A constructive method to determine the total vertex irregularity strength of two flower graph variants

Abstract

Graph labeling is a well-established area of research in discrete mathematics. One notable variant is total vertex irregular labeling, which assigns labels to both vertices and edges such that the resulting vertex weights are all distinct. In this paper, we propose a constructive method to determine the total vertex irregularity strength of two flower graph variants, each characterized by only two distinct vertex degrees. Our approach involves assigning explicit labels and verifying that the resulting weighted degrees differ at every vertex. This method not only determines the exact total vertex irregularity strength for each graph but also introduces a replicable labeling strategy applicable to other graphs with similar structural features.

The main results in this paper are as follows:

• •The total vertex irregularity strength of the modified sunflower graph is $\frac{n}{2}+1$;
• •The total vertex irregularity strength of the flower petal graph is $\lceil \frac{n^2-2n+2}{3}\rceil$;
• •These values differ due to the fundamentally distinct structural properties of the to graphs.

The proposed method contributes to the broader study of irregular graph labeling and offers potential applications in network analysis and graph-based modeling.

Graphical abstract

Specifications table

Subject area	Mathematics and Statistics
More specific subject area	Graph Theory; Combinatoric; Network System
Name of your method	Graph-Based Modelling Techniques (G-BMT)
Name and reference of original method	On irregularity strength of diamond network (2018), Hinding, N., Firmayasari, D., Basir, H., Bača, M., Semaničová-Feňovčíková, A, AKCE International Journal of Graphs and Combinatorics, 2018, 15(3), pp. 291–297. Vertex-irregular labeling and vertex-irregular total labeling on caterpillar graph, Nurdin, Zakir, M., Firman, International Journal of Applied Mathematics and Statistics, 2013, 40(10), pp. 99–105. Total edge irregularity strength of the disjoint union of helm graphs, Siddiqui, M.K., Nurdin, Baskoro, E.T., Journal of Mathematical and Fundamental Sciences, 2013, 45(2), pp. 163–171.
Resource availability	no

Background

Graph labeling is an established area of research in discrete mathematics, with numerous variants and applications. One particular type, the total vertex irregular labeling, assigns weights to both the vertices and the edges of a graph in such a way that the total weighted degrees of all vertices become distinct. The smallest maximum weight required to achieve such a labeling for a graph $\mathit{G}$ is known as its total vertex irregularity strength, denoted by $tvs\left(G\right)$. This concept plays an important role in analyzing structural irregularities and has applications in areas such as network design, data communication, and coding theory.

Although the total vertex irregularity strength has been investigated for various classes of graphs, many graph families remain unexplored. In particular, graphs that exhibit structural symmetry or have a limited number of vertex degrees present both challenges and opportunities in constructing such labelings. Flower graphs composed of a central vertex connected to multiple petal, like structures, serve as an interesting subject due to their compact form and repetitive substructures. Variants of flower graphs with only two distinct vertex degrees allow for methodical exploration of labeling strategies that may be extended to other structurally similar graphs. The total vertex irregular labeling for two types of flower graphs, namely the modified sunflower graph and the flower petal graph, has not previously been studied. In this paper, we determine the total vertex irregularity strength for both graph types.

The graph structure generally influences the construction of a total vertex irregular labeling. One factor determining a graph's structure is the degree of the graph's vertices. A graph can have varying vertex degrees. For graphs with a single point degree, this has been studied by other researchers. The methods used in constructing a label are similar. However, the labeling function for graphs with a single degree cannot be used for graphs whose vertex degrees are not single. This is behind the emergence of this research, namely how to construct a total vertex irregular labeling for graphs whose vertex degrees are not single. Several definitions are written as follows to provide a complete description related to this article.

Definition 1

[2] Let $G=\left(V,E\right)$ be a connected graph without isolated vertices or not a graph of order two and a function $f:V\cup E\to \left\{1,2,3,\cdots ,k\right\}$. A function $f$ is called total vertex irregular $k-labeling$ $on$ $G$ if the weight of each vertex, the assignment of a vertex plus the assignments of all edges incident to that vertex, are different.

What is interesting about Definition 1 is that the smallest value of $k$ is such that a graph $G$ has a total vertex irregular $k-labeling.$

Definition 2

[2] Let $f:V\cup E\to \left\{1,2,3,\cdots ,k\right\}$ be a total vertex irregular $k$-labeling on a graph $G$. Then, the smallest integer $k$ is called the total vertex irregularity strength of $G=\left(V,E\right)$ and is denoted by $tvs\left(G\right).$

In 2010, Nurdin et al. provided a lower bound on the total value of vertex irregularity strength for any graph as follows.

Theorem 3

[5] Let $G$ be a connected graph having $n_i$ vertices of degree $i\left(i=\delta ,\delta +1,\delta +2,\cdots ,\Delta \right),$where$\delta$ and$\Delta$ are the minimum and the maximum degree of $G$, respectively. Then

$$max\left\{\lceil \frac{\delta +n_{\delta }}{\delta +1}\rceil ,\lceil \frac{\delta +n_{\delta }+n_{\delta +1}}{\delta +2}\rceil ,\cdots ,\lceil \frac{\delta +{\Sigma }_{i=\delta }^{\Delta }{n}_i}{\Delta +1}\rceil \right\}\le tvs\left(G\right).$$

Previously, Bača et al. gave lower and upper bounds on the total vertex irregularity strength for any graph, as in Theorem 4.

Theorem 4

[2] Let $G\left(V,E\right)$ be a graph with $n$ vertices, $\delta$ minimum degree, and $\Delta$maximum degree. Then

$$\lceil \frac{n+\delta }{\Delta +1}\rceil \le tvs\left(G\right)\le n+\Delta -2\delta +1.$$

The study of total vertex irregularity strength in graph has seen substantial developments. In 2018, Nurdin et al. focused on diamond graphs, investigating their total vertex irregularity strength. This work laid the groundwork for exploring irregularities in complex graph structures, setting the stage for further studies. Diamond graphs, known for their unique properties, provided a significant benchmark for subsequent research [6]. Besides that, Nurdin have studied about caterpillar and helm graphs [13].

