaBrie, 2017). IBL does not indicate less guidance from the teacher but rather delivers instruction in such a way that the student constructs their meaning (Pitman di Caswell and
1036 LaBrie,2017. The teacher serves as the facilitator who
plans, instigates, and observes the student's learning process. Currently, there are many definitions of IBL and a variety of approaches. The Academy of Inquiry- Based Learning states that IBL engages students and requires them to: solve problems, conjecture, experiment, explore, create, and communicate (Ernst di Caswell and LaBrie, 2017).
Inquiry-Based Learning (IBL) is a learning practice in which students explore content by posing, investigating, and answering questions. Student is at the center of the learning experience and takes ownership of their learning (Wells in Caswell and LaBrie, 2017). Students can work independently and in small collaborative groups.
The role of the Teacher in Inquiry-Based Learning abyss guide, students and promote thinking and curiosity. It takes planned planning to manage many student investigations simultaneously. Teachers monitor the progress of each student and provide immediate feedback. Lecturers serve as facilitators who plan, incite, and observe the learning process.
Inquiry-Based Learning (IBL) involves students and requires them to: solve problems, speculate, experiment, explore, create, and communicate (Wells in Caswell and LaBrie, 2017).
According to the National Research Council (1996) in Ismail (2005), inquisition generally intends to find information, question, and investigate phenomena that occur around. Through inquiry, lessons explain objects of questionable processes, conducting experiments to share discoveries or solutions.
Scientific inquiry refers to the various ways that scientists use to study nature and suggests an explanation based on evidence of the results rather than their effort. Inquiry in science refers to the activities of students allowing them to expand knowledge and understand scientific ideas and understanding of how designers study the universe.
So, student inquiry involves observing, raising questions, listening to books and other sources of information about known litigants based on experimental evidence, using tools to collect, analyze and interpret data, suggest answers, and explanations, and sharing decisions or opinions. Inquiry requires a kind of assumption, the use of critical and logical thinking, and the development of alternative explanations.
According to DoBoer (1991), the process of inquiry is modeled through the methods used by scientists in
making discoveries. Science is seen as a set of theories and ideas built on the physical world, and not a collection of pertinent and indisputable facts, inquiry is a complex process and inquiry lessons will lead teachers to experience for themselves scientific inquiry.
The inquiry can also be interpreted as follows:
1. Inquiry is the process of finding and investigating problems, constructing hypotheses, forming experiments, collecting data and making experiments, and making conclusions for problem-solving.
2. Inquiry is defined as the process of seeking truth, information, or knowledge through the rules of silliness. The process of inquiry begins with the collection of information through the senses of sight, hearing, touch, and sense of smell (what school and Disney learning 2000).
3. Inquiry is defined as a technique of socializing a thing and finding answers to the spoken silliness.
It involves careful observation and measurement, making hypotheses, translating, and constructing theories. Inquiry requires the skills of experimenting, reflecting, and taking into account the strengths and weaknesses of the methods used (Herank, 2000).
In the scientific Inquiry model, lecturers use knowledge, imagination, superstition, and process skills to actively build scientific understanding, in addition to the Scientific Inquiry using thought and process skills to build an understanding of scientific knowledge actively. Through inquiry, students practice the necessary skills days life. Skills are the beliefs learned to do things well. Life skills are characterized as skills that help individuals to succeed through productive and fulfilling lives, such as thinking, concern, and so on (Hendrick, 1996).
Mathematics in the 21st century requires lecturers to teach students how to learn and how process information. Furthermore, this can be detailed into what will be taught, how it is taught, how the condition of the student and what new views can be given.
Gulo (2002) stated that the inquiry learning model is a series of learning activities that involve to the maximum the ability of students to seek and investigate systematically, critically, and logically.
analytically so that they can formulate their findings with confidence. According to Majir (2017), The Inquiry-Based Learning Model is a teaching technique where lecturers involve students in the learning
1037 process through the use of ways of asking questions,
problem-solving activities, and critical thinking. This will take a lot of time to prepare. Inquiry-based learning is usually collaborative work. Classes are divided into small groups. Each group is given a question or problem that will lead all group members to work together to develop a project based on that question to find the answer. Because inquiry-based learning is based on questions, lecturers must prepare questions that are open so that students can develop their minds. Mahasiswa should be allowed to try to discover for themselves the concepts taught.
