PROSIDING
Peranan Matematika dan Statistika dalam
Menyikapi Perubahan Iklim
VOLUME 2/NO.1/2014
ISN : 2337-392X
SEMINAR NASIONAL MATEMATIKA,
STATISTIKA, PENDIDIKAN MATEMATIKA,
DAN KOMPUTASI
Jurusan Matematika
Fakultas Matematika dan Ilmu Pengetahuan Alam
Universitas Sebelas Maret Surakarta
Jl. Ir. Sutami 36 A Solo - Jawa Tengah
ii
Tim Prosiding
Editor
Purnami Widyaningsih, Respatiwulan, Sri Kuntari, Nughthoh Arfawi Kurdhi, Putranto Hadi Utomo, dan Bowo Winarno Tim Teknis
Hamdani Citra Pradana, Ibnu Paxibrata, Ahmad Dimyathi, Eka Ferawati, Meta Ilafiani, Dwi Ardian Syah, dan Yosef Ronaldo Lete B.
Layout & Cover
iii
Tim Reviewer
Drs. H. Tri Atmojo Kusmayadi, M.Sc., Ph.D. Dr. Sri Subanti, M.Si.
Dr. Dewi Retno Sari Saputro, MKom. Drs. Muslich, M.Si.
Dra. Mania Roswitha, M.Si.
Dra. Purnami Widyaningsih, M.App.Sc. Drs. Pangadi, M.Si.
Drs. Sutrima, M.Si. Drs. Sugiyanto, M.Si. Dra Etik Zukhronah, M.Si. Dra Respatiwulan, M.Si. Dra. Sri Sulistijowati H., M.Si. Irwan Susanto, DEA
Winita Wulandari, M.Si. Sri Kuntari, M.Si.
Titin Sri Martini, M.Kom. Ira Kurniawati, M.Pd.
iv
Steering Committee
Prof. Drs.Tri Atmojo Kusmayadi, M.Sc., Ph.D. Prof. Dr. Budi Murtiyasa, M.Kom.
Prof. Dr. Dedi Rosadi, M.Sc.
Prof. Dr. Ir. I Wayan Mangku, M.Sc. Prof. Dr. Budi Nurani, M.S.
Dr. Titin Siswantining, DEA Dr. Mardiyana, M.Si.
v
KATA PENGANTAR
Puji syukur dipanjatkan kepada Tuhan Yang Maha Esa sehingga prosiding seminar nasional Statistika, Pendidikan Matematika dan Komputasi ini dapat diselesaikan.
Prosiding ini bertujuan mendokumentasikan dan mengkomunika-sikan hasil presentasi paper pada seminar nasional dan terdiri atas 95 paper dari para pemakalah yang berasal dari 30 perguruan tinggi/politeknik dan institusi terkait. Paper tersebut telah dipresentasikan di seminar nasional pada tanggal 18 Oktober 2014. Paper didistribusikan dalam 7 kategori yang meliputi kategori Aljabar 14%, Analisis 9%, Kombinatorik 8%, Matematika Terapan 14%, Komputasi 7%, Statistika Terapan 27%, dan Pendidikan Matematika 19%.
Terima kasih disampaikan kepada pemakalah yang telah berpartisipasi pada desiminasi hasil kajian/penelitian yang dimuat pada prosiding ini. Terimakasih juga disampaikan kepada tim reviewer, tim prosiding, dan steering committee.
Semoga prosiding ini bermanfaat.
vi DAFTAR ISI
Halaman
Halaman Judul ………..……….. i
Tim Prosiding ………..…………. ii
Tim Reviewer ………..…………. iii
Steering Committee ………..…… iv
Kata Pengantar ………... v
Daftar Isi ………..………. vi
BIDANG ALJABAR 1 Bentuk-Bentuk Ideal pada Semiring (DDian Winda Setyawati ………..nxn(Z+), +, ) 1 2 Penentuan Lintasan Kapasitas Interval Maksimum dengan Pendekatan Aljabar Max-Min Interval M. Andy Rudhito dan D. Arif Budi Prasetyo ...………. 8
3 Karakterisasi Aljabar Pada Graf Bipartit Soleha, Dian W. Setyawati ………..………. 18
4 Semigrup Bentuk Bilinear Terurut Parsial Reguler Lengkap dalam Batasan Quasi-Ideal Fuzzy Karyati, Dhoriva Urwatul Wutsqa ………... 26
5 Syarat Perlu dan Cukup Ring Lokal Komutatif Agar Ring Matriksnya Bersih Kuat (-Regular Kuat) Anas Yoga Nugroho, Budi Surodjo ……….. 34
6 Sifat-sifat Modul Komultiplikasi Bertingkat Putri Widi Susanti, Indah Emilia Wijayanti ……….. 42
7 Ideal dari Ring Polinomial F2n[x] mod(xn-1) untuk Kontrol Kesalahan dalam Aplikasi Komputer Komar Baihaqi dan Iis Herisman ……….……… 49
9 Submodul Hampir Prima Dyana Patty, Sri Wahyuni ….………….……….………. 55
Subgrup Normal suatu Grup Perkalian dari Ring Pembagian yang Radikal atas Subring Pembagian Sejati Juli Loisiana Butarbutar dan Budi Surodjo ……….. 64
Sifat dan Karakterisasi Submodul Prima Lemah S(N) Rosi Widia Asiani, Sri Wahyuni ……….. 73
Modul Distributif dan Multiplikasi Lina Dwi Khusnawati, Indah Emilia Wijayanti ……… 83
vii
Penjadwalan Keberangkatan Kereta Api di Jawa Timur dengan Menggunakan Model Petrinet dan Aljabar Max-plus
Ahmad Afif, Subiono ……… 92
\
Minimalisasi Norm Daerah Hasil dari Himpunan Bayangan Matriks Aljabar Maks-Plus dengan Sebagian Elemen Ditentukan
Antin Utami Dewi, Siswanto, dan Respatiwulan ……… 107
Himpunan Bayangan Bilangan Bulat Matriks Dua Kolom dalam Aljabar Maks-Plus
Nafi Nur Khasana, Siswanto, dan Purnami Widyaningsih .……… 112
BIDANG ANALISIS Ruang 2-Norma Selisih
Sadjidon, Mahmud Yunus, dan Sunarsini …….……….. 120
Teorema Titik Tetap Pemetaan Kontraktif pada Ruang C[a,b]-Metrik (ℓp ,
dC[a,b])
