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All Journal International Journal of Electrical and Computer Engineering CAUCHY: Jurnal Matematika Murni dan Aplikasi Prosiding Seminar Matematika dan Pendidikan Matematik UNEJ e-Proceeding Jurnal Matematika: MANTIK Jurnal Ilmiah Soulmath : Jurnal Edukasi Pendidikan Matematika BAREKENG: Jurnal Ilmu Matematika dan Terapan JTAM (Jurnal Teori dan Aplikasi Matematika) Indonesian Journal of Combinatorics CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Dedication : Jurnal Pengabdian Masyarakat

Ika Hesti Agustin, Ika Hesti

Department Of Mathematics, University Of Jember, Indonesia.

Author-ID : 488507

Computer Science & IT Control & Systems Engineering Decision Sciences, Operations Research & Management Economics, Econometrics & Finance Education Electrical & Electronics Engineering Energy Engineering Mathematics Mechanical Engineering Physics Transportation Other

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On r-dynamic coloring of some graph operations Ika Hesti Agustin; D. Dafik; A. Y. Harsya
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Let $G$ be a simple, connected and undirected graph. Let $r,k$ be natural number. By a proper $k$-coloring of a graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $|S| = k$, such that any two adjacent vertices receive different colors. An $r$-dynamic $k$-coloring is a proper $k$-coloring $c$ of $G$ such that $|c(N (v))| \geq min\{r, d(v)\}$ for each vertex $v$ in $V(G)$, where $N (v)$ is the neighborhood of $v$ and $c(S) = \{c(v) : v \in S\}$ for a vertex subset $S$ . The $r$-dynamic chromatic number, written as $\chi_r(G)$, is the minimum $k$ such that $G$ has an $r$-dynamic $k$-coloring. Note that the $1$-dynamic chromatic number of graph is equal to its chromatic number, denoted by $\chi(G)$, and the $2$-dynamic chromatic number of graph has been studied under the name a dynamic chromatic number, denoted by $\chi_d(G)$. By simple observation it is easy to see that $\chi_r(G)\le \chi_{r+1}(G)$, however $\chi_{r+1}(G)-\chi_r(G)$ can be arbitrarily large, for example $\chi(Petersen)=2, \chi_d(Petersen)=3$, but $\chi_3(Petersen)=10$. Thus, finding an exact values of $\chi_r(G)$ is significantly useful. In this paper, we will show some exact values of $\chi_r(G)$ when $G$ is an operation of special graphs.

Chromatics Number of Operation Graphs Kiki Kurdianto; Ika Hesti Agustin; Dafik Dafik
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 1, No 1 (2020): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Let G = (V (G); E(G)) be connected nontrivial graph. Edge coloring is de-¯ned as c : E(G) ! f1; 2; :::; kg; k 2 N, with the conditions no edges adja-cent having the same color. Coloring k-color edges r-dynamic is edges color-ing as much as k color such that every edges in E(G) with adjacent at least minfr; d(u) + d(v) ¡ 2g have di®erent color. An Edge r dynamic is a proper c of E(G) such that jc(N(uv))j = minfr; d(u) + d(v) ¡ 2g, for each edge N(uv) is the neighborhood of uv and c(N(uv)) is color used to with adjacent edges of uv. the edge r-dynamic chromatic number, written as ¸(G), is the minimum k such that G has an edge r-dynamic k-coloring. chromatic number 1-dynamic writ-ten as ¸(G), chromatic number 2-dynamic written as ¸d(G) And for chromatic number r-dynamic written as ¸(G). A graph is used in this research namely gshack(H3; e; n), amal(Bt3; v; n) and amal(S4; v; n). Keywords: r-dynamic coloring, r-dynamic chromatic number, graph operations.