Moreover, if students are also allowed to measure their learning progress, then ha] this will help them learn. According to Syarifuddin (2018), the Inquiry Learning Inquiry Model Inkuiri means the process of learning is based on search and discovery through a systematic thought process (lstarani, 2016). While Basyiruddin Usman (2005) said that inquiry is a way of sharing lessons by studying something that is critically searching, analytical, dun argumentative human using certain steps towards the conclusion in learning.
RESEARCH METHODS
Writing is done through the study of literature. The procedures used in this study are proving the properties of symmetrical circular matrix operations, proving a single solution of a linear equation system using matrix inverses, and proving matrix inverses of two special types of the symmetrical circular matrix over skew fields.
To achieve the research objectives stated in the introduction, the steps are taken for n = 2, 3, 4, ... , 7 in determining the complete graph spectrum using the circular matrix are as follows:
1. Draw a Kn,n graph
2. Draw isomorphic from graph Kn,n
3. Looking for the efficiency matrix of the Kn,n graph whose finite matrix must be a circular matrix
4. Find the eigenvalue of the circular matrix by using Theorem 2.2: λr = ∑ajw (j−1) r n j=2 5. Forming the spectrum of a complete bipartite
graph using the equation 2.9: SpecG = [ λ0 λ1 ...
λs−1 m (λ0) m (λ1) ... m (λs−1)]
6. Get a complete graph spectrum Research Results and Simplifiers
One of the basic studies in studying mathematics about algebra is the matrix. A matrix is a right
rectangular arrangement of numbers, which is then called an entry or element of the matrix. Many things can be calculated from a matrix, such as matrix multiplication, addition, determinant, inverse, trace matrix, and so on.
The matrix is a rectangular-shaped array arranged based on rows and columns placed between two parentheses (Fitriyani et al, 2018). There are various types of matrices, one of which is the circular matrix.
The circular matrix generally meets the same operating axiom as the square matrix, it's just that the circular matrix has special properties, namely the result of its commutative multiplication and well- defined closeness properties (Fahlevi, 2020). A circular matrix is a matrix with entries on each row identical to those in the previous row but moved one position to the right to circle it. The overall entry on the circular matrix is determined by the entry on the first line (Davis, 1979). There are several types of circular matrix, namely semi-circular matrix, reverse circular matrix (left circular), and k−circular matrix.
One of the important discussions in matrix theory is determining the determinants of the matrix.
Determinants have an important role in solving several problems in the matrix and are widely used in mathematics and applied sciences (Rahma et al, 2019). Matrix determinants have a long history, even longer than matrix science itself (Goldberg, 1991). The concept of determinants is very useful in the development of Mathematics as well as applications across sciences. In determining determinant values several methods have been used, namely the Sarrus method, cofactor expansion, line reduction, Gbemi's method, Jacobi method, and condensation method.
In linear algebra, it is studied a wide variety of matrices, one of the matrices whose shape is very unique is the circulant matrix. The circulant matrix is an n × n ordo matrix formed from n vectors and has only one input on the first row. Each entry from the previous row shifts one position to the right resulting in the next row and the entry along the diagonal of its matrix is the same. The circulant matrix is commonly used to solve polynomial equations.
The circulant matrix is often discussed in several scientific fields. In 2018, the circulant matrix was used to reconstruct or transform an image, which significantly reduced computational time compared to using common means of reconstructing images.
Furthermore, in the field of Signal processing (Compressed Sensing), the circulant matrix is used to
1038 efficiently reconstruct the signal. Furthermore,
DeVille, L. and Nijholt, E. discovered the eigenvalue formula for solving network map problems. In the field of algebraic mathematics, special techniques were found in determining the inverse group of the circulant matrix and some classes of circulant matrices. The circulant matrix was also once discussed by Aryani, F. et al who invented the closed formula for the trace matrix Toeplitz complex special form 3 × 3 ranked n positive integers. With the same discussion, found the formula of the trace matrix complex 2 × 2 specially shaped positive
integer rank [3]. Then Olii, I. et al discovered the trace matrix formula Toeplitz 2-tridiagonal 3 × 3 with the rank of a positive integer. Finally, a common form of trace matrix real 3 × 3 is found to be positive and negative integers.
The graph spectrum is the value of the Eigen and the value of the repeating eigence (multiplicity). This paper used a chain matrix in the form of a circular matrix pattern. The graphs to be sought for eigenvalues and their multiplicity are circle graphs 𝐶𝑛,, and cubic graphs 𝑄𝑛 that are part of regular graphs.