Sunarsini, Sadjidon, Mahmud Yunus ………..……….. 124
Generalisasi Ruang Barisan Yang Dibangkitkan Oleh Fungsi Orlicz
Nur Khusnussa’adahdan Supaman ………..……….. 132
Gerakan Kurva Parameterisasi Pada Ruang Euclidean
Iis Herisman dan Komar Baihaqi …….……….. 141
Penggunaan Metode Transformasi Diferensial Fraksional dalam
Penyelesaian Masalah Sturm-Liouville Fraksional untuk Persamaan Bessel Fraksional
Marifatun, Sutrima, dan Isnandar Slamet……….……….. 148
Konsep Topologi Pada Ruang C[a,b]
Muslich ……….……….. 155
Kekompakan Terkait Koleksi Terindeks Kontinu dan Ruang Topologis Produk
Hadrian Andradi, Atok Zulijanto
……….……….. 162
A Problem On Measures In Infinite Dimensional Spaces
Herry Pribawanto Suryawan ..……….. 171
Masalah Syarat Batas Sturm-Liouville Singular Fraksional untuk Persamaan Bessel
Nisa Karunia, Sutrima, Sri Sulistijowati H ……… 179
BIDANG KOMBINATORIK
Pelabelan Selimut (a,d)-H-Anti Ajaib Super pada Graf Buku
viii
Digraf Eksentrik Dari Graf Hasil Korona Graf Path Dengan Graf Path
Putranto Hadi Utomo, Sri Kuntari, Tri Atmojo Kusmayadi ……… 193
Super (a, d)-H-Antimagic Covering On Union Of Stars Graph
Dwi Suraningsih, Mania Roswitha, Sri Kuntari ……… 198
Dimensi Metrik pada Graf Umbrella
Hamdani Citra Pradana dan Tri Atmojo Kusmayadi ……… 202
Dimensi Metrik pada Graf Closed Helm
Deddy Rahmadi dan Tri Atmojo Kusmayadi . ……….. 210
Pelabelan Selimut (a,b)-Cs+2-Anti Ajaib Super pada Graf Generalized
Jahangir
Anna Amandha, Mania Roswitha, dan Bowo Winarno ……… 215
Super (a,d)-H-Antimagic Total Labeling On Sun Graph
Marwah Wulan Mulia, Mania Roswitha, and Putranto Hadi Utomo …… 223
Maksimum dan Minimum Pelabelan pada Graf Flower
Tri Endah Puspitosari, Mania Roswitha, Sri Kuntari …………...……….. 231
BIDANG MATEMATIKA TERAPAN
2 Penghitungan Volume Konstruksi dengan Potongan Melintang
Mutia Lina Dewi ………... 238
4 Pola Pengubinan Parabolis
Theresia Veni Dwi Lestari dan Yuliana Pebri Heriawati ……….. 247
5 Analisis Kestabilan Model Mangsa Pemangsa Hutchinson dengan Waktu Tunda dan Pemanenan Konstan
Ali Kusnanto, Lilis Saodah, Jaharuddin……….. 257
6 Susceptible Infected Zombie Removed (SIZR) Model with Quarantine and Antivirus
Lilik Prasetiyo Pratama, Purnami Widyaningsih, and Sutanto ………. 264
7 Model Endemik Susceptible Exposed Infected Recovered Susceptible (SEIRS) pada Penyakit Influenza
Edwin Kristianto dan Purnami Widyaningsih ………... 272
9 Churn Phenomenon Pengguna Kartu Seluler dengan Model Predator-Prey
Rizza Muamar As-Shidiq, Sutanto, dan Purnami Widyaningsih ………… 279
Pemodelan Permainan Flow Colors dengan Integer Programming
Irfan Chahyadi, Amril Aman, dan Farida Hanum ……….. 283
Optimasi Dividen Perusahaan Asuransi dengan Besarnya Klaim Berdistribusi Eksponensial
ix
Permasalahan Kontrol Optimal Dalam Pemodelan Penyebaran Penyakit
Rubono Setiawan ……….. 300 Model Pengoptimuman Dispatching Bus pada Transportasi Perkotaan:
Studi Kasus pada Beberapa Koridor Trans Jakarta
Farida Hanum, Amril Aman, Toni Bakhtiar, Irfan Chahyadi ……….. 306
Model Pengendalian Epidemi dengan Vaksinasi dan Pengobatan
Toni Bachtiar dan Farida Hanum ……….. 315
How Realistic The Well-Known Lotka-Volterra Predator-Prey Equations Are
Sudi Mungkasi ……….. 323 Aplikasi Kekongruenan Modulo pada Algoritma Freund dalam
Penjadwalan Turnamen Round Robin
Esthi Putri Hapsari, Ira Kurniawati ……….. 334
BIDANG KOMPUTASI
Aplikasi Algoritma Enkripsi Citra Digital Berbasis Chaos Menggunakan Three Logistic Map
Suryadi MT, Dhian Widya ……… 344
Implementasi Jaringan Syaraf Tiruan Untuk Mengklasifikasi Kualitas Citra Ikan
Muhammad Jumnahdi ……….…. 352
Sistem Pengkonversi Dokumen eKTP/SIM Menjadi Suatu Tabel Nurul Hidayat, Ikhwan Muhammad Iqbal, dan Muhammad Mushonnif
Junaidi ………... 360
Kriptografi Kurva Eliptik Elgamal Untuk Proses Enkripsi-Dekripsi Citra Digital Berwarna
Daryono Budi Utomo, Dian Winda Setyawati dan Gestihayu Romadhoni F.R 373
Penerapan Assosiation Rule dengan Algoritma Apriori untuk Mengetahui Pola Hubungan Tingkat Pendidikan Orang Tua terhadap Indeks Prestasi Kumulatif Mahasiswa
Kuswari Hernawati ………... 384
Perancangan Sistem Pakar Fuzzy Untuk Pengenalan Dini Potensi Terserang Stroke
Alvida Mustika R., M Isa Irawan dan Harmuda Pandiangan……….. 394
Miniatur Sistem Portal Semiotomatis Berbasis Sidik Jari pada Area Perpakiran
x BIDANG STATISTIKA