Analisa Himpunan Dominasi Lokasi pada Model Topologi Graf Khusus dan Operasinya Reyka Bella Desvandai; Ika Hesti Agustin; Kusbudiono Kusbudiono
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 2, No 2 (2021): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Misalkan $G=(V,E)$ adalah graf sederhana tidak berarah dan terhubung dengan himpunan titik $V$ dan himpunan sisi $E$. Himpunan $D\in V(G)$ dikatakan himpunan dominasi lokasi dari suatu graf terhubung $G$ jika setiap dua titik yang berbeda $u,v \in V(G)\ D$, $N(u)\cap D\neq N(v)\cap D$. Kardinalitas minimal dari himpunan dominasi lokasi disebut nilai himpunan dominasi lokasi dari graf $G$ yang disimbolkan dengan $\gamma_L(G)$. Penelitian ini menghasilkan nilai himpunan dominasi lokasi pada beberapa graf khusus dan operasinya.

Kajian Rainbow 2-Connected Pada Graf Eksponensial dan Beberapa Operasi Graf Herninda Lucky Oktaviana; Ika Hesti Agustin; Dafik Dafik
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 2, No 2 (2021): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Let $G=(V(G),E(G))$ is a graph connected non-trivial. \textit{Rainbow connection} is edge coloring on the graph defined as $f:E(G)\rightarrow \{1,2,...,r|r \in N\}$, for every two distinct vertices in $G$ have at least one \textit{rainbow path}. The graph $G$ says \textit{rainbow connected} if every two vertices are different in $G$ associated with \textit{rainbow path}. A path $u-v$ in $G$ says \textit{rainbow path} if there are no two edges in the trajectory of the same color. The edge coloring sisi cause $G$ to be \textit{rainbow connected} called \textit{rainbow coloring}. Minimum coloring in a graph $G$ called \textit{rainbow connection number} which is denoted by $rc(G)$. If the graph $G$ has at least two \textit{disjoint rainbow path} connecting two distinct vertices in $G$. So graph $G$ is called \textit{rainbow 2-connected} which is denoted by $rc_2(G)$. The purpose of this research is to determine \textit{rainbow 2-connected} of some resulting graph operations. This research study \textit{rainbow 2-connected} on the graph (${C_4}^{K_n}$ and $Wd_{(3,2)}\square K_n$).

Pewarnaan Sisi r-Dinamis pada Graf Khusus dan Graf Operasi Sakel Viqedina Rizky Noviyanti; Kusbudiono Kusbudiono; Ika Hesti Agustin; Dafik Dafik
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 2, No 1 (2021): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Let $G=(V(G),E(G))$ be a nontrivial connected graph. The edge coloring is defined as $c:E(G) \rightarrow \{1,2,...,k\}, k \in N$, with the condition that no adjacent edges have the same color. \emph{k}-color \emph{r}-dynamic is an edge coloring of \emph{k}-colors such that each edge in neighboring $E(G)$ is at least min $\{r,d( u)+d(v)-2\}$ has a different color. The dynamic \emph{r}-edge coloring is defined as a mapping of $c$ from $E(G)$ such that $|c(N(uv))|$ = min$\{r,d(u)+d(v)- 2\}$, where $N(uv)$ is the neighbor of $uv$ and $c(N(uv))$ is the color used by the neighboring side of $uv$. The minimum value of $k$ so that the graph $G$ satisfies the \emph{k}-coloring \emph{r}-dynamic edges is called the dynamic \emph{r}-edge chromatic number. 1-dynamic chromatic number is denoted by $\lambda(G)$, 2-dynamic chromatic number is denoted by $\lambda_d(G)$ and for dynamic \emph{r}-chromatic number is denoted by $\lambda_r(G)$. The graphs that used in this study are graph $TL_n$, $TCL_n$ and the switch operation graph $shack(H_{2,2},v,n)$.

