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All Journal Desimal: Jurnal Matematika Jurnal Matematika UNAND
Haryeni, Debi Oktia
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Computer Science & IT Education Mathematics Social Sciences

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Partition dimension of disjoint union of complete bipartite graphs Haryeni, Debi Oktia; Baskoro, Edy Tri; Saputro, Suhadi Wido
Desimal: Jurnal Matematika Vol. 4 No. 2 (2021): Desimal: Jurnal Matematika
Publisher : Universitas Islam Negeri Raden Intan Lampung

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24042/djm.v4i2.10190

Abstract

For any (not necessary connected) graph $G(V,E)$ and $A\subseteq V(G)$, the distance of a vertex $x\in V(G)$ and $A$ is $d(x,A)=\min\{d(x,a): a\in A\}$. A subset of vertices $A$ resolves two vertices $x,y \in V(G)$ if $d(x,A)\neq d(y,A)$. For an ordered partition $\Lambda=\{A_1, A_2,\ldots, A_k\}$ of $V(G)$, if all $d(x,A_i)<\infty$ for all $x\in V(G)$, then the representation of $x$ under $\Lambda$ is $r(x|\Lambda)=(d(x,A_1), d(x,A_2), \ldots, d(x,A_k))$. Such a partition $\Lambda$ is a resolving partition of $G$ if every two distinct vertices $x,y\in V(G)$ are resolved by $A_i$ for some $i\in [1,k]$. The smallest cardinality of a resolving partition $\Lambda$ is called a partition dimension of $G$ and denoted by $pd(G)$ or $pdd(G)$ for connected or disconnected $G$, respectively. If $G$ have no resolving partition, then $pdd(G)=\infty$. In this paper, we studied the partition dimension of disjoint union of complete bipartite graph, namely $tK_{m,n}$ where $t\geq 1$ and $m\geq n\geq 2$. We gave the necessary condition such that the partition dimension of $tK_{m,n}$ are finite. Furthermore, we also derived the necessary and sufficient conditions such that $pdd(tK_{m,n})$ is either equal to $m$ or $m+1$.
Inclusive Distance Antimagic Labeling of Shadow Graph of Complete and Circulant Graph Arafah, Siti Hafshah Nurul; Sugeng, Kiki Ariyanti; Haryeni, Debi Oktia
Jurnal Matematika UNAND Vol. 15 No. 1 (2026)
Publisher : Departemen Matematika dan Sains Data FMIPA Universitas Andalas Padang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.25077/jmua.15.1.57-62.2026

Abstract

Consider a graph $G = (V, E)$ with order $n$. Suppose that we have a bijection $f: V(G) \to \{1, 2, ..., n\}$. A graph $G$ is said to admit an inclusive distance antimagic labeling if every pair of distinct vertices has different weights, with a vertex weight is defined by $w(v) = \sum_{u \in N(v)} f(u) + f(v)$. Furthermore, if the vertex weights form an arithmetic progression with the first term $a$ and the common difference $d$, then $G$ is said to admit an $(a,d)$-inclusive distance antimagic labeling. This paper investigates the inclusive distance antimagic labeling of the shadow graph of the complete and circulant graph.
Co-Authors Arafah, Siti Hafshah Nurul Edy Tri Baskoro Kiki Ariyanti Sugeng Saputro, Suhadi Wido