1 Uji Van Der Waerden Sebagai Alternatif Analisis Ragam Satu Arah Tanti Nawangsari……….. 417
2
Analisis Faktor-Faktor Yang Mempengaruhi Keberhasilan Mahasiswa Politeknik (Studi Kasus Mahasiswa Polban)
Euis Sartika………...….. 425 3
Distribusi Prior Dirichlet yang Diperumum sebagai Prior Sekawan dalam Analisis Bayesian
Feri Handayani, Dewi Retno Sari Saputro …………...……… 439 5
Pemodelan Curah Hujan Dengan Metode Robust Kriging Di Kabupaten Sukoharjo
Citra Panindah Sari, Dewi Retno Sari S, dan Muslich ……… 444
6
Premi Tunggal Bersih Asuransi Jiwa Endowment Unit Link Dengan Metode Annual Ratchet
Ari Cahyani, Sri Subanti, Yuliana Susanti……….………. 453 7
Uji Siegel-Tukey untuk Pengujian Efektifitas Obat Depresan pada Dua Sampel Independen
David Pratama dan Getut Pramesti ………..……… 462 8
Aplikasi Almost Stochastic Dominance dalam Evaluasi Hasil Produksi Padi di Indonesia
Kurnia Hari Kusuma, Isnandar Slamet, dan Sri Kuntari ……….. 470
9
Pendeteksian Krisis Keuangan Di Indonesia Berdasarkan Indikator Nilai Tukar Riil
Dewi Retnosari, Sugiyanto, Tri Atmojo ……….... 475 Pendekatan Cross-Validation untuk Pendugaan Data Tidak Lengkap pada
Pemodelan AMMI Hasil Penelitian Kuantitatif
Gusti Ngurah Adhi Wibawa dan Agusrawati……… 483
11
Aplikasi Regresi Nonparametrik Menggunakan Estimator Triangle pada Data Meteo Vertical dan Ozon Vertikal, Tanggal 30 Januari 2013
Nanang Widodo, Tony Subiakto, Dian Yudha R, Lalu Husnan W ………. 493
12 Pemodelan Indeks Harga Saham Gabungan dan Penentuan Rank Correlation dengan Menggunakan Copula
Ika Syattwa Bramantya, Retno Budiarti, dan I Gusti Putu Purnaba …….. 502
13
Identifikasi Perubahan Iklim di Sentra Produksi Padi Jawa Timur dengan Pendekatan Extreme Value Theory
Sutikno dan Yustika Desi Wulan Sari……….. 513
15
Analisis Data Radiasi Surya dengan Pendekatan Regresi Nonparametrik Menggunakan Estimator Kernel Cosinus
xi
16
Pengujian Hipotesis pada Regresi Poisson Multivariate dengan Kovariansi Merupakan Fungsi dari Variabel Bebas
Triyanto, Purhadi, Bambang Widjanarko Otok, dan Santi Wulan Purnami 533
17
Perbandingan Metode Ordinary Least Squares (OLS), Seemingly
Unrelated Regression (SUR) dan Bayesian SUR pada Pemodelan PDRB Sektor Utama di Jawa Timur
Santosa, AB, Iriawan, N, Setiawan, Dohki, M ……… 544
19 Studi Model Antrian M/G/1: Pendekatan Baru
Isnandar Slamet ……… 557
20
Pengaruh Pertumbuhan Ekonomi dan Konsumsi Energi Terhadap Emisi CO2 di Indonesia: Pendekatan Model Vector Autoregressive (VAR)
Fitri Kartiasih ………...………. 567
21
Estimasi Parameter Model Epidemi Susceptible Infected Susceptible (SIS) dengan Proses Kelahiran dan Kematian
Pratiwi Rahayu Ningtyas, Respatiwulan, dan Siswanto .………….……… 578
22
Pendeteksian Krisis Keuangan di Indonesia Berdasarkan Indikator Harga Saham
Tri Marlina, Sugiyanto, dan Santosa Budi Wiyono ………. 584
23
Pemilihan Model Terbaik untuk Meramalkan Kejadian Banjir di Kecamatan Rancaekek, Kabupaten Bandung
Gumgum Darmawan, Restu Arisanti, Triyani Hendrawati, Ade Supriatna 592
24
Model Markov Switching Autoregressive (MSAR) dan Aplikasinya pada Nilai Tukar Rupiah terhadap Yen
Desy Kurniasari, Sugiyanto, dan Sutanto ………. 602
25
Pendeteksian Krisis Keuangan di Indonesia Berdasarkan Indikator Pertumbuhan Kredit Domestik
Pitaningsih, Sugiyanto, dan Purnami Widyaningsih ……… 608
26 Pemilihan Model Terbaik untuk Meramalkan Kejadian Banjir di Bandung dan Sekitarnya
Gumgum Darmawan, Triyani Hendrawati, Restu Arisanti ……… 615
27 Model Probit Spasial
Yuanita Kusuma Wardani, Dewi Retno Sari Saputro ……… 623
28
Peramalan Jumlah Pengunjung Pariwisata di Kabupaten Boyolali dengan Perbandingan Metode Terbaik
Indiawati Ayik Imaya, Sri Subanti ………..……... 628
29
Pemodelan Banyaknya Penderita Demam Berdarah Dengue (DBD) dengan Regresi Kriging di Kabupaten Sukoharjo
xii
Ekspektasi Durasi Model Epidemi Susceptible Infected (SI)
Sri Kuntari, Respatiwulan, Intan Permatasari………. 646
BIDANG PENDIDIKAN
3
Konsep Pembelajaran Integratif dengan Matematika Sebagai Bahasa Komunikasi dalam Menyongsong Kurikulum 2013
Surya Rosa Putra, Darmaji, Soleha, Suhud Wahyudi, ………. 653
4
Penerapan Pendidikan Lingkungan Hidup Berbasis Pendidikan Karakter dalam Pembelajaran Matematika
Urip Tisngati ………. 664
5
Studi Respon Siswa dalam Menyelesaikan Masalah Matematika
Berdasarkan Taksonomi SOLO (Structure of Observed Learned Outcome) Herlin Widia, Urip Tisngati, Hari Purnomo Susanto ……….. 677
