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All Journal Pythagoras: Jurnal Matematika dan Pendidikan Matematika BIMASTER Epsilon: Jurnal Matematika Murni dan Terapan Edumatsains Jurnal Matematika Sains dan Teknologi Zero : Jurnal Sains, Matematika, dan Terapan Teorema: Teori dan Riset Matematika Jambura Journal of Mathematics Mathvision : Jurnal Matematika Variabel Jurnal Pengabdian kepada Masyarakat Nusantara Jurnal Abdimas Bina Bangsa
Fransiskus Fran
Jurusan Matematika, Fakultas MIPA, Universitas Tanjungpura, Jl. Prof. Dr. Hadari Nawawi, Kota Pontianak 78124, Kalimantan Barat
Agriculture, Biological Sciences & Forestry Biochemistry, Genetics & Molecular Biology Chemistry Computer Science & IT Decision Sciences, Operations Research & Management Economics, Econometrics & Finance Education Languange, Linguistic, Communication & Media Mathematics Physics Social Sciences Transportation Other

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BILANGAN KROMATIK EQUITABLE PADA GRAF BINTANG, GRAF LOLIPOP, DAN GRAF PERSAHABATAN Yuda Praja; Fransiskus Fran; Nilamsari Kusumastuti
EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN Vol 17, No 1 (2023)
Publisher : Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Lambung Mangkurat

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20527/epsilon.v17i1.8869

Abstract

Let G be a connected and undirected graph. Vertex coloring in a graph G is a mapping from the set of vertices in G to the set of colors such that every two adjacent vertices have different colors. There are many types of vertex coloring, such as complete coloring, k-differential coloring, and equitable coloring. Equitable coloring of G is a vertex coloring of G that satisfies the condition that for each induced color class it has an equitable cardinality with difference 0 or 1. The minimum number of colors used for such coloring of G is called the equitable chromatic number of G, denoted by χe(G). In this study, we only concern with graphs that have a central vertex, which means a vertex that is adjacent to every other vertex, in particular on the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). This research aims to formulate the equitable chromatic number of the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). The first step taken in this research is to apply vertex coloring to Sn, Ln, and fn. After that, the color classes of the vertex set are obtained and its cardinality is determined. Next, analyze that the applied vertex coloring meets the definition of equitable coloring. Then, prove that the number of colors used is minimum. Thus, the chromatic number for each graph is obtained and proved. Based on this research, the equitable chromatic number of Sn is ⌈n/2⌉ + 1, the equitable chromatic number of Ln is n, and the equitable chromatic number of fn is 3, for n = 1 and n + 1, for n ≥ 2.
PENDEKATAN PEMBELAJARAN BERBASIS PERMAINAN SEBAGAI UPAYA MENINGKATKAN KECERDASAN MATEMATIKA SISWA Mariatul Kiftiah; Nilamsari Kusumastuti; Bayu Prihandono; Yundari Yundari; Helmi Helmi; Evi Noviani; Fransiskus Fran; Yudhi Yudhi; Meliana Pasaribu; Nur’ainul Miftahul Huda
Jurnal Abdimas Bina Bangsa Vol. 5 No. 1 (2024): Jurnal Abdimas Bina Bangsa
Publisher : LPPM Universitas Bina Bangsa

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.46306/jabb.v5i1.979

Abstract

The perception of mathematics as a compulsory subject in school is often negative. To address this challenge, a service-learning initiative was implemented at SMAN 1 Sambas, employing a game-based pedagogy. Mathematics seminars and exhibitions were organized by Mathematics Study Program to enhance students' engagement and comprehension of mathematical concepts. The effectiveness of this approach was evaluated through a questionnaire, which revealed a high level of approval among students regarding the relevance, motivation, understanding of mathematical principles, problem-solving abilities, and playing skills in mathematics learning. This approach is expected to change students’ perception of mathematics and improve their learning outcomes
The Complexity of Octopus Graph, Friendship Graph, and Snail Graph Fransiskus Fran; Alexander; Yundari; Putri Romanda; Ervina Febyolga
EduMatSains : Jurnal Pendidikan, Matematika dan Sains Vol 9 No 1 (2024): July
Publisher : Fakultas Keguruan dan Ilmu Pendidikan, Universitas Kristen Indonesia

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.33541/edumatsains.v9i1.6042

Abstract

Graphs are basic structures that represent objects with nodes and relationships between objects with edges. Trees are one of the parts studied in graph theory along with finding the number of spanning trees of a graph such as octopus graph, friendship graph, and snail graph. The complexity of an octopus graph is strongly dependent on the number and length of tentacles, the complexity of a friendship graph is dependent on the number of triangle cycles, and the complexity of a snail graph is dependent on the number of edges and vertices located in the shell-like part of the snail. To calculate the number of spanning trees (τ(G)) of a graph, various calculations can be used, such as the extension of Kirchhoff's formula. The extension of Kirchhoff's formula uses the determinant of the adjacency matrix and degree matrix of the graph complement of a graph. Therefore, this research applies the extension of Kirchhoff's formula to obtain the complexity of octopus graph, friendship graph, and snail graph. From the analysis, it is obtained that for any n≥2, the number of spanning trees of octopus graph and friendship graph are τ(On )=1/5 √5 [((3+√5)/2)^n-((3-√5)/2)^n ] and τ(Fn )=3^n and the number of spanning trees of snail graph is τ(SIn )=2^(n+2)+3n∙2^(n-1) for n≥1.
MENENTUKAN INVERS DRAZIN DENGAN TEOREMA CAYLEY HAMILTON Nora Yoshinta Sigalingging; Fransiskus Fran; Nilamsari Kusumastuti
MathVisioN Vol 6 No 1 (2024): Maret 2024
Publisher : Prodi Matematika FMIPA Unirow Tuban

