The local multiset dimension of unicyclic graph

Adawiyah, Robiatul; Dafik, Dafik; Prihandini, R. M.; Albirri, Ermita Rizki; Agustin, Ika Hesti; Alfarisi, Ridho

doi:10.1088/1755-1315/243/1/012075

Cited by 7 publications

(6 citation statements)

References 2 publications

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“…Agustin & Dafik, 2022;Dafik et al, 2021). This research also the development of super (a, d) -antimagic labelling which have been observed by (Adawiyah et al, 2019;Agustin et al, 2017;Figueroa-Centeno & Muntaner-Batle, 2021;Ming-ju, 2013;Moussa & Badr, 2016;Purnapraja et al, 2014). Another research about the labelling of graph have been observed by (Gallian, 2022;Prihandini et al, , 2020Prihandini et al, 2019).…”

Section: Introductionsupporting

confidence: 61%

“…In this topic to determine or prove the local (a, d) -antimagic chromatic number Xle(a,d) by using the lemma or theorem that has been studied previously. Pattern detection method (pattern recognition) is a method used to determine patterns in Local (a, d) -antimagic coloring so that a local (a, d) -antimagic chromatic number will be denoted by Xle(a,d) (Adawiyah et al, 2019). Next apply local edge (a, d) -antimagic coloring to design batik.…”

Section: Methodsmentioning

confidence: 99%

“…This This is the development of local edge antimagic coloring of graphs study which had been observed by (Adawiyah et al, 2019;Agustin, 2017;I. Agustin & Dafik, 2022;Dafik et al, 2021).…”

Section: Introductionmentioning

confidence: 93%

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Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

Adawiyah,

Makhfudloh,

Prihandini

2023

Alifmatika J. Pendidik. dan Pembelajaran Mat.

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Suppose a graph G = (V, E) is a simple, connected and finite graph with vertex set V(G) and an edge set E(G). The local edge antimagic coloring is a combination of local antimagic labelling and edge coloring. A mapping f∶ V (G)→ {1, 2, ..., |V (G)|} is called local edge antimagic coloring if every two incident edges e_1and e_2, then the edge weights of e_1and e_2 maynot be the same, w(e_1) ≠ w(e_2), with e = uv ∈ G, w(e) = f(u)+ f(v) with the rule that the edges e are colored according to their weights, w_e. Local edge antimagic coloring has developed into local (a,d)-antimagic coloring. Local antimagic coloring is called local (a,d)-antimagic coloring if the set of edge weights forms an arithmetic sequence with a as an initial value and d as a difference value. The graphs used in this study are sunflower graphs and umbrella graphs. This research will also discuss one of the applications of local edge (a,d)-antimagic coloring, namely the design of the Sidoarjo line batik motif. The result show that χ_le(3,1) (Sf_n) = 3n and χ_le(3n/2,1) (U_(m,n) ) = m+1 . The local (a,d)-antimagic coloring is formed into a batik motif design with characteristics from the Sidoarjo region.

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Section: Introductionsupporting

confidence: 61%

Section: Methodsmentioning

confidence: 99%

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Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

Adawiyah,

Makhfudloh,

Prihandini

2023

Alifmatika J. Pendidik. dan Pembelajaran Mat.

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“…If G has a multiset basis, then its cardinality is called the multiset dimension of G, denoted by md(G). There are some related research about this topic in [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

A Note on Multiset Dimension and Local Multiset Dimension of Graphs

Alfarisi

Lin²,

Ryan³

et al. 2020

Stat., optim. inf. comput.

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All graphs in this paper are nontrivial and connected simple graphs. For a set W = {s1,s2,...,sk} of verticesof G, the multiset representation of a vertex v of G with respect to W is r(v|W) = {d(v,s1),d(v,s2),...,d(v,sk)} whered(v,si) is the distance between of v and si. If the representation r(v|W)̸= r(u|W) for every pair of vertices u,v of a graph G, the W is called the resolving set of G, and the cardinality of a minimum resolving set is called the multiset dimension, denoted by md(G). A set W is a local resolving set of G if r(v|W) ̸= r(u|W) for every pair of adjacent vertices u,v of a graph G. The cardinality of a minimum local resolving set W is called local multiset dimension, denoted by µl(G). In our paper, we discuss the relationship between the multiset dimension and local multiset dimension of graphs and establish bounds of local multiset dimension for some families of graph.

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“…9 also studied local multiset dimension of unicyclic graphs. The followings are some results which is used for proving the new results in this study.Lemma 0.1 Let G be a connected graphs and W ⊂ V (G).…”

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confidence: 99%

Local Multiset Dimension of Amalgamation Graphs

Alfarisi

Dafik

Prabhu³

2023

F1000Res

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\textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs\\

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The local multiset dimension of unicyclic graph

Cited by 7 publications

References 2 publications

Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

A Note on Multiset Dimension and Local Multiset Dimension of Graphs

Local Multiset Dimension of Amalgamation Graphs

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