In 2019, research expanded to encompass various other graph types. Koam and Ahmad analyzed theta graphs, adding depth to understanding irregularity strengths in this category [11]. Concurrently, Rosyida and Indriati explored the total vertex irregularity strength of the tadpole chain graph [16], and Haryanti et al. delved into the firecracker graph, uncovering distinct irregular patterns [4]. These studies enriched the field by identifying how graph structures influence vertex irregularity strength.

Continuing the momentum in 2019, Dewi et al. examined the $C_n{K}_n$ graph, contributing insights into the total vertex irregularity strength properties of combined graph forms [3]. Indriati further contributed to the discourse by analyzing lollipop graphs [10]. At the same time, Ramdani et al. extended the exploration by determining the total vertex irregularity strength of several graphs derived from star operations [17]. Additionally, Ramdani and his team investigated the comb product of cycles and paths, highlighting the intricacies of these interconnected graph structures [15].

The year 2020 marked another leap in this research domain. Ramdani focused on the comb product of two cycles and two stars, broadening the understanding of total vertex irregularity strength in composite graphs [14]. Rosyida et al. investigated generalized uniform cactus and pentagon cactus chain graphs, revealing intriguing total vertex irregularity strength patterns [16]. Simanjuntak's analysis of trees with numerous vertices of degree two further diversified the scope of the graph types studied [19].

In 2021, research efforts shifted towards more specialized and complex graph structures. Meanwhile, Hinding et al. examined hexagonal cluster graphs [7]. These studies added a layer of complexity to understanding a graph's total vertex irregularity strength by introducing less conventional graph types into the discourse.

Furthering these advancements in 2021, Imran et al. discussed generalized prism graphs, offering insights into their total vertex irregularity strength [9]. Ahmad explored graphs associated with the zero-divisor graph of commutative rings, bridging algebra, and graph theory [1]. Hinding et al. also analyzed a dodecahedral-modified generalization graph, contributing to the growing catalog of studied graphs with high structural diversity [8].

In addition, a few years earlier, Nurdin et al. determined a caterpillar graph's total vertex irregularity strength [13]. Furthermore, Siddiqui et al. also studied a graph's total edge irregular labeling. They determined the total edge irregularity strength of the disjoint union of a helm graph [18].

Finally, studies in 2021 also included explorations of real-world applications and specialized graph categories. Rusdi and Syahrir examined the Twitter network, demonstrating the practical relevance of graph theory in social media analysis [17]. Yanti et al. focused on symmetric cubic graphs from Foster's census [20]. At the same time, Mazuki et al. discussed series-parallel graphs, rounding out the year with comprehensive research on theoretical and applied aspects of total vertex irregularity strength [12].

Method details

This section describes the approach and techniques used to determine the total vertex irregularity strength of the modified sunflower (MSF) and the flower petal (FP) graph. The methods are outlined as follows:

Graph definition and structure

The modified sunflower graph (MSF) is derived from the sunflower graph by removing the central vertex and adding a cycle and additional edges. The graph's vertex set and edge set were systematically described.

The flower petal graph (FP) is constructed from the sunflower graph by replacing paths and creating a more intricate structure, as detailed in Definition 5.

Mathematical framework

Key definitions and theorems relevant to total vertex irregular labeling were adopted, including those by Bača et al. and subsequent extensions.

Lower and upper bounds for the total vertex irregularity strength $\left(tvs\right)$ were established using Theorem 1 and Theorem 2. These bounds provided a foundation for the proposed labeling functions.

Labeling construction

A specific total vertex irregular labeling function was constructed for each graph. The function ensures unique vertex weights by combining the assigned labels of vertices and incident edges.

Detailed steps of the labeling construction were outlined, supported by illustrative examples for both MSF and FP graphs.

Method validation

The proofs for the $tvs$ relied on the validity of the constructed labeling functions. For the $MSF$ graph, the total vertex irregularity strength was shown to be through the direct application of labeling rules and theoretical bounds. For the $FP$ graph, the $tvs$ were demonstrated as leveraging specific properties of the graph’s structure and degree distribution.

Results

The research results in this paper are written in two theorems. In Theorem 7, we study modified sunflower graphs, and in Theorem 9, we study flower petal graphs.

Modified sunflower graph

A modified sunflower graph is a graph constructed from a sunflower graph.

Definition 5

Let $W_n$be a wheel graph with the center vertex $c$ and a cycle of order $n,$ having vertices $c_1,c_2,\dots ,c_n,$ where $c_i{c}_{i+1}$ for$1\le i\le n-1$ represents an edge in $W_n$. A sunflower graph, denoted as $S{F}_n,$ is formed by extending $W_n$ with $n$additional vertices $b_1,b_2,\cdots ,b_n,$ such that each $b_i$ is adjacent to$c_i$ and $c_{i+1}$for$1\le i\le n-1,$ and $b_n$is adjacent $to{c}_n$and$c_1.$

In Fig. 1, the following is an illustration of the $S{F}_8$ graph constructed from the wheel graph $W_8$ with vertex set $V\left(W_8\right)=\left\{c,c_1,c_2,c_3,c_4,c_5,c_6,c_7,c_8\right\}$ plus 8 new vertices $b_1,b_2,b_3,b_4,b_5,b_6,b_7,b_8.$ Thus, the vertex set and edge set of the graph $F_8$ are

$$V\left(S{F}_8\right)=\left\{c,c_1,c_2,\cdots ,c_8,b_1,b_2,\cdots ,b_8\right\}$$

and

$$E\left(S{F}_8\right)=\left\{c{c}_1,c{c}_2,\cdots ,c{c}_8,c_1{c}_2,c_2{c}_3,\cdots ,c_7{c}_8,c_8{c}_1,c_1{b}_1,c_2{b}_2,\cdots ,c_8{c}_8,b_1{c}_2,b_2{c}_3,\cdots ,b_7{c}_8,b_8{c}_1\right\},$$

respectively.