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.55719/mv.v6i1.998

Abstract

Setiap matriks tidak selalu memiliki invers. Matriks yang memiliki invers disebut matriks non-singular dan matriks yang tidak memiliki invers disebut matriks singular. Tetapi, matriks singular dapat ditentukan invers tergeneralisasinya. Invers tergeneralisasi adalah konsep aljabar linear yang digunakan dalam menentukan invers dari matriks singular. Salah satu invers tergeneralisasi yaitu invers Drazin dari suatu matriks singular  dilambangkan . Pada penelitian ini membahas cara menentukan invers Drazin yang merupakan salah satu invers tergeneralisasi dengan teorema Cayley Hamilton. Langkah-langkah untuk menentukan invers Drazin menggunakan teorema Cayley Hamilton, dimulai dengan mencari indeks suatu matriks singular . Indeks suatu matriks merupakan bilangan bulat non-negatif terkecil  yang memenuhi kondisi . Selanjutnya, dengan diperoleh indeks matriks dapat ditentukan matriks  dan  dengan menggunakan koefisien polinomial karakteristik dari matriks . Matriks  adalah matriks yang diperoleh dari  dan  adalah matriks yang diperoleh dari . Untuk menentukan invers Drazin dapat dihitung dengan .
Exploring the Metric Chromatic Number of Uniform, Centralized Uniform, and Cycle Uniform Theta Graphs Raventino Raventino; Fransiskus Fran
ZERO: Jurnal Sains, Matematika dan Terapan Vol 10, No 1 (2026): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v10i1.27103

Abstract

Metric coloring allows adjacent vertices of a graph to share the same color provided that their associated distance vectors are distinct, leading to the concept of the metric chromatic number. This notion is closely related to problems of vertex distinguishability and resource allocation in network-like structures. In this paper, we present the first exact determination of the metric chromatic number for three families of theta type graphs: uniform theta graphs, centralized uniform theta graphs, and a newly introduced class called the cycle uniform theta graph, obtained by cyclically arranging uniform theta subgraphs. The proposed construction enables an investigation of how cyclic configurations influence metric coloring behavior. Using a constructive metric coloring approach, exact values of the metric chromatic number are obtained. It is shown that the uniform theta graph  and the centralized uniform theta graph  both satisfy  for all positive integers  and . For the cycle uniform theta graph , the metric chromatic number equals  when  and  have the same parity or when  is odd and  is even. In contrast,  when  is even and  is odd. This latter case arises because the longest path in the cyclic structure has odd length, forcing the graph to have chromatic number three. Since the graph is connected and its chromatic number is at most three, this structural constraint directly implies that three colors are also necessary for a valid metric coloring.
Kompleksitas Graf Cocktail Party dan Graf Sandat Berdasarkan Spektrum Laplacian Fransiskus Fran; Suryani Suryani; Bayu Prihandono
PYTHAGORAS Jurnal Matematika dan Pendidikan Matematika Vol. 21 No. 1 (2026)
Publisher : Department of Mathematics Education, Faculty of Mathematics and Natural Sciences, UNY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.21831/pythagoras.v21i1.87662

Abstract

Misalkan adalah graf terhubung dan tidak berarah. Setiap graf terhubung memiliki pohon perentang, yaitu subgraf yang terdiri dari seluruh simpul dalam graf dan membentuk pohon. Kompleksitas graf , yang dinotasikan dengan , merupakan banyaknya pohon perentang pada graf . Dalam artikel ini, digunakan pendekatan spektrum Laplacian untuk menentukan kompleksitas graf, khususnya graf Cocktail Party dan graf Sandat. Spektrum Laplacian merupakan matriks yang dibentuk dari susunan nilai eigen matriks Laplacian dan multiplisitasnya. Penelitian diawali dengan menentukan matriks adjacency dan matriks degree dari graf Cocktail Party untuk dan graf Sandat untuk , kemudian dibentuk matriks Laplacian serta polinomial karakteristik matriks Laplaciannya. Selanjutnya, dirumuskan formula polinomial karakteristik ke- matriks Laplacian masing-masing graf. Pembuktian kebenaran formula polinomial karakteristik dilakukan dengan memanfaatkan matriks blok. Berdasarkan hasil tersebut, dirumuskan dan dibuktikan formula spektrum Laplacian, serta formula kompleksitas graf untuk graf Cocktail Party dan Sandat. Hasil dari penelitian ini adalah rumusan spektrum Laplacian dan kompleksitas dari graf Cocktail Party serta graf Sandat.
Route Optimization in Asymmetric Capacitated Vehicle Routing Problem (ACVRP) Model using Tabu Search Algorithm (Case Study: Car Oil Distribution of PT. Kencana Central Mobil) Thedorus Junjun; Mariatul Kiftiah; Fransiskus Fran
Jurnal Matematika Sains dan Teknologi Vol. 25 No. 2 (2024)
Publisher : LPPM Universitas Terbuka