Pay attention to the following circle graph
Figure 1. Graph 𝐶3
with a matrix of constancy 𝐴(𝐶3) = [
0 1 1
1 0 1
1 1 0
]
The values of the graph eigen with r = 1,2,3 and n = 3 are obtained as follows:
𝑟 = 1, 𝜆1= ∑3𝑗=2𝑐𝑗𝜔(𝑗−1)1= 𝑐2𝜔1+ 𝑐3𝜔2= 1. 𝜔1+ 1. 𝜔2
𝜔1= 𝑒2𝜋𝑖⁄3= 𝑐𝑜𝑠2
3𝜋 + 𝑖𝑠𝑖𝑛2
3𝜋 = −1
2+
1 2√3𝑖
𝜔2= 𝑒4𝜋𝑖⁄3= 𝑐𝑜𝑠4
3𝜋 + 𝑖𝑠𝑖𝑛4
3𝜋 = −1
2−
1 2√3𝑖
𝜆1= 1 (−1
2+1
2√3𝑖) + 1 (−1
2−
1
2√3𝑖) = −1
For, 𝑟 = 2, 𝜆2= ∑3𝑗=2𝑐𝑗𝜔(𝑗−1)2= 𝑐2𝜔2+ 𝑐3𝜔4 = 1. 𝜔2+ 1. 𝜔4
To, 𝑟 = 2, 𝜆2= ∑3𝑗=2𝑐𝑗𝜔(𝑗−1)2= 𝑐2𝜔2+ 𝑐3𝜔4= 1. 𝜔2+ 1. 𝜔4
In the same way obtained: 𝜆2= −1.
for, 𝑟 = 3 by the same analogy obtained. 𝜆3= 2.
Based on the definition, the graph spectrum is obtained 𝐶3
1 2
1 ) 2
( C
3Sp
.Using the same method, is
𝑆𝑝(𝐶4), 𝑆𝑝(𝐶5), 𝑆𝑝(𝐶6), ⋯ , 𝑆𝑝(𝐶10) presented in table 3.1 below:
Table 1: Cn Circle Graph Spectrum
Circle Grap h
Graph Spectrum
𝐶3
1 2 1 ) 2
(C3 Sp
C4
1 2 1
2 0 ) 2
( C
4Sp
𝐶5
1 2 2
6 , 1 6 , 0 ) 2
( C
5Sp
𝐶6
1
2 2
1 2 1 1 ) 2 ( C
6Sp
𝐶7
2
8 , 1 2
4 , 0 2
2 , 1 1 ) 2 ( C
7Sp
𝐶8
1 2 2 2 1
2 2 0
2 ) 2
( C
8Sp
𝐶9
1 2 2 2 2
86 , 1 1 34 , 0 52 , 1 ) 2
( C
9Sp
𝐶10
1 2 2 2 2 1
2 6 , 1 6 , 0 6 , 0 6 , 1 ) 2 (C10 Sp
1039 In general, the graph's 𝐶𝑛 efficiency matrix is:
0 0
0 0 1
0 0
1 0 0
0 1
0 1 0
0 0
1 0 1
1 0
0 1 0
) (
C
nA
with as
0,1,0,0....,1
the first row of the chain matrix in the form of a circular matrix pattern. According to (Garrett 2007), suppose there is a matrix with the eigenvalues of the matrix are 𝐻𝑛𝑥𝑛 with and 1, 𝜔, 𝜔2, ⋯ , 𝜔𝑛−1 is e2i/n many nodes on the circle graph. Based on Theorem 2.1 obtained thevalue of Eigen, namely:
nj
r j j
r
s r n
1
) 1
(
, 0 , 1 ,...., 1
, that is
( 1)0 0 0
0 n 1 j j 2 n 2
j c c c
( 1)1 1 1 1 1 1
1 n1 j j 2 n n n n.
j c c c
e n e
n i n
i
1
2 cos 2
.
22 1
( 1)2 2 2( 1) 1 2( 1) 2 2 2
2 n1 j j 2 n n n n.
j c c c
n e n
e n
i n
i
2.2 2cos 4
cos 2 .
2 2 2 2
2 2
( 1)3 3 3( 1) 3 3( 1) 3 3 3
3 n1 j j 2 n n n n.
j c c c
n e n
e n
i n
i
6
cos 2 2
. cos 3 2 .