Fig. 1.

The Graph $\mathrm{S}{\mathrm{F}}_8$ according to Definition 5.

Based on Definition 5, the sets of vertex and edge of graph $S{F}_n$ are as follows. The vertex set is

$$V\left(S{F}_n\right)=\{c,c_i,b_i|1\le i\le n\}$$

and edge set is

$$E\left(S{F}_n\right)=\{c{c}_i,b_i{c}_i|1\le i\le n\}\cup \left\{c_i{c}_{i+1}\right\},b_i{c}_{i+1}\left\}\right|1\le i\le n-1\}\cup \left\{c_n{c}_1,b_n{c}_1\right\}.$$

Definition 6

Let $S{F}_n$ be a sunflower graph with its vertex and edge sets as defined in Definition 5. A modified sunflower graph for $n\ge 4$ and$n$being an even integer, denoted by $MS{F}_n,$ is constructed from $S{F}_n$ by removing the center vertex $c,$ adding a cycle $C_n:a_1{a}_2,a_2{a}_3,\dots ,a_{n-1}{a}_n,a_n{a}_1,$ where each $a_i$is connected to $c_i$ for $1\le i\le n,$ and adding edges $b_i{b}_{i+1}$ for every odd $1\le i\le n-1.$

In Fig. 2, an illustration of the $MS{F}_8$ graph is given.

Fig. 2.

The Graph $\text{SM}{\mathrm{F}}_8$ according to Definition 6.

Here is a theorem from research related to the total vertex irregular labeling of a modified sunflower graph.

Theorem 7

For $n\ge 4$ and $n$being an even integer. Then $tvs\left(MS{F}_n\right)=\frac{n}{2}+1.$

Proof.

To prove that $\frac{n}{2}+1\le tvs\left(MS{F}_n\right),$ just use the Theorem 3. Since the number of vertices of $MS{F}_n$ is $3n$ and the maximum degree is 5, then $\frac{n}{2}+1\le tvs\left(MS{F}_n\right).$

To prove that $tvs\left(MS{F}_n\right)\le \frac{n}{2}+1,$ we construct a total vertex irregular labeling on $MS{F}_n.$ For this purpose, we redefine the edge set of $MS{F}_n$ as follows

$$E\left(MS{F}_n\right)=\left\{a_1{a}_2,c_1{c}_2\right\}\cup \{a_i{a}_{i+2},c_i{c}_{i+2}|1\le i\le n-2\}\cup \left\{a_{n-1}{a}_n,c_{n-1}{c}_n\right\}\cup \left\{b_i{b}_{i+1}\right|iodd\},$$

as illustrated in Fig. 3. for $MS{F}_8.$

Fig. 3.

The Graph $\text{MS}{\mathrm{F}}_8$ according to the definition in proof of the Theorem 7.

For some $k,$ define the total labeling $f:V\cup E\to \left\{1,2,3,\cdots ,k\right\}$on$MS{F}_n$ in 3 cases.

Case 1. For $n=6s$ for some positive integer number $s$, as follows.

$$f\left(a_i\right)=1,fori=1,2,3,4,5,$$

$$f\left(a_i\right)=\frac{i}{6},fori=6,12,18,\cdots$$

$$f\left(a_i\right)=\frac{i-1}{6},fori=7,13,19,\cdots$$

$$f\left(a_i\right)=\frac{i-2}{6},fori=8,14,20,\cdots$$

$$f\left(a_i\right)=\frac{i-3}{6}+1,fori=9,15,21,\cdots$$

$$f\left(a_i\right)=\frac{i-4}{6}+1,fori=10,16,22,\cdots$$

$$f\left(a_i\right)=\frac{i-5}{6}+1,fori=11,17,23,\cdots$$

$$f\left(a_{n-1}\right)=f\left(a_n\right)=\{\begin{array}{cc}\frac{n}{6},\hfill & n=6sforsomepositiveintegers\\ \frac{n-2}{6},\hfill & n=6s+2forsomepositiveintegers\\ \frac{n-4}{6}+1,\hfill & n=6s+4forsomepositiveintegers\end{array}$$

$$f\left(b_i\right)=\{\begin{array}{cc}1\hfill & fori=1\\ 2\hfill & fori=2,3\\ 3\hfill & fori=4,\end{array}$$

$$f\left(b_i\right)=\frac{i}{2}+1,\text{for}i=6,12,18,\cdots$$

$$f\left(b_i\right)=\frac{i}{2}\text{for}i=7,13,19,\cdots$$

$$f\left(b_i\right)=\frac{i}{2}+1\text{for}i=8,14,20,\cdots$$

$$f\left(b_i\right)=\frac{i-3}{2}+2\text{for}i=9,15,21,\cdots$$

$$f\left(b_i\right)=\frac{i}{2}+1\text{for}i=10,16,22,\cdots$$

$$f\left(b_i\right)=\frac{i-5}{2}+3\text{for}i=11,17,23,\cdots$$

$$f\left(b_{n-1}\right)=\frac{n}{2},$$

$$\mathrm{f}\left({\mathrm{b}}_{\mathrm{n}}\right)=\frac{\mathrm{n}}{2}+1,$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\{\begin{array}{ccc}1\hfill & \text{for}& i=1,2,3,4\\ 2\hfill & \text{for}& i=5\end{array}$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\frac{\mathrm{i}}{2}-1\text{for}i=6,12,18,\cdots ,$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\frac{i-1}{2}\text{for}i=7,13,19,\cdots ,$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\frac{\mathrm{i}}{2}-1\text{for}i=8,14,20,\cdots ,$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\frac{i-3}{2}+1\text{for}i=9,15,21,\cdots ,$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\frac{\mathrm{i}}{2}-1,\text{for}i=10,16,22,\cdots ,$$