7 Desain Model Discovery Learning pada Mata Kuliah Persamaan Diferensial
Rita Pramujiyanti Khotimah, Masduki ……….. 684
8 Efektivitas Pembelajaran Berbasis Media Tutorial Interaktif Materi Geometri
Joko Purnomo, Agung Handayanto, Rina Dwi Setyawati ……… 693
10
Pengembangan Modul Pembelajaran Matematika Menggunakan
Pendekatan Problem Based Learning (PBL) Pada Materi Peluang Kelas VII SMP
Putri Nurika Anggraini, Imam Sujadi, Yemi Kuswardi ……… 703
11 Pengembangan Bahan Ajar Dalam Pembelajaran Geometri Analitik Untuk Meningkatkan Kemandirian Mahasiswa
Sugiyono\, Himmawati Puji Lestari ………..……… 711
13
Pengembangan Strategi Pembelajaran Info Search Berbasis PMR untuk Meningkatkan Pemahaman Mata Kuliah Statistika Dasar 2
Joko Sungkono, Yuliana, M. Wahid Syaifuddin ……… 724
14
Analisis Miskonsepsi Mahasiswa Program Studi Pendidikan Matematika Pada Mata Kuliah Kalkulus I
Sintha Sih Dewanti ………..………..……. 731 Kemampuan Berpikir Logis Mahasiswa yang Bergaya Kognitif Reflektif
vs Impulsif
Warli ………. 742 Model Pembelajaran Berbasis Mobile
xiii
Profil Gaya Belajar Myers-Briggs Tipe Sensing-Intuition dan Strateginya Dalam Pemecahan Masalah Matematika
Rini Dwi Astuti, Urip Tisngati, Hari Purnomo Susanto ……….. 760 Penggunaan Permainan Matematika Berbasis Lingkungan Hidup untuk
Menningkatkan Minat dan Keterampilan Matematis Peserta Didik
Rita Yuliastuti ……….. 772
Tingkat Pemahaman Peserta PLPG Matematika Rayon 138 Yogyakarta Tahun 2014 Terhadap Pendekatan Saintifik Pada Kurikulum 2013 Berdasarkan Kuesioner Awal dan Akhir Pelatihan
Beni Utomo, V. Fitri Rianasari dan M. Andy Rudhito ……….. 784
Pengembangan Perangkat Pembelajaran Matematika Melalui Pendekatan RME dengan CD Interaktif Berbasis Pendidikan Karakter Materi Soal Cerita Kelas III
Sri Surtini, Ismartoyo, dan Sri Kadarwati ………. 791
E-Learning Readiness Score Sebagai Pedoman Implementasi E-Learning
Nur Hadi Waryanto ……….. 805 Pengembangan Lembar Kerja Siswa (LKS) Matematika Realistik di SMP
Berbasis Online Interaktif
Riawan Yudi Purwoko, Endro Purnomo ……….. 817 IbM APE Matematika Bagi TK Pinggiran Di Kota Malang
Marwah Wulan Mulia, Mania Roswitha, and Putranto Hadi Utomo
Department of Mathematics
Faculty of Mathematics and Natural Sciences Sebelas Maret University
Abstract. A simple graph G = (V (G), E(G)) admits an H-covering if every edge in
E(G) belongs to a subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total
la-beling of G is a bijective function ξ : V (G)∪E(G) → {1, 2, ..., |V (G)+E(G)|} such that for all subgraph H′ isomorphic to H, the weight w(H′) = Σv∈V (H′)ξ(v) + Σe∈E(H′)ξ(e)
con-stitute an arithmetic progression a, a + d, a + 2d, ..., a + (t−1)d where a and d are positive integers and t is the number of subgraph of G isomorphic to H. Additionally, the labeling
ξ is called a super (a, d)-H-antimagic total labeling, if ξ(V (G)) ={1, 2, ..., |V (G)|}.
This research has found (a, d)-H-antimagic total labeling of wheel graph Wn, sun
graph Sn and Cn⊙ Km graph.
Keywords: (a, d)-H-antimagic total labeling, wheel graph, sun graph, Cn⊙ Km graph
1. Introduction
Graph labeling is an interested topic in graph theory. According to Wallis [13], a labeling of a graph is a map that carries graph elements (vertex, edge or vertex and edge) to numbers (usually to the positive or non-negative integers). The most common choices of domain are the set of all vertices and edges (total labeling), the vertex-set alone (vertex labeling), or the edge-set alone (edge labeling).
One type of graph labeling is a magic labeling which was formulated by Kotzig and Rosa [7]. The most simple magic labeling is edge magic total labeling, which is a one-to-one map λ from V (G)∪ E(G) → 1, 2, ..., |V (G)| + |E(G)|, with the property that, given any edge xy, λ(x) + λ(xy) + λ(y) = k for some constant k. In 2005, Guti´errez and Llad´o [1] defined H-magic labeling which is a generalization of Kotzig and Rosa’s edge-magic total labeling [7]. According to Llad´o and Moragas [8], let G = (V, E) be finite simple graph. An edge covering on Graph G is the family of different subgraphs H1, H2, ..., Hk such that for any kind of edge of E belongs to at
least one of subgraphs Hi for 1≤ i ≤ k. If every Hi is isomorphic to a given graph
H, the we say that G admits an H-covering.
