A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph of order is a mapping of the set of edges of the graph to 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 . 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.

This research is the development from Siddiqui's research on edge irregularity strength of subdivision of star. Siddiqui [3] had determined the total edge irregularity strength of sub-division of star S m for 1≤m≤8 and n≥3. However, the total edge of irregularity strength for star subdivision S m for m≥9 and n≥3 are still in an open problem and also the total edge irregularity strength of subdivision of star by inserting mi vertices on different to every edge of a star Sn yet to be determined.In this paper, it will be proven that the total edge irregularity strength of subdivision of star S m for m≥9 and n≥3 and the total edge irregularity strength of subdivision of star S_n^{m₁, m₂, ⋯ ,m_n} for m_i≥1, m_i<m_i+1, 1≤i≤n, and n≥3.

An antimagic labeling of a graph G is a bijection $f:E(G)\to \{1,\dots ,|E(G)\left|\right\}$ such that the weights $w(x)={\sum }_{y\sim x}f(y)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than $K_2$ admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection $f:V(G)\to \{1,\dots ,|V(G)\left|\right\}$ such that the weight$\omega (x)={\sum }_{y\in N_D(x)}f(y)$ is distinct for each vertex x, where $N_D(x)=\{y\in V(G)|d(x,y)\in D\}$ is the D-neigbourhood set of a vertex x. If $N_D(x)=r$, for every vertex x in G, a graph G is said to be $(D,r)$-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for $D=\left\{1\right\}$, all graphs of order up to 8 concur with the conjecture. We prove that the set of $(D,r)$-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric $(D,r)$-regular that are D-antimagic and examples of disjoint union of non-symmetric non-$(D,r)$-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

For a simple graph G with a vertex set V(G) and an edge set E(G), a labeling f : V(G)∪^​E(G)⟶{1,2, ⋯, k} is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V(G) we have wt(x) ≠ wt(y) where wt(x) = f(x) + ∑_u∈V(G)f(xu). The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G, denoted by tvs(G). The lower bound of tvs(G) for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2. Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is (3n² + 1)/2.