$$\mathrm{f}\left({\mathrm{c}}_{\mathrm{i}}\right)=\frac{i-5}{2}+2\text{for}i=11,17,23,\cdots ,$$

$$\mathrm{f}\left({\mathrm{c}}_{n-1}\right)=f\left({\mathrm{c}}_{\mathrm{n}}\right)=\frac{\mathrm{n}}{2}-1,$$

$$\mathrm{f}\left({\mathrm{a}}_1{\mathrm{a}}_2\right)=1,$$

$$\mathrm{f}\left({\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{i+2}\right)=\{\begin{array}{ccc}1\hfill & \text{for}& i=1\\ 2\hfill & \text{for}& i=2,3,\end{array}$$

$$f\left(a_i{a}_{i+2}\right)=\frac{i+2}{6}+1\text{for}i=4,10,16,22,\cdots ,$$

$$\mathrm{f}\left({\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{i+2}\right)=\frac{i+1}{6}+1\text{for}n=6\mathrm{s}\text{and}i=5,11,17,23,\cdots ,$$

$$\mathrm{f}\left({\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{i+2}\right)=\frac{i+1}{6}+2\text{for}\mathrm{n}\text{other}\text{and}i=5,11,17,23,\cdots ,$$

$$\mathrm{f}\left({\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{i+2}\right)=\frac{\mathrm{i}}{6}+2\text{for}i=6,12,18,\cdots ,$$

$$\mathrm{f}\left({\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{i+2}\right)=\frac{i-1}{6}+2\text{for}i=7,13,19,\cdots ,$$

$$\mathrm{f}\left({\mathrm{a}}_{\mathrm{i}}{\mathrm{a}}_{i+2}\right)=\frac{i-2}{6}+2\text{for}i=8,14,20,\cdots ,$$

$$f\left(a_i{a}_{i+2}\right)=\frac{i-3}{6}+2\text{for}i=9,15,21,\cdots ,$$

$$f\left(a_{n-2}{a}_n\right)=f\left(a_{n-1}{a}_n\right)=\{\begin{array}{cc}\frac{n}{6},\hfill & n=6sforsomepositiveintegers\\ \frac{n-2}{6},\hfill & n=6s+2forsomepositiveintegers\\ \frac{n-4}{6},\hfill & n=6s+4forsomepositiveintegers\end{array}$$

$$f\left(c_1{c}_2\right)=f\left(c_1{c}_3\right)=\frac{n}{2},$$

$$f\left(c_i{c}_j\right)=\frac{n}{2}+1\text{for}anyi\ne j,$$

$$f\left(b_i{b}_{i+1}\right)=\{\begin{array}{ccc}1& for& i=1\\ 2& for& i=3\end{array}$$

$$f\left(b_i{b}_{i+1}\right)=\frac{i+1}{2}\text{for}i=5,11,17,\dots ,$$

$$f\left(b_i{b}_{i+1}\right)=\frac{i-1}{2}+1\text{for}i=7,13,20,\dots ,$$

$$f\left(b_i{b}_{i+1}\right)=\frac{i-3}{2}+2\text{for}i=9,15,22,\dots ,$$

$$f\left(b_{n-1}{b}_n\right)=\frac{n}{2},$$

$$f\left(a_i{c}_i\right)=\{\begin{array}{ccc}1& for& i=1,2\\ 2& for& i=3\end{array}$$

$$f\left(a_i{c}_i\right)=\frac{i}{2}\text{for}i=4,10,16,\cdots ,$$

$$f\left(a_i{c}_i\right)=\frac{i-1}{2}\text{for}i=5,11,17,\cdots ,$$

$$f\left(a_i{c}_i\right)=\frac{i}{2}\text{for}i=6,12,18,\cdots ,$$

$$f\left(a_i{c}_i\right)=\frac{i-1}{2}\text{for}i=7,13,19,\cdots ,$$

$$f\left(a_i{c}_i\right)=\frac{i}{2}\text{for}i=8,14,20,\cdots ,$$

$$f\left(a_i{c}_i\right)=\frac{i-3}{2}+1\text{for}i=9,15,21,\cdots ,$$

$$f\left(a_{n-2}{c}_{n-2}\right)=f\left(a_{n-1}{c}_{n-1}\right)=\frac{n}{2}-1,$$

$$f\left(a_n{c}_n\right)=\frac{n}{2},$$

$$f\left(b_1{c}_1\right)=f\left(b_1{c}_2\right)=f\left(b_3{c}_1\right)=f\left(b_3{c}_3\right)=f\left(b_{n-4}{c}_{n-4}\right)=f\left(b_{n-4}{c}_{n-2}\right)=f\left(b_{n-2}{c}_{n-3}\right)=f\left(b_{n-2}{c}_{n-1}\right)=f\left(b_{n-1}{c}_{n-2}\right)=f\left(b_{n-1}{c}_n\right)=f\left(b_{n-1}{c}_{n-1}\right)=f\left(b_{n-1}{c}_{n-3}\right)=f\left(b_n{c}_{n-1}\right)=f\left(b_n{c}_n\right)=\frac{n}{2}+1.$$

Based on this definition, it is obtained that the codomain of the function $f$ is $\frac{n}{2}+1.$

It will be shown next that $f$is a total vertex irregular labeling. Based on the definition of $f$, the weight of each vertices of $MS{F}_n$ is obtained as follows, where the weight of vertex $x$ is, $wt\left(x\right)=\sum _{xy\in E\left(MS{F}_n\right)}f\left(xy\right).$

$$wt\left(a_i\right)=3+ifori=1,2,3,\cdots ,n,$$

$$wt\left(b_i\right)=n+3+i,fori=1,2,3,\cdots ,n,$$

$$wt\left(c_i\right)=2n+3+i,fori=1,2,3,\cdots ,n.$$

Based on the definition of $f$, it is obtained that it is a total vertex irregular $k-$labeling with $k=\frac{n}{2}+1.$ Therefore $tvs\le \frac{n}{2}+1.$ ♦

For illustration, in Fig. 4 a total vertex irregular $5-$labeling of $MS{F}_8$ based on the previous construction of the function f.