According to Inayah et al. [4], an (a, d)-H-antimagic total labeling of G is bijective function ξ : V (G) ∪ E(G) → {1, 2, ..., |V (G) + E(G)|} such that for all subgraphs H′ isomorphic to H, the weight w(H′) = Σv∈V (H′)ξ(v) + Σe∈E(H′)ξ(e) constitutes an arithmetic progression a, a + d, a + 2d, ..., a + (t− 1)d where a and d are positive integers and t is the number of subgraphs of G isomorphic to H. Such a labeling is called super if the smallest possible labels appear on the vertices and G is super (a, d)-H-antimagic. Inayah [3] developed (a, d)-Cn-antimagic total labeling of
wheel graph. Furthermore, Karyanti [6] provided (a, d)-K1,3-antimagic total labeling of sun graph Snwhere n is odd. This research aims to determine (a, d)-Fn-antimagic
total labeling of wheel graph, (a, d)-K1,3-antimagic total labeling of sun graph where
n is even and (a, d)-G + e-antimagic total labeling of Cn⊙ Km graph. In section
2 and 3 we introduced a technique of partitioning a multiset that will be used to prove our new results.
2. k-balanced (k, δ)-Anti Balanced Multiset
Maryati et al. [10] defined k-balanced multiset as follows. Multiset is a set that allows the existence of the same elements in it. Let k ∈ N and Y be a multiset that contains positive integers. Y is said to be k-balanced if there exist k subsets of Y , say Y1, Y2, ..., Yk, such that for every i ∈ [1, k], |Yi| = |Y |k ,
∑ Yi = ∑ Y k ∈ N, and ⊎k
i=1Yi = Y . If this is the case for every i ∈ [1, k], then Yi is called a balanced
subset of Y . Roswitha et al. [11] have proved a lemma that refers to Maryati [9] and Maryati et al. [10]. Lemma 2.1 has proved by Roswitha et al. [11].
Lemma 2.1. Let x and y be non-negative integers. Let X = [x + 1, x(y + 1)] with |X| = xy and Y = [x(y + 2), 2x(y + 1) − 1] where |Y | = xy. Then, the multiset
K = X ⊎ Y is xy-balanced with all its subsets are 2-sets.
Then, we have the following lemma.
Lemma 2.2. Let x,y be positive integers. If X = [1, x(y + 1)] and Y = [x(y + 1) + 1, 2x(y + 1)] with |X| = |Y | = xy. Then, the multiset K = X ⊎ Y is 2xy-balanced with all its subsets are 2-sets.
Proof. For every i ∈ [1, 2xy], we define Ki ={i, 2x(y + 1) + 1 − i}. Then,
∑ Ki =
2x(y + 1) + 1 for every i ∈ [1, 2xy] and K is 2xy-balanced with all its subsets are
2-sets.
Lemma 2.3. Let x and y be positive integers. If X = {1, 3, ..., 2x(y + 1) − 1} and
Y ={2, 4, 6, ..., 2x(y + 1)} with |X| = |Y | = 2xy. Then, the multiset K = X ⊎ Y is
2xy-balanced with all its subsets are 2-sets.
Proof. For every i∈ [1, 2xy], we define Ki ={2i − 1, 2x(y + 1) + 1 − (2i − 1)}. Then,
∑
Ki = 2x(y + 1) + 1 for every i∈ [1, 2xy] and K is 2xy-balanced with all its subsets
are 2-sets.
Let k ∈ N and let X be a multiset containing positive integers. Then X is said to be (k, δ)-anti balanced if there exist k subsets of X, say X1, X2, ..., Xk, such that
for every i ∈ [1, k], |Xi| = |X|k , ⊎ki=1Xi = X, and for i∈ [1, k − 1],
∑
Xi+1− Xi = δ
is satisfied [see Inayah et al. [5]].
Lemma 2.4. Let x and y be positive integers. Then, the set A = [x + 1, x(y + 1)] with |A| = xy is (xy, 1)-anti balanced.
Proof. For i ∈ [1, xy], we define set Ai = {x + i}. Then,
∑
Ai = x + i for every
i ∈ [1, xy] and ∑Ai+1−
∑
Ai = 1 for every i ∈ [1, xy − 1] and A is (xy, 1)-anti
balanced.
Lemma 2.5. Let x and y be positive integers with x ≥ 3. Then, the set A =
[1, x(y + 1)]\ {2i − 1} for i = [1, xy]] with |A| = xy is (xy, 2)-anti balanced. Proof. For i∈ [1, xy], we define set Ai ={2i}. Then,
∑
Ai = 2i for every i∈ [1, xy]
and ∑Ai+1−
∑
Ai = 2 for every i∈ [1, xy − 1] and A is (xy, 2)-anti balanced.
Lemma 2.6. Let x≥ 3 be positive integers. Then, the multiset X = [2x + 2, 3x + 1] where |X| = x − 1 is (x, 1)-anti balanced.
Proof. For j = [1, x] define Aj = {2x + 1 + j} and for every i = [1, x] define
Xi = Aj \ {3x + 1 − i}. Then, ∑ Xi = 12(5x2 − 3x) − 2 + i for i ∈ [1, x] and ∑ Xi+1− ∑
Xi = 1 for i = [1, x− 1] and X is (x, 1)-anti balanced.
Lemma 2.7. Let x ≥ 3 be positive integers. Then, the multiset X = [x + 2i, 3x + 1] \ {x + 2i} where |X| = x − 1 is (x, 2)-anti balanced.
Proof. For j = [1, x] define Aj = {x + 1 + 2j} and for every i = [1, x] define
Xi = Aj \ {3x + 1 − 2i}. Then, ∑ Xi = 2x2 − x − 3 + 2i for i ∈ [1, x] and ∑ Xi+1− ∑
Xi = 2 for i = [1, x− 1] and X is (x, 2)-anti balanced.
Lemma 2.8. Let x ≥ 3 be positive integers. Then, the multiset X = [4, 7, ..., 3x+1] where |X| = x − 1 is (x, 3)-anti balanced.