2 3 2 3
3 3
n1
n1
n n( 1).
(n 1)
n e n
e
n n n i n
i n
n
. 2cos 2( 1)
) 1 2 ( 2 1
) 1 ( 1
So that the value of the Eigen to-𝐶𝑛, 𝑛 ≥ 3 graph is
𝜆𝑟= 2𝑐𝑜𝑠2𝜋𝑟
𝑛 , for each. 𝑟 = 0,1, ⋯ , 𝑛 − 1.
𝑆𝑝(𝐶𝑛)
= [ 2 2𝑐𝑜𝑠 2𝜋
𝑛 ⋯ 2𝑐𝑜𝑠(𝑛 − 1)𝜋 𝑛 1 2 ⋯ 2
] (𝑛
≥ 3, 𝑛 odd) 𝑆𝑝(𝐶𝑛) = [2 2𝑐𝑜𝑠2𝜋
𝑛 ⋯ 2𝑐𝑜𝑠(𝑛−2)𝜋
𝑛 −2 1 2 ⋯ 2 1 ] (𝑛 ≥ 4, 𝑛 even)
Furthermore, the spectrum of cubic graph 3(𝑄𝑛) is determined in the same way as in the graph. 𝐶𝑛 Cubic graph spectrum, 𝑄𝑛, with 𝑛 = 4,6, ⋯ ,20 presented in table 2 below
Table 2 Cubic Graph Spectroctor Cubic
Graph
Graph Spectrum
𝑸𝟒
1 3
1 ) 3
( Q
4Sp
𝑄6
1 4 1
3 0 ) 3
( Q
6Sp
𝑸𝟖
1 2 2 1 2
4 , 2 1 4 , 0 1 ) 3
( Q
8Sp
𝑸𝟏𝟎
1 2 2 2 2 1
3 6 , 1 6 , 0 6 , 0 6 , 1 ) 3 (Q10 Sp
𝑄12
1 2 2 2 3 2
7 , 2 1 0 7 , 0 2 ) 3
( Q
12Sp
𝑄14
1 2 2 2 2 2 2 1
3 2 , 2 8 , 0 56 , 0 56 , 0 8 , 0 2 , 2 ) 3 (Q14 Sp 𝑸𝟏𝟔
1 2 2 2 2 2 1 2 2
8 , 2 76 , 1 1 4 , 0 24 , 0 8 , 0 1 4 , 2 ) 3 (Q16 Sp 𝑸𝟏𝟖
1 2 2 2 4 2 2 2 1
3 5 , 2 3 , 1 8 , 0 0 8 , 0 3 , 1 5 , 2 ) 3 (Q18 Sp 𝑸𝟐𝟎
1 2 2 2 2 2 2 3 2 2
9 , 2 16 , 2 1 6 , 0 16 , 0 4 , 0 9 , 0 6 , 1 6 , 2 ) 3 (Q20 Sp
Because the regular graf is 3 or the cubic graph patterned with a circular matrix is a graph in the form of a Mobius ladder graph has n vertex with a value of 2h or h n
2
1 so that the graph spectrum of the cubic graph is
1040 ( 1) 1
3 2 cos ( 1) 2 cos ( 1) 3
( )
1 2 2 1
h
n
h
Sp Q h h
, for ℎ odd and
1 2
2 1
1 ) 1 ) ( 1 cos( 2 ) 1 ( cos 2 ) 3 (
1
h
n h
h Q h
Sp
, for ℎ even
Conclusion
Based on the calculation of the value of the eigence, then by using the circular matrix obtained the following conclusions
1. The Circle Graph Spectrum (Cn) is
2 2
1
) 1 cos ( 2 2
cos 2 ) 2
(
n
n C n
Sp
n
(
,
3
n
for 𝑛 odd)
1 2
2 1
) 2 2 cos ( 2 2
cos 2 ) 2
(
n
n C n
Sp
n
(for 𝑛 even)
2. Cubic Graph Spectrum (𝑄𝑛) ( i.e.
( 1)
13 2cos ( 1) 2cos ( 1) 3 ( )
1 2 2 1
h n
h
Sp Q h h
, for ℎ odd and
1 2
2 1
1 ) 1 ) ( 1 cos ( 2 ) 1 ( cos 2 ) 3 (
1
hn
h
h Q h
Sp
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