Fig. 4.

Total vertex irregular $5-\text{labeling}$ on $\text{MS}{\mathrm{F}}_8$.

The following result is related to the total vertex irregular labeling on the flower petal graph.

Flower petal graph

A flower petal graph is a graph constructed from a sunflower graph.

Definition 8

A flower petal graph is a graph, denoted by $F{P}_n,$ constructed from an $S{F}_n$ graph, with vertex set and edge set as in Definition 7, where the center vertex $c$ of the $S{F}_n$ is removed. The path $c_i{b}_i{c}_{i+1}$ for $1\le i\le n-1$ and $c_n{b}_n{c}_1$ are replaced by the path $P_n$ for $n\ge 4.$ After that, vertex $c_i$ replaced by $x_i$ for $1\le i\le n,$ such that the sets of vertex and edge of $F{P}_n$ are as follows

$$V\left(F{P}_n\right)=\{x_i|1\le i\le n\}\cup \left\{v_{i,j}|1\le i\le 2n,1\le j\le \frac{n}{2}-2\right\}\cup \left\{y_i|1\le i\le 2n\right\},$$

and

$$E\left(F{P}_n\right)=\left\{x_i{x}_{i+1}|1\le i\le n-1\right\}\cup \left\{x_1{x}_n\right\}\cup \{y_i{y}_{i+1}|i=1,3,5,\cdots ,2n-1\}\cup \left\{v_{2i-2,\frac{n}{2}-2}{x}_i|2\le i\le n\right\}$$

$$\cup \left\{v_{1,\frac{n}{2}-2}{x}_1,v_{2n,\frac{n}{2}-2}{x}_1\right\}\cup \{v_{i,1}{y}_i|1\le i\le 2n\}\cup \left\{v_{i,j}{v}_{i,j+1}|1\le i\le 2n,1\le j\le \frac{n}{2}-3\right\}.$$

As an illustration, in Fig. 5 we are given an $F{P}_8$ graph with vertices and edges as in Definition 8.

Fig. 5.

A Flower Petal Graph $\mathrm{F}{\mathrm{P}}_8$ according to Definition 8.

Here is a theorem from research related to the total vertex irregular labeling of a flower petal graph.

Theorem 9

Suppose $F{P}_n$ is a flower petal graph where $n\ge 4$ and $n$ an even positive integer. Then $tvs\left(F{P}_n\right)=\lceil \frac{n^2-2n+2}{3}\rceil .$

Proof.

To prove that $\lceil \frac{n^2-2n+2}{3}\rceil \le tvs\left(F{P}_n\right),$ just use the Theorem 2. Note that the vertices of $F{P}_n$ are only degrees 2 and 4. The number of vertices of degree 2 is $n\left(n-2\right),$ and the number of vertices of degree 4 is $n.$ Using Theorem 3, we obtain $max\left\{\lceil \frac{2+n\left(n-2\right)}{3}\rceil ,\lceil \frac{2+n\left(n-2\right)+n}{5}\rceil \right\}=\lceil \frac{n^2-2n+2}{3}\rceil \le tvs\left(F{P}_n\right).$

To prove that $tvs\left(F{P}_n\right)\le \lceil \frac{n^2-2n+2}{3}\rceil ,$ we construct a total vertex irregular labeling on $F{P}_n$.

Define the total labeling $f:V\cup E\to \left\{1,2,3,\cdots ,k\right\}$ on $F{P}_n$ as follows.

Case I. For $n=6,12,18,\cdots$

$$f\left(x_i\right)=i,fori=1,2,3,\cdots ,n,$$

$$f\left(y_i\right)=1,fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{i,1}\right)=\frac{n-3}{3}fori=1,2,3,4,$$

$$f\left(v_{i,1}\right)=\frac{i-3}{2}n-\frac{n-3}{3},fori=5,7,9,\cdots ,2n-1,$$

$$f\left(v_{i,1}\right)=\frac{i-4}{2}+n-\frac{n-3}{3},fori=6,8,10,\cdots ,2n,$$

$$f\left(v_{i,j}\right)=i+n+\frac{n}{3},fori=1,2,3,\cdots ,2nandj=2,3,$$

$$f\left(v_{i,j}\right)=i+\frac{2j-5}{3}n+\frac{n}{3},fori=1,2,3,\cdots ,2nandj=4,7,10,\cdots ,3m-5,$$

$$f\left(v_{i,j}\right)=i+\frac{2j-4}{3}n+\frac{n}{3},fori=1,2,3,\cdots ,2nandj=5,8,11,\cdots ,3m-4,$$

$$f\left(v_{i,j}\right)=i+\frac{2j-6}{3}n+\frac{n}{3},fori=1,2,3,\cdots ,2nandj=6,9,12,\cdots ,3m-3,$$

$$f\left(v_{i,\frac{n}{2}-2}\right)=i+\left(\frac{n}{3}-3\right)n+\frac{n}{3},fori=1,2,3,\cdots ,2n,$$

$$f\left(x_i{x}_{i+1}\right)=\frac{n}{6}\left(n-2\right),fori=1,2,3,\cdots ,n-1,$$