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.33830/jmst.v25i2.6575.2024

Abstract

PT. Kencana Central Mobil is an automotive workshop that distributes oil to nine regular customers spread across Pontianak and its surroundings. The route from the depot to the customer and between customers has a different length, because when making a round trip using different roads. From this information, the author conducted observations in the field to measure the distance between customers with the help of the Avanza application. Thus, the problem in this company is included in the Asymmetric Capacitated Vehicle Routing Problem. The ACVRP model is a problem where the route from location  to  is not the same as the route from location  to . Based on the problems that have been explained, this study uses the Tabu Search algorithm to solve it. The Tabu Search algorithm works by moving from one route to another, so that when related to a problem the company can find a trip by choosing the shortest route. There are six steps in solving this problem, namely determining the initial route, finding alternative routes by swapping two node positions so that the routes that can be formed each iteration are  routes (nine is the number of customers), choosing the best route among alternative routes, determining the new best route, updating the tabu list and checking the stopping criteria. From the calculation results, there is a difference in the distance traveled from the initial route which is 63.16 km long, while when calculated using the Tabu Search Algorithm, it can be seen that the tabu search criteria stops at the 7th iteration with a route length of 50.26 km so that it differs by 12.9 km from the initial route. The length of the route is optimal because it has the shortest route length of all the literature that has been traced.
Inovasi Media Pembelajaran Realistik Interaktif bagi Siswa Sekolah Dasar di Pulau Lemukutan Nur'ainul Miftahul Huda; Evi Noviani; Bayu Prihandono; Nilamsari Kusumastuti; Yundari Yundari; Helmi Helmi; Yudhi Yudhi; Fransiskus Fran; Meliana Pasaribu; Onelia Rochmah; Asri Rahmawati
Jurnal Pengabdian kepada Masyarakat Nusantara Vol. 7 No. 2 (2026): Edisi Mei - Agustus
Publisher : Lembaga Dongan Dosen

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.55338/jpkmn.v7i2.8731

Abstract

Kesenjangan mutu pendidikan di wilayah kepulauan, seperti di Pulau Lemukutan, diperparah oleh keterbatasan sarana pembelajaran matematika yang kontekstual. Pengabdian ini bertujuan mengatasi rendahnya pemahaman konsep matematika melalui implementasi pendekatan Realistic Mathematics Education (RME) berbasis media interaktif. Metode pelaksanaan meliputi sosialisasi, pendampingan belajar bagi siswa kelas IV-VI di SDN 06 Pulau Lemukutan, serta evaluasi hasil belajar. Hasil kegiatan menunjukkan dampak positif yang konkret bagi mitra. Pada kelas V dan VI, terjadi peningkatan skor rata-rata yang signifikan sebesar 12,31 poin, di mana mayoritas siswa berhasil mencapai kategori nilai tinggi (≥80) dengan sebaran kemampuan yang lebih merata. Meskipun peningkatan di kelas IV cenderung kecil (2,86 poin) karena adanya disparitas kemampuan awal, penggunaan media realistik terbukti meningkatkan motivasi dan keterlibatan aktif siswa dalam memecahkan masalah sehari-hari. Simpulan dari pengabdian ini adalah media realistik interaktif efektif menjadi solusi pembelajaran di wilayah pesisir, namun memerlukan strategi pendampingan diferensiasi tambahan untuk jenjang kelas yang lebih rendah guna memastikan pemerataan capaian belajar.
Co-Authors Alexander Andriko Andriko Annisatul Khairiah Asri Rahmawati Ayu Fitriani Bambang Poniman Bayu Prihandono Brella Glysentia Vilgalita Dany Riansyah Putra Dorotea Rahmawati Dwi Oktaviana Ervina Febyolga Esi Sari Evi Noviani Evi Utami Febby Desy Lia Felicia Yulita Kartika Sari Fransiska Evalia Helmi Helmi Ilhamsyah Lidwina Evi Purwanti Lili Surai’ya Mariatul Kiftiah Meliana Pasaribu Neno Juli Triami Nicko Nicko Nilamsari Kusumastuti Nora Yoshinta Sigalingging Novia Kristefany Kabang Novita Indah Saputri Nurhamzah Nurhamzah NUR’AINUL MIFTAHUL HUDA Onelia Rochmah Putri Romanda Raventino Raventino Resti Julia Susanti Rizky Oktaviani Robiandi Robiandi Robiandi Suryani Suryani Thedorus Junjun Umi Salmah Yuda Praja Yudhi Yundari, Yundari Yundari, Yundari