Proof. For j = [1, x] define Aj = {3x + 4 − 3j} and for every i = [1, x] define
Xi = Aj \ {3x + 1 − 2i}. Then, ∑ Xi = 12(3x2 − x) − 4 + 3i for i ∈ [1, x] and ∑ Xi+1− ∑
Xi = 3 for i = [1, x− 1] and X is (x, 3)-anti balanced.
Lemma 2.9. Let x, y be positive integers with x ≥ 3 and y = {1, 2}. Let X =
{1, 3, ..., 2xy − 1} and Y = [x(y + 2) + 1, 2x(y + 1)] with |X| = |Y | = xy. Then, multiset K = X ⊎ Y is (xy, δ) anti balanced with all its subsets are 2-sets.
Proof. (i) For every i ∈ [1, xy], we define Ki = {2i − 1, 2x(y + 1) + 1 − i}. Then
∑
Ki = 2x(y + 1) + i for every i = [1, xy] and
∑
Ki+1−
∑
Ki = 1 for i = [1, xy− 1]
and K is (xy, 1)-anti balanced with all its are 2-sets.
(ii) For every i ∈ [1, xy], we define Ki = {2i − 1, x(y + 2) + i}. Then
∑
Ki =
x(y + 2)− 1 + 3i for every i = [1, xy] and ∑Ki+1−
∑
Ki = 3 for i = [1, xy− 1]
andK is (xy, 3)-anti balanced with all its are 2-sets.
Lemma 2.10. Let x, y be positive integers with x ≥ 3. If X = [1, xy] and Y = [x(y + 2) + 1, 2n(m + 1)] then K = X⊎ Y is (xy, 2)-anti balanced with all its subsets are 2 sets.
Proof. Let x, y be positive integers with x ≥ 3. For i ∈ [1, xy], we define Ki =
{i, x(y +2)+i} and |Ki| = 2. Then
∑xy
i=1Ki = x(y + 2) + 2i and
∑
Ki+1−
∑
Ki = 2
for every i ∈ [1, xy − 1]. Then K is (xy, 2)-anti balanced with all its subsets are 2
sets.
Lemma 2.11. Let x, y be positive integers with x≥ 3. If X = [1, x(y + 3)] then X is (xy, 4)-anti balanced with all its subsets are 2 sets.
Proof. Let x, y be positive integers with x ≥ 3. For i ∈ [1, xy], we define Xi =
{2i−1, 2i} and |Xi| = 2. Then
∑xy i=1Xi = 4i−1 and ∑ Xi+ 1− ∑ Xi = 4 for every
i∈ [1, xy − 1]. Then X is (xy, 4)-anti balanced with all its subsets are 2 sets.
3. (a, d)-K1,3-Anti Magic Total Labeling of Sun Graph
Sun graph is Cn⊙ K1 graph. It obtained by taking one copy of Cn and n copy
of K1 and joining the ith vertex of Cn with an edge to every vertex in the ith copy
of Cn [see corona definition by Harary [2]]. Let {u1, u2, ..., un} be vertices on cycle
graph Cn. Sun graph Sn has a set vertices V (G) = {u1, u2, ..., un, v1, v2, ..., vn}
and edges E(G) = {(u1, v1), (u2, v2), ..., (un, vn), (u1, u2), ...(un−1, un), (un, u1)} and
|V (G)| + |E(G)| = 2n and H be a complete bipartite graph K1,3 called also star
graph S4.
Lemma 3.1. If G is (a, d)-H-antimagic then d ≤ ⌊32(nn−2−1)⌋.
Proof. Let G ∼= Sn and H ∼= K1,3. Let |V (G)| = vG, |E(G)| = eG,|V (H)| = vH and
|E(H)| = eH. The maximum possible H-weight is vG+(vG−1)+(vG−2)+...+(vG−
vH+ 1) + (vG+ eG) + (vG+ eG−1)+...+(vG+ eG−eH+ 1) or a + (n−1)d ≤ 4(8n−7).
On the other hand, the least possible H-weight is 1 + 2 + ... + (vG+ eH) or a≥ 36.
Then we have, d≤ 32(nn−2−1). Since d is positive integer, so d≤ ⌊32(nn−2−1)⌋.
Theorem 3.2. Let G be a sun graph Sn where n ≥ 3. Then, there is a super
(a, d)-K1,3-anti magic total labeling for d ={1, 2}.
Proof. Let G be a sun graph Sn where n ≥ 3. Let R = [1, 4n].Partition R into 4
sets, namely R = X∪ A ∪ B ∪ Y where X = [1, n], A = [n + 1, 2n], B = [2n + 1, 3n] and Y = [3n + 1, 4n]. Now define a total labeling ξ on G as follows. Label edges {(u1, u2), ..., (ui, ui+1), (un, u1)} by elements of B and edges (ui, vi) by elements of
Y from right to left (counter-clockwise). Then, label vertices vi by elements of X
start from v2 and label vertices ui by elements of B start from u2 from left to right (clockwise). Define K = X ⊎ Y , by Lemma 2.2 where x = n and y = 1, K is (2n)-balanced and∑Ki = 4n + 1. Apply Lemma 2.4, where x = n and y = 1, then
A is (n, 1)-anti balanced and ∑Ai = n + i. Let H = K1,3, for i = [1, n], ω(K1,3i ) = 3∑ni=1Ki+
∑n
i=1A = 13n + 3 + i. Then we have, d = ω(K
i+1
1,3 )− ω(K1,3i ) = 1 and
a = ω(K1
1,3) = 13n + 4. Sun graph Sn admits a super (13n + 4, 1)-K1,3-anti magic total labeling. For d = 2 is similar to proof of d = 1. So we obtain, sun graph Sn
admits a super (12n + 5, 2)-K1,3-anti magic total labeling. 4. (a, d)-Fn-Anti Magic Total Labeling of Wheel Graph Wn
A wheel graph Wn is a graph that obtained by joining n vertices of Cn with one
vertex in central (Wallis et al [12]). Let v1, v2, ..., vnbe vertices of Cnand c be central
vertex of wheel graph. A wheel graph has a set of vertices V (G) ={c, v1, v2, ..., vn}
and edges E(G) = {(c, v1), ..., (c, vn), (v1, v2), ..., (vn−1, vn), (vn, v1)} and |V (G)| =
n + 1 and |E(G)| = 2n and H be a fan graph Fn.