$$f\left(x_1{x}_n\right)=\frac{n}{6}\left(n-2\right),$$

$$f\left(y_i{y}_{i+1}\right)=1,fori=1,3,$$

$$f\left(y_i{y}_{i+1}\right)=\frac{i-1}{2}fori=5,7,9,\cdots ,2n-1,$$

$$f\left(y_{2n-1}{y}_{2n}\right)=n-1,$$

$$f\left(v_{i,j}{v}_{i,j+1}\right)=n+\frac{n}{3}+1fori=1,2,3,\cdots ,2nandj=1,2,$$

$$f\left(v_{i,j}{v}_{i,j+1}\right)=\frac{2j+1}{3}n+\frac{n}{3}+1fori=1,2,3,\cdots ,2nandj=3,6,9,\cdots ,3m-6,$$

$$f\left(v_{i,j}{v}_{i,j+1}\right)=\frac{2j+1}{3}n+\frac{n}{3}+1,fori=1,2,3,\cdots ,2nandj=4,7,10,\cdots ,3m-5,$$

$$f\left(v_{i,j}{v}_{i,j+1}\right)=\frac{2j+2}{3}n+\frac{n}{3}1,fori=1,2,3,\cdots ,2nandj=5,8,11,\cdots ,3m-4,$$

$$f\left(v_{i,\frac{n}{2}-3}{v}_{i,\frac{n}{2}-2}\right)=\left(\frac{n}{3}-1\right)n+\frac{n}{3}+1,fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{1,\frac{n}{2}-2}{x}_1\right)=f\left(v_{2n,\frac{n}{2}-2}{x}_1\right)=\frac{n}{3}\left(n+1\right)-n+1,$$

$$f\left(v_{2i-2,\frac{n}{2}-2}{x}_i\right)=f\left(v_{2i-1,\frac{n}{2}-2}{x}_i\right)=\frac{n}{3}\left(n+1\right)-n+1,fori=2,3,4,\cdots ,n,$$

$$f\left(v_{i,1}{y}_i\right)=i,fori=1,2,3,4,$$

$$f\left(v_{i,1}{y}_i\right)=\frac{i+3}{2},fori=5,7,9,\cdots ,2n-3,$$

$$f\left(v_{i,1}{y}_i\right)=\frac{i}{2}2,fori=6,8,10,\cdots ,2n-2,$$

$$f\left(v_{2n,1}{y}_{2n}\right)=n+2.$$

Based on this definition, it is obtained that the codomain of the function $f$ is $\lceil \frac{n^2-2n+2}{3}\rceil .$

It will be shown next that $f$ is a total vertex irregular labeling. Based on the definition of $f,$ the weight of each vertices of $F{P}_n$ is obtained as follows, where the weight of vertex $x$ is, $wt\left(x\right)=\sum _{xy\in E\left(F{P}_n\right)}f\left(xy\right).$

$$wt\left(x_i\right)=n^2-2n+2+i,fori=1,2,3,\cdots ,n,$$

$$wt\left(y_i\right)=2+i,fori=1,2,3,\cdots ,2n,$$

$$wt\left(v_{i,j}\right)=2+2jn+i,fori=1,2,3,\cdots ,2n,$$

$$wt\left(v_{i,j}\right)=2+2jn+i,fori=1,2,3,\cdots ,2n.$$

Case II. For $\mathit{n}=8,14,20,\cdots$

$$f\left(x_i\right)=i,fori=1,2,3,\cdots ,n,$$

$$f\left(y_i\right)=i,fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{i,1}\right)=1,fori=1,2,3,4,$$

$$f\left(v_{i,1}\right)=\frac{i-1}{2},fori=5,7,9,\cdots ,2n-1,$$

$$f\left(v_{i,1}\right)=\frac{i-2}{2},fori=6,8,10,\cdots ,2n,$$

$$f\left(v_{i,2}\right)=i,fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{i,2}\right)=i+\frac{2j}{3}-1,fori=1,2,3,\cdots ,2nandj=3,6,9,\cdots ,3m-3,$$

$$f\left(v_{i,2}\right)=i+\frac{2j-5}{3},fori=1,2,3,\cdots ,2nandj=4,7,10,\cdots ,3m-2,$$

$$f\left(v_{i,2}\right)=i+\frac{2j-4}{3},fori=1,2,3,\cdots ,2n,andj=5,8,11,\cdots ,3m-4,$$

$$f\left(v_{i,\frac{n}{2}-2}\right)=i+\frac{n-8}{3},fori=1,2,3,\cdots ,2n,$$

$$f\left(x_i{x}_{i+1}\right)=\frac{n-2}{6}n,fori=1,2,3,\cdots ,n-1,$$

$$f\left(x_1{x}_n\right)=\frac{n-2}{6}n,$$

$$f\left(y_i{y}_{i+1}\right)=1,fori=1,3,$$

$$f\left(y_i{y}_{i+1}\right)=\frac{i-1}{2},fori=5,7,9,\cdots ,2n-1,$$

$$f\left(y_{2n-1}{y}_{2n}\right)=n-1,$$

$$f\left(v_{i,j}{v}_{i,i+1}\right)=2n\frac{j+2}{3}+1,fori=1,2,3,\cdots ,2nandj=1,4,7,\cdots ,3m-5,$$

$$f\left(v_{i,j}{v}_{i,i+1}\right)=2n\frac{j+1}{3}+1,fori=1,2,3,\cdots ,2nandj=2,5,8,\cdots ,3m-4,$$

$$f\left(v_{i,j}{v}_{i,i+1}\right)=n\left(\frac{2j}{3}+1\right)+1,fori=1,2,3,\cdots ,2nandj=3,6,9,\cdots ,3m-3,$$

$$f\left(v_{i,\frac{n}{2}-3}{v}_{i,\frac{n}{2}-2}\right)=n\frac{n-2}{3}1,fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{1,\frac{n}{2}-2}{x}_1\right)=f\left(v_{2n,\frac{n}{2}-2}{x}_1\right)=n\frac{n-2}{3}+1,$$