Nilai Ketidakteraturan Total Selimut pada Graf Yessy Eki Fajar Reksi; Dafik Dafik; Ika Hesti Agustin
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 2, No 2 (2021): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Misal $G$ dan $K$ adalah graf sederhana, nontrivial dan graf tak berarah. Operasi \emph{total comb product} menghasilkan graf baru dengan mengoperasikan dua buah graf. Misalkan \emph{G} dan \emph{K} adalah graf terhubung dan \emph{v} $\in$ \emph{V(K}) dan \emph{e} $ \in $ \emph{E(K)}. Operasi \emph{total comb product} dari graf \emph{G} dan \emph{K} yang dinotasikan ($G$ $\dot{\unrhd}$ $K$) merupakan operasi graf yang diperoleh dengan mengambil salinan satu graf \emph{G} dan $|V(G)|+|E(G)|$ salinan \emph{K}, kemudian merekatkan salinan ke-\emph{i} dari graf \emph{K} di titik cangkok \emph{v} pada titik ke-\emph{i} dari graf \emph{G} dan merekatkan salinan ke-\emph{j} dari graf \emph{K} di sisi cangkok \emph{e} pada sisi ke-\emph{j} dari graf \emph{G}. Pelabelan total didefinisikan suatu fungsi $f : V(G) \cup E(G) \rightarrow \{1,2,3,...,k\}$ merupakan pelabelan \emph{k-total} pada graf $G$. Pelabelan \emph{k-total} dikatakan pelabelan total ketidakteraturan selimut pada graf $G$ jika untuk $H \subseteq G$ dengan kata lain $H$ merupakan selimut dari suatu graf $G$, bobot total selimut $W(H)=\Sigma_{v\in V(H)}f(v)+\Sigma_{e\in E(H)}f(e)$ berbeda. Nilai minimum $k$ pada pelabelan total ketidakteraturan selimut disebut dengan \emph{total H-Irregularity Strength} dari suatu graf $G$ yang dinotasikan dengan $tHs(G)$. Pada artikel ini dilakukan penelitian tentang pelabelan total ketidakteraturan selimut yaitu mencari nilai ketidakteraturan total selimut pada graf hasil operasi \emph{total comb product} dari graf khusus.

Analisa Antimagic Total Covering Super pada Eksponensial Graf Khusus dan Aplikasinya dalam Mengembangkan Chipertext Hani'ah Zakin; Ika Hesti Agustin; Kusbudiono Kusbudiono; Dafik Dafik
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 2, No 1 (2021): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Let ${H_i}$ be a finite collection of simple, nontrivial and undirected graphs and let each $H_i$ have a fixed vertex $v_j$ called a terminal. The amalgamation $H_i$ as $v_j$ as a terminal is formed by taking all the $H_i$'s and identifying their terminal. When $H_i$ are all isomorphic graphs, for any positif integer $n$, we denote such amalgamation by $G={\rm Amal}(H,v,n)$, where $n$ denotes the number of copies of $H$. The graph $G$ is said to be an $(a, d)$-$H$-antimagic total graph if there exist a bijective function $f: V(G) \cup E(G) \rightarrow \{1, 2,\dots ,|V (G)| + |E(G)|\}$ such that for all subgraphs isomorphic to $H$, the total $H$-weights $w(H)= \sum_{v\in V(H)}f(v)+\sum_{e\in E(H)}f(e)$ form an arithmetic sequence $\{a, a + d, a +2d,...,a+(t - 1)d\}$, where $a$ and $d$ are positive integers and $t$ is the number of all subgraphs isomorphic to $H$. An $(a,d)$-$H$-antimagic total labeling $f$ is called super if the smallest labels appear in the vertices. In this paper, we study a super $(a, d)$-$H$ antimagic total labeling of $G={\rm Amal}(H,v,n)$ and its disjoint union when $H$ is a complete graph.

Independent Domination Number of Operation Graph Siti Aminatus Solehah; Ika Hesti Agustin; Dafik Dafik
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 1, No 1 (2020): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Let G be a simple, undirected and connected graph. An independent set or stable set is a set of vertices in a graph in which no two of vertices are adjacent. A set D of vertices of graph G is called a dominating set if every vertex u ∈ V (G) − D is adjacent to some vertex v ∈ D. A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. A minimum independent dominating set is an independent set of smallest possible size for a given graph G. This size is called the independence number of G, and denoted i(G). Operation Graph is a technical to get a new graph types by performing the operation of two or more graphs. Power Graph is a operation graph where let the graph G and H , notation of the power graph is (GH ). Keywords: r-dynamic coloring, r-dynamic chromatic number, graph operations.

