ResearchArticle Modular Irregular Labeling on Double-Star and Friendship Graphs K. A. Sugeng , 1 Z. Z. Barack, 1 N. Hinding , 2 and R. Simanjuntak 3 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok, Indonesia 2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Hasanuddin, Makassar, Indonesia 3 Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural, Sciences, Institute Technology Bandung, Bandung, Indonesia Correspondence should be addressed to K. A. Sugeng; kiki@sci.ui.ac.id Received 17 September 2021; Revised 8 October 2021; Accepted 23 November 2021; Published 28 December 2021 Academic Editor: Ali Jaballah Copyright © 2021 K. A. Sugeng et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n isamappingofthesetofedgesofthegraphto 1, 2, ... ,k { } suchthattheweightsofallverticesaredifferent.evertexweightisthe sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n. e modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case. 1. Introduction Graph labeling is a mapping of a set of numbers, called the labels, to the graph elements, usually vertices or edges [1]. Generally, the label is a positive integer. ere are several labelings that have been developed; among them are ir- regular labeling and modular irregular labeling. e reader can check the dynamic survey of graph labeling by Gallian to obtain more information on various labeling [1]. In 1988, irregular labeling was first introduced by Chartrand et al. [2]. To date, there have been studies on the irregular labelings of certain graphs. e terminology not included in this paper can be found in [3]. An irregular labeling is defined as a labeling f: E ⟶ 1, 2, ... ,k { } with k as a positive integer, such that wt f(x) �  (y∈N(x)) f(xy) is different for all vertices, where N(x) is a neighbour of vertex x. e irregularity strength s(G) of a graph G is the minimum value of k for which G has irregular labeling with labels at most k. e irregularity strength s(G) of a graph G is defined only for graphs containing at most one isolated vertex and no connected component of order 2. e lower bound of the irregularity strength of a graph G is s(G) ≥ max 1≤i≤△ n i + i − 1/i  , where n i vertices with degree i, as stated in eorem 1. For a regular graph G, Przyboylo [4] has proved an upper bound of an irregularity strength is s(G) < 16n/d + 6. For tree graphs, Aigner and Triesch [5] proved that the irregularity strength of any tree with no vertices of degree two is equal to the number of its leaves. Ferrara et al. [6] later proved that if the tree T has every two vertices of degree not equal to two at a distance of at least eight with number of leaves at least three, then s(T)� n 1 + n 2 /2, where n 1 is the number of leaves and n 2 is the number of vertices of degree two. e survey of irregular labeling has been done by Baˇ ca et al. [7]. After this survey paper, there are still many results which have been found. See Gallian’s survey, for the update [1]. Modular irregular labeling of a graph is a mapping φ: E(G) ⟶ 1, 2, ... ,k { } so that a bijective function wt φ(x) �  (y∈N(x)) φ(xy) can be defined and has different values. e set of the weights of the vertices is a group of integers modulo n. e minimum k such that this kind of labeling exists is called the modular irregularity strength of G Hindawi Journal of Mathematics Volume 2021, Article ID 4746609, 6 pages https://doi.org/10.1155/2021/4746609