$$f\left(v_{2i-2,\frac{n}{2}-2}{x}_i\right)=f\left(v_{2i-1,\frac{n}{2}-2}{x}_i\right)=n\frac{n-2}{3}+1,fori=2,3,4,\cdots ,n,$$

$$f\left(v_{i,1}{y}_i\right)=i,fori=1,2,3,4,$$

$$f\left(v_{i,1}{y}_i\right)=\frac{i+3}{2},fori=5,7,9,\cdots ,2n-1,$$

$$f\left(v_{i,1}{y}_i\right)=\frac{i}{2}+2,fori=6,8,10,\cdots ,2n-2,$$

$$f\left(v_{2n,1}{y}_{2n}\right)=n+2.$$

Case III. For$\mathit{n}=10,16,22,\cdots$

$$f\left(x_i\right)=i,fori=1,2,3,\cdots ,n,$$

$$f\left(y_i\right)=i,fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{i,1}\right)=n+\frac{n+2}{3},fori=1,2,3,4,$$

$$f\left(v_{i,1}\right)=n+\frac{i-3}{2}+\frac{n+2}{3},fori=5,7,9,\cdots ,2n-1,$$

$$f\left(v_{i,1}\right)=n+\frac{i-4}{2}+\frac{n+2}{3}fori=6,8,10,\cdots ,2n,$$

$$f\left(v_{i,}\right)=n+\frac{2n+1}{3}+i-1,fori=1,2,3,\cdots ,2nandj=2,3,$$

$$f\left(v_{i,j}\right)=\frac{2j-5}{3}n+\frac{2n+1}{3}+i-1,fori=1,2,3,\cdots ,2nandj=4,7,10,\cdots ,3m-2,$$

$$f\left(v_{i,j}\right)=\frac{2j-7}{3}n+\frac{2n+1}{3}+i-1fori=1,2,3,\cdots ,2nandj=5,8,11,\cdots ,3m-1,$$

$$f\left(v_{i,j}\right)=\frac{2j-6}{3}n+\frac{2n+1}{3}+i-1,fori=1,2,3,\cdots ,2nandj=6,9,12,\cdots ,3m-3,$$

$$f\left(v_{i,\frac{n}{2}-2}\right)=\frac{n-10}{3}n+\frac{2n+1}{3}+i-1,fori=1,2,3,\cdots ,2n,$$

for $n=10,$

$$f\left(v_{i,j}\right)=\frac{i+1}{2}+n+\frac{2n+1}{3}-1,fori=1,3,5\cdots ,2n-1andj=2,3,$$

$$f\left(v_{i,j}\right)=\frac{i}{2}+n+\frac{2n+1}{3}-1,fori=2,4,6\cdots ,2nandj=2,3,$$

$$f\left(x_i{x}_{i+1}\right)=\frac{n-4}{6}\left(n+2\right)+1,fori=1,2,3,\cdots ,n-1,$$

$$f\left(x_1{x}_n\right)=\frac{n-4}{6}\left(n+2\right)+1,$$

$$f\left(y_i{y}_{i+1}\right)=1,fori=1,3,$$

$$f\left(y_i{y}_{i+1}\right)=\frac{i-1}{2},fori=5,7,9,\cdots ,2n-1,$$

$$f\left(y_{2n-1}{y}_{2n}\right)=n-1,$$

$$f\left(v_{i,1}{v}_{i,2}\right)=n-\frac{n-4}{3},fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{i,2}{v}_{i,3}\right)=n+\frac{2n+4}{3},fori=1,2,3,\cdots ,2n,$$

$$f\left(v_{i,j}{v}_{i,j+1}\right)=\frac{2j}{3}n+\frac{2n+4}{3},fori=1,2,3,\cdots ,2nandj=3,6,9,\cdots ,3m-3,$$

$$f\left(v_{i,j}{v}_{i,+1}\right)=\frac{2j+1}{3}n+\frac{2n+4}{3},fori=1,2,3,\cdots ,2nandj=4,7,10,\cdots ,3m-2.$$

Based on the definition of the function $f$, it is easy to see that

$$f:V\cup E\to \left\{1,2,3,\cdots ,\lceil \frac{n^2-2n+3}{3}\rceil \right\}.$$

To show that $f$ is a total vertex irregular labeling, it must be shown that the weights of all the vertices are different, that is, $w\left(x\right)=f\left(x\right)+f\left(e\right)$ for all edges $e$ incident to a vertex $x.$ Note that:

$$wt\left(y_i\right)=i+2fori=1,2,3,\cdots ,2n,$$

$$wt\left(v_{i,j}\right)=i+2nj+2fori=1,2,3,\cdots ,2n,andj=1,2,3,\cdots ,\frac{n}{2}-2,$$

$$wt\left(x_i\right)=i+n^2-2n+2fori=1,2,3,\cdots ,n.$$

Based on the weight formula, the following sequence of vertex weights is obtained in $F{P}_n$:

$$wt\left(y_1\right)<wt\left(y_2\right)<wt\left(y_3\right)<\cdots <wt\left(y_{2n}\right)<wt\left(v_{1,1}\right)<wt\left(v_{2,1}\right)<wt\left(v_{3,1}\right)<\cdots <wt\left(v_{2n,1}\right)<wt\left(v_{1,2}\right)<wt\left(v_{2,2}\right)<wt\left(v_{3,2}\right)<\cdots <wt\left(v_{2n,2}\right)<wt\left(v_{1,3}\right)<wt\left(v_{2,3}\right)<wt\left(v_{3,3}\right)<\cdots <wt\left(v_{2n,3}\right)<\cdots <wt\left(v_{1,\frac{n}{2}-2}\right)<wt\left(v_{2,\frac{n}{2}-2}\right)<wt\left(v_{3,\frac{n}{2}-2}\right)<wt\left(v_{2n,\frac{n}{2}-2}\right)<wt\left(x_1\right)<wt\left(x_2\right)<wt\left(x_3\right)<\cdots <wt\left(x_n\right).$$

Based on the definition of $f$, it is obtained that it is a total vertex irregular $k-$labeling with $k=\lceil \frac{n^2-2n+3}{3}\rceil .$ Therefore $tvs\left(F{P}_n\right)\le \lceil \frac{n^2-2n+3}{3}\rceil .$ ♦

For illustration, in Fig. 6 a total vertex irregular $17-$labeling of $F{P}_8$ based on the previous construction of the function $f$.