Analisis Locating Dominating Set pada Graf Khusus dan Hasil Operasi Comb Sisi Imro’atun Rofikah; Ika Hesti Agustin; Dafik Dafik
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 1, No 2 (2020): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Assume that G = (V;E) is an undirected and connected graph with vertex set V and edge set E. D is called a dominating set of the vertex in G such that for each vertex v 2 V one of: v 2 D or a neighbor u of v in D with u 2 D. While locating dominating set of G is a dominating set D of G when satisfy this condition: for every two vertices u; v 2 (V ???? D);N(u) \ DN(v) \ D. The minimum cardinality of a locating dominating set of G is the location domination number L(G). In this paper, locating dominating set and location domination number of special graph and edge comb product operation result will be determined. Location domination number theorem on triangular book graph Btn and edge comb product operation result that is Cm D Btn and Sm D Btn are the results from this experiment.

Analisis Rainbow Vertex Connection pada Beberapa Graf Khusus dan Operasinya Ida Ariska; Dafik Dafik; Ika Hesti Agustin
CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS Vol 2, No 1 (2021): CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS
Publisher : jcgant

Suppose $G=(V(G),E(G))$ is a non-trivial connected graph with edge coloring defined as $c:E(G) \rightarrow \{1,2,...,k\} ,k \in N$, with the condition that neighboring edges can be the same color. An original path is {\it rainbow path} if there are no two edges in the path of the same color. The graph $G$ is called rainbow connected if every two vertices in $G$ with rainbow path in $G$. The coloring here is called rainbow coloring, and the minimal coloring in a graph $G$ rainbow connection number is denoted by $rc(G)$. Suppose $G=(V(G),E(G))$ is a non-trivial connected graph with a vertex coloring defined as $c':V(G) \rightarrow \{1,2,...,k\},k \in N$, with the condition that neighboring interior vertex may have the same color. An original path is rainbow vertex path if there are no two vertices in the path of the same color. The graph $G$ is called rainbow vertex connected if every two vertices in $G$ with rainbow vertex path in $G$. The $G$ coloring is called rainbow vertex coloring, and the minimal coloring in a $G$ graph is called rainbow vertex connection number which is denoted by $rvc(G)$. This research produces rainbow vertex connection number on the graph resulting from the operation \emph{amal}($Bt_{m}$, $v$, $n$), $Wd_{3,m}$ $\Box$ $ P_n$, $P_m$ $\odot$ $Wd_{3,n}$, $Wd_{3,m}$ $+$ $C_n$, and \emph{shack}($Bt_{m}$, $v $, $n$).

Co-Authors A. Y. Harsya Agustina Muharromah, Agustina Alfian Futuhul Hadi Alfian Yulia Harsya, Alfian Yulia Antonius Cahya Prihandoko Arika Indah Kriatiana Cangul, Ismail Naci D. Dafik Dafik Dafik Dwi Agustin Retnowardani Eric Dwi Putra Firdaus Ubaidillah, Firdaus Firdausiyah, Iftitahul Gembong A. W. Hani'ah Zakin Herninda Lucky Oktaviana Ida Ariska Ika Nur Maylisa Imro’atun Rofikah Ismail Naci Cangul Jannah, Excelsa Suli Wildhatul Karinda Rizqy Aprilia, Karinda Rizqy Kiki Kurdianto Kosala Dwidja Purnomo Kurniawati, Elsa Yuli Kusbudiono Kusbudiono, Kusbudiono Larasati, Indry Lusia Herni Sullystiawati Mardiyah, Fitriyatul Marsidi Marsidi Marsidi Marsidi Md. Saidur Rahman Mohanapriya, N. Mursyidah, Indah Lutfiyatul N. Mohanapriya N. Mohanapriya Novian Nur Fatihah Nur Asia Jamil, Nur Asia Nuraeni Mauliddatul Fikria Prihandini, Rafiantika Megahnia Putri Rizky H.P, Putri Rizky Qonita Ilmi Awalin R. Sunder R. Sunder Rafiantika Megahnia Prihandini Reyka Bella Desvandai Ridho Alfarisi, Ridho Riza Nurfadila Robiatul Adawiyah Rosita, Merlinda S Slamin Sherly Citra Wuni, Sherly Citra Sih Muhni Yunika, Sih Muhni Siswono, Hendrik Siti Aminatus Solehah Siti Latifah Sulisawati, Dwi Noviani Ukas Alif Fuadi Viqedina Rizky Noviyanti Yessy Eki Fajar Reksi

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