Lemma 4.1. If G is (a, d)-H-antimagic then d ≤ ⌊ 3n
n−1⌋.
Proof. Let G ∼= Wn and H ∼= Fn. Let |V (G)| = vG, |E(G)| = eG, |V (H)| = vH
and |E(H)| = eH. The maximum possible H-weight is vG + (vG − 1) + (vG −
2) + ... + (vG− vH + 1) + (vG+ eG) + (vG + eG− 1) + ... + (vG+ eG− eH + 1) or
a + (n− 1)d ≤ 12(3n)(3n + 3). On the other hand, the least possible H-weight is
1 + 2 + ... + (vG + eH) or a ≥ 12(3n)(3n + 1). Then we have, d ≤ n3n−1. Since d is
positive integer, so d≤ ⌊n3n−1⌋.
Theorem 4.2. Let G be a wheel graph Wn. Then, there is (32(3n2 + n), d)-Fn-anti
magic total labeling for d = [1, 3].
Proof. Let G be a wheel graph Wn where n≥ 3. Let C = 1 and R = [2, 3n + 1].
(i) Partition R into 3 sets, namely R = A∪ B ∪ X, where A = [2, n + 1], B = [n + 2, 2n + 1] and X = [2n + 2, 3n + 1]. Then, for i = [1, x] by Lemma 2.6 where x = n we have|Xi| = n−1, X is (n, 1)-anti balanced and
∑
Xi = 12(5n2−3n)−2+i.
Next, we define a total labeling ξ on graph G as follows. Label the vertex center of wheel with C ={1} and vertices vi with elements of A. Then label the edges (c, vi)
with elements of B. Label edges {(v1, v2), ..., (vi, vi+1), (vn, v1)} with elements of X
in counter-clockwise. For i = [1, n], ω(Fni) = 32(3n2 + n)− 1 + i. Then we have, d = ω(Fni+1)− ω(Fni) = 1 and a = ω(Fn1) = 32(3n2+ n). Wheel graph Wn admits a
super (32(3n2+ n), 1)-F
n-antimagic total labeling.
(ii) Partition R into 2 sets, namely R = A∪ X, where A = [2, n + 1] and X = [n + 2, 3n + 1]. Then, for i = [1, x] by Lemma 2.7 where x = n we have|Xi| = n − 1,
X is (n, 2)-anti balanced and ∑Xi = 12(5n2− 3n) − 2 + 2i. Next, we define a total
labeling ξ on graph G as follows. Label the vertex center of wheel with C ={1} and vertices vi with elements of A. For n is odd, label the edges (c, vi) with odd numbers
of elements of X and edges (vi, vi+1) with even numbers of elements of X in
counter-clockwise. For n is even, label the edges (c, vi) with even numbers of elements of X
and edges {(v1, v2), ..., (vi, vi+1), (vn, v1)} with odd numbers of elements of X. For
i = [1, n], ω(Fi
n) =
3
2(3n
2 + n)− 2 + 2i. Then we have, d = ω(Fi+1
n )− ω(Fni) = 2 and a = ω(F1 n) = 3 2(3n 2 + n). Wheel graph W n admits a super (32(3n2 + n), 2)-Fn
-antimagic total labeling.
(iii) Partition R into 3 sets, namely R = A∪ B ∪ X, where A = {2, 5, 3n − 1}, B ={3, 6, ..., 3n}and X = [4, 7, ..., 3n + 1]. Then, for i = [1, x] by Lemma 2.8 where x = n we have|Xi| = n−1, X is (n, 2)-anti balanced and
∑
Xi = 12(3n2−n)−4+3i.
Next, we define a total labeling ξ on graph G as follows. Label the vertex center of wheel with C = {1} and vertices vi with elements of A. Label the edges (c, vi)
with elements of B and edges {(v1, v2), ..., (vi, vi+1), (vn, v1)} with even numbers of elements of X in counter-clockwise. For i = [1, n], ω(Fi
n) =
3
2(3n
2 + n)− 3 + 3i.
Then we have, d = ω(Fi+1
n )− ω(Fni) = 3 and a = ω(Fn1) =
3
2(3n
2+ n). Wheel graph
Wn admits a super (32(3n2+ n), 3)-Fn-antimagic total labeling.
5. (a, d)-G + e-Anti Magic Total Labeling of Cn⊙ Km Graph
Let G′ ∼= Cn ⊙ Km graph where n ≥ 3 and m ∈ N. Cn ⊙ Km graph
ob-tained by taking one copy of Cn and n copy of Km and joining the ith vertex
of Cn with an edge to every vertex in the ith copy of Cn [see corona definition
by Harary [2]]. Let ui where i ∈ [1, n] be vertices on Cn and Cn ⊙ Km graph
has a set of vertices V (G′) = {ui, vj} where j ∈ [1, mn] and edges E(G′) =
{(u1, u2), ..., (un−1, un), (un, u1), (u1, v1), ..., (u1, vm), (u2, vm+1), ..., (u2, v2m), (un, vnm)}
and |V (G′)| = |E(G′)| = n(m + 1) and H be a G + e graph. Lemma 5.1. If G′ is (a, d)-H-antimagic then d ≤ 4(n + 1).
Proof. Let G′ ∼= Cn ⊙ Km and H ∼= G + e. Let |V (G′)| = vG′, |E(G′)| = eG′,
|V (H)| = vH and |E(H)| = eH. The maximum possible H-weight is vG′ + (vG′ −
1)+(vG′−2)+...+(vG′−vH+1)+(vG′+eG′)+(vG′+eG′−1)+...+(vG′+eG′−eH+1)
or a + (nm− 1)d ≤ (n + 1)(4nm + 2n − 1). On the other hand, the least possible
H-weight is 1+2+...+(vG′+eH) or a ≥ (n+1)(2n+3). So we have, d ≤ 4(n+1).