Fig. 6.

Total vertex irregular $17-\text{labeling}$ on $\mathrm{F}{\mathrm{P}}_8$.

Discussion

Analysis of results

This study determines the total vertex irregularity strength $\left(tvs\right)$ of two variants of flower graphs: the modified sunflower graph and flower petal graph. The novelty of this work lies in the constructive method developed to assign total labels that ensure all vertex weight are distinct. The labeling procedure is explicit, deterministic, and repeatable. To the best of our knowledge, this is the first report to provide exact $tvs$ values for these two specifics flower-like graph structures.

The differences in structural configuration between the modified sunflower graph and the flower petal graph lead to different $TVIS$ values, despite both graphs having similar degree constraints. The modified sunflower graph benefits from central symmetry, allowing for straightforward label distribution. In contrast, the flower petal graph requires a more careful assignment due to its cyclic and even-petal construction. These differences emphasize the importance of local subgraph configuration in determining labeling complexity.

Comparative insights

Compared to simpler graphs studied in earlier research, such as caterpillar and diamond graphs, the results for $MSF$ and $FP$ graphs indicate an increasing trend in$tvs$ with growing structural complexity.

The findings align with the established bounds provided in the literature, validating the approach.

Implications and applications

Theoretical results on the total vertex irregularity strength of graph classes, such as flower graph variants, can have meaningful implications in real-world scenarios involving networks with hub-and-spoke configurations.

One practical scenario where the concept of total vertex irregularity strength can be applied is in the design of sensor networks with heterogeneous data transmission rates. Consider a circular garden with central control units and sensor nodes arranged in a radial pattern, similar to the structure of a flower graph. Each sensor transmits data to the central node, and interference between nodes needs to be minimized.

By assigning different weights (labels) to sensor nodes and their connections based on the total vertex irregularity strength, we can ensure that each node has a distinct overall transmission load. This helps in avoiding communication collisions and managing energy usage more effectively.

For example, if the flower graph represents the physical topology of the sensor layout, then a labeling that satisfies the total vertex irregularity condition helps to assign unique bandwidth or duty cycle settings to each sensor, thereby optimizing the network's operation.

Conclusion

In this paper, we investigated the total vertex irregular labeling of two variants of flower graphs: the modified sunflower graph $\left(MS{F}_n\right)$ and the flower petal graph $\left(F{P}_n\right).$ Our primary objective was to determine their total vertex irregularity strength $\left(TVIS\right),$ which ensures that the weighted degree of every vertex is distinct.

The main findings are summarized as follows:

• For the modified sunflower graph $MS{F}_n,$ where $n$ is the number of vertices with minimum degree 3 and a maximum degree 5, the total vertex irregularity strength is $tvs\left(MS{F}_n\right)=\frac{n}{2}+1.$

• For the flower petal graph $F{P}_n,$ defined for even integer $n\ge 4,$ the total vertex irregularity strength is $tvs\left(F{P}_n\right)=\lceil \frac{n^2-2n+2}{3}\rceil .$

These results highlight a structured approach to computing $TVIS$ in graphs with limited and regular degree distributions. The labeling constructions presented in this work assign explicit values to both vertices and edges, guaranteeing that all vertex weights are distinct. These constructions not only verify the computed $TVIS$ values but also introduce a systematic, repeatable method that can be adapted to other graphs with similar structures.

Our approach contributes to the ongoing development of irregular graph labeling by offering a replicable and extensible technique. Future directions include applying this method to more complex or asymmetrical graph families, and potentially automating the labeling process through algorithmic or computational techniques.

In particular, the labeling strategies developed here can be extended to other graph families that—like the modified sunflower and flower petal graphs—feature a small number of distinct vertex degrees and a symmetric or repetitive structure. Identifying such structural patterns enables the design of similar constructive labelings, paving the way for applications to wheel-like graphs, expanded petal structures, and other generalized flower graph configurations.

Limitations

The study is limited to graphs with specific structural properties and degree distributions. Extending the approach to graphs with arbitrary structures remains a challenge.

Future research can focus on applying these techniques to hybrid or dynamic graphs and exploring computational algorithms for efficient labeling.

Ethics statements

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CRediT author statement

Nurdin Hinding: Conceptualization, Methodology, Formal analysis, Writing – original draft, Supervision.

Nurtiti Sunusi : Methodology, Formal analysis, Visualization.

Sarti Mutmainnah: Methodology, Investigation, Writing – review & editing.

Ika Indriani: Methodology, Investigation, Writing – review & editing.

Syafrizal Sy: Validation, Formal analysis, Visualization, Writing – review & editing.

Kiki A. Sugeng: Software, Writing – review & editing, Visualization.

Rinovia G. Simanjuntak: Conseptualization, Writing-review & editing.

Supplementary material and/or additional information [OPTIONAL]

None

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author(s) used ChatGPT in order to improve language and readability, with caution. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

Acknowledgments

This research is partly supported by the Indonesian Collaborative Research $\left(RKI\right)$ based on the Decree of the Rector of Hasanuddin University Number $03976/UN4.1/KEP/2024$ dated April 4, 2024, and Agreement/Contract Number $01369/UN4.22/PT.01.03/2024$ dated April 5, 2024. We would like to thank the editors and referees.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.