Modular irregularity strength of graphs

Bača, Martin; Muthugurupackiam, Kadarkarai; Kathiresan, K.; Ramya, S.

doi:10.5614/ejgta.2020.8.2.19

Cited by 16 publications

(33 citation statements)

References 12 publications

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“…Theorem 3 (see [8]). If G is a graph of order n, n ≡ 2(mod 4), then G has no modular irregular k-labeling, i.e., ms(G) � ∞.…”

Section: Known Resultsmentioning

confidence: 99%

“…Aman and Togni [5] and Ferrara et al [6] proved the irregularity strength of trees family. e modular irregularity strength of the family of trees which is already known is path and star [8]. Since we consider biregular graphs in this paper, then the regular trees family that we choose is regular doublestars.…”

Section: New Resultsmentioning

confidence: 99%

“…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 and denoted by ms(G). Bača et al [8] determined the modular irregularity strength of path, star, triangular graph, cycle, and gear graphs. Muthugurupackiam et al [9] proved the modular irregularity of the tadpole graph and doublecycle graph.…”

Section: Introductionmentioning

confidence: 99%

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Modular Irregular Labeling on Double-Star and Friendship Graphs

Sugeng

Barack

Hinding

et al. 2021

Journal of Mathematics

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A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The 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.

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“…Theorem 3 (see [8]). If G is a graph of order n, n ≡ 2(mod 4), then G has no modular irregular k-labeling, i.e., ms(G) � ∞.…”

Section: Known Resultsmentioning

confidence: 99%

Section: New Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modular Irregular Labeling on Double-Star and Friendship Graphs

Sugeng

Barack

Hinding

et al. 2021

Journal of Mathematics

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“…The exact values of the modular irregularity strength for certain families of graphs, namely paths, cycles, stars, triangular graphs and gear graphs, are determined in [12]. A fan graph F n , n ≥ 2, is a graph obtained by joining all vertices of a path P n on n vertices to a further vertex, called the center.…”

Section: Theorem 1 ([12]mentioning

confidence: 99%

“…In [12], the authors introduced a modification of an irregular assignment known as a modular irregular assignment. A function ψ : E(G) → {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Irregularity and Modular Irregularity Strength of Wheels

Bača

Imran

Semaničová-Feňovčíková

2021

Mathematics

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It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

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