Theorem 5.2. Let n and m be positive integers with n≥ 3, there is (a, d)-G+e-anti magic total labeling on graph Cn⊙ Km.
Proof. Let G′ be Cn⊙ Km graph where n≥ 3 and m ∈ N. Let Z = [1, 2n(m + 1)].
Partition Z into 4 sets, namely Z = A ∪ B ∪ C ∪ D where A = [1, nm], B = [nm+1, n(m+1)], C = [n(m+1)+1, n(m+2)] and D = [n(m+2)+1, 2n(m+1)]. We define K = A∪D. Apply Lemma 2.10 where x = n and m = y,∑Xi = n(m+1)+2i,
so X is (mn, 2)-anti balanced. Next, we define a total labeling ξ on graph G as follows. Label the vertices vj with elements of A from right to left and edges (ui, vj)
with elements of D. Then, label the edges (ui, vj) with elements of C and edges
(ui, ui+1) with elements of C in any manner. Let H is G + e, for i ∈ [1, nm],
ω(G+ei) = 2n2m+nm+2n2+3n+2i. Then we have, d = ω(G+ei+1)−ω(G+ei) = 2
and a = ω(G + e1) = 2n2m + nm + 2n2+ 3n + 2 so Cn⊙ Km graph admits a super
(2n2m + nm + 2n2+ 3n + 2, 2)-G + e-anti magic total labeling. For d = 4 is similar to proof of d = 2. So we obtain, Cn⊙ Km graph admits a (2n2m + 8n2+ n + 3,
4)-G + e-anti magic total labeling.
Theorem 5.3. Let n and m be positive integers with n ≥ 3 and m = {1, 2}, there is (a, d)-G + e-anti magic total labeling on graph Cn⊙ Km for d ={1, 3}.
Proof. Let G′ be CnKm graph where n ≥ 3 and m ∈ N. Let Z = [1, 2n(m + 1)].
Partition Z into 4 sets, namely Z = A∪ B ∪ C ∪ D where A = {1, 3, ..., 2nm − 1}, B = [n(m + 2) + 1, 2n(m + 1)], C = {2, 4, ..., 2n}. Partition D into 2 sets, for m = 1,
d1 = {2n + j} and for m = 2, d2 ={2n + 2j} where j = [1, n]. Apply Lemma 2.9
where x = n and m = y,∑Ki = 2n(m + 1) + i, so X is (mn, 4)-anti balanced. Next,
we define a total labeling ξ on graph G as follows. Label the vertices vj with odd
numbers of elements of A from left to right and edges (ui, vj) with even numbers of
elements of D from left to right. Then, label the vertices (ui) with elements of B
and edges {(u1, u2, ..., (ui, ui+1), (un, u1)} with elements of C in any manner. Let H
is G + e, for i ∈ [1, nm], ω(G + ei) = 3n2 + 12n2m + 52 + 3n2 + i. Then we have,
d = ω(G + ei+1)− ω(G + ei) = 1 and a = ω(G + e1) = 3n2+ 12n2m + 52 + 3n + 1, so Cn⊙ Km graph admits an super (3n2+ 12n2m + 52 + 3n + 1, 1)-G + e-anti magic
total labeling. For d = 3 is similar to proof of d = 1. So we obtain, Cn⊙ Km graph
admits a (3n2+1 2n
2m + 3
2nm + 3n + 2, 3)-G + e-anti magic total labeling
6. Conclusion
This research has conclusions as follows. There are super (a, d)-K1,3-anti magic total labeling on sun graph Sn for d ={1, 2}, (a, d)-Fn-anti magic total labeling on
wheel graph Wn for d = [1, 3] and (a, d)-G + e-anti magic total labeling on Cn⊙ Km
graph for d = [1, 4]. Moreover, this research can be developed for other covers or d values for all graphs.
References
1. Gutierrez, A and A.Llado, Magic Coverings, J. Combin. Math. Combin. Comput 55 (2005), 43–46.
2. Harary, J. A., Graph Theory, Addison-Wesley Publishing Company Inc., Canada, 1969. 3. Inayah, N., Pelabelan (a, d)− H−Anti Ajaib Super Pada Beberapa Kelas Graf, Disertasi, ITB,
Bandung, 2013.
4. Inayah, N., A. N. M.Salman, and R. Simanjuntak, On (a, d)−h−antimagic coverings of graphs, J. Combin. Math. Combin. Comput 71 (2009), 273–281.
5. Inayah, N., A.N.M. Salman, and K.I.A Syuhada, Super (a, d)-H-Antimagic Total Labeling for
Shackles of a Connected Graph H, Australasian Journal of Combinatorics 57 (2013), 127–138.
6. Karyanti, Pelabelan Selimut (a, d)−h−Anti Ajaib Super Pada graf Fan, Sun, dan Generalized
Petersen, Skripsi, UNS, Surakarta, 2012.
7. Kotzig, A and A. Rosa, Magic Valuation of Finite Graph, Canada Math Bull 13 (1970), 451–461.
8. Llado, A. and J. Moragas, Cycle Magic Graphs, (2006).
9. Maryati, T.K., Karakteristik Graf H-Ajaib and Graf H-Ajaib Super, Disertasi, ITB, Bandung, 2011.
10. Maryati, T.K. , A. N. M. Salman, E. T. Baskoro, J. Ryan, and M. Miller, On H-Supermagic
Labelings for Certain Shackles and Amalgations of a Connected graph, Utilitas Mathematica
83 (2010), 333–342.
11. Roswitha, M. and E.T. Baskoro, H-Magic Covering om Some Classes of Graphs, AIP Confer-ence Proceeding on ICREM5 1450 (2012), 135–138.
12. Wallis, W.D., E.T. Baskoro, M. Miller and Slamin, Edge Magic Total Labelings, Australasian J. Combin 22 (2000), 177–190.
13. WallisW. D. , Magic Graph, Birkh¨auser Boston, New York, 2001.
