On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k � max ke, 2kv 􏼈 􏼉. A vertex irregular reflexive k-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wt(x)≠wt(x′), where wt(x) � fv(x) + Σxy∈E(G)fe(xy). )e minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvs(G). In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.


Introduction
We consider a simple and finite graph G � (V, E) with vertex set V(G) and edge set E(G). We motivate the readers to refer Chartrand et al. [1], for detailed definition of the graph. A topic in graph theory which has grown fast is the labeling of graphs. e concept of graph labeling, firstly, was introduced by Wallis in [2]. He defined a labeling of G is a mapping that carries a set of graph elements into a set of integers called labels. By this definition, we can have a vertex label, edge label, or both of them. Baca et al. [3] introduced the total labeling, and they defined the vertex weight as the sum of all incident edge labels along with the label of the vertices. Many types of labeling have been studied by researchers, namely, graceful labeling, magic labeling, antimagic labeling, irregular labeling, and irregular reflexive labeling.
Furthermore, labeling known as a vertex irregular total k-labeling and total vertex irregularity strength of graph is the minimum k for which the graph has a vertex irregular total k-labeling. e bounds for the total vertex irregularity strength are given in [3]. In [4], Tanna et al. identified the concept of vertex irregular reflexive labeling of graphs. In this paper, we continue to study a vertex irregular reflexive labeling as there are still many open problems. By irregular reflexive labeling, we mean a labeling of graph which the vertex labels are assigned by even numbers from 0, 2, . . . , 2k and the edge labels are assigned by 1, 2, 3, . . . , k, where k is positive integer. e weight of each vertex, under a total labeling, is determined by summing the incident edge labels and the label of the vertex itself.
A k-labeling assigns numbers 1, 2, . . . , k { } to the elements of graph. Let k be a natural number, a function f: V(G) ∪ E(G) ⟶ 1, 2, 3, . . . , k { } is called total k-irregular labeling. Hinding et al. [5] defined that a total labeling ϕ: xy) is distinct for every two different vertices, w t ϕ (x) ≠ w t ϕ (y) for x, y ∈ V(G), x ≠ y. e minimum k for which graph G has a vertex irregular total k-labeling is called total vertex irregularity strength, denoted by tvs(G). e concept of vertex irregular total k-labeling extends to a vertex irregular reflexive total k-labeling. e definition of total k-labeling is a function f e from the edge set to the first natural number k e and a function f v from the vertex set to the nonnegative even number up to 2k v , where k � max k e , 2k v . A vertex irregular reflexive k-labeling of the graph G is the total k-labeling, for every two different vertices x and x ′ of G, e minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvs(G).
Some results related to vertex irregular reflexive labeling have been studied by several researchers. Tanna et al. [4] have studied the vertex irregular reflexive of prism and wheel graphs, Ahmad and Bacȃ [6] have studied the total vertex irregularity strength for two families of graphs, namely, Jahangir graphs and circulant graphs, and Agustin et al. [7] also study the concept of vertex irregular reflexive labeling of cycle graph and generalized friendship. Another results of irregular labeling can be seen on [8][9][10][11][12][13][14][15]. In this paper, we have found the lower bound of vertex irregular reflexive strength of any graph G and determined the vertex irregular reflexive strength of graphs with pendant vertex. Our results are started by showing one lemma and theorem which describe a general construction of the existence of vertex irregular reflexive k-labeling of graph with pendant vertex.

Result and Discussion
e following lemma and theorem will be used as a base construction of analysing the reflexive vertex strength of any graph with pendant vertex, namely, sunlet graph, helm graph, subdivided star graph, and broom graph.

Lemma 1.
For any graph G of order p, the minimum degree δ, and the maximum degree Δ, Proof. Let G be a graph of order p, the minimum degree δ, and the maximum degree Δ. e total k-labeling which labeling . Furthermore, since we require k-minimum for the graph G which has a vertex irregular reflexive labeling, the set of a vertex weight should be consecutive, otherwise it will not give a minimum rvs. us, the set of a vertex weight is Wt(x) � δ, δ + 1, δ + 2, . . . , Since the minimum k � max k e , 2k v is the reflexive vertex strength, the maximum possible vertex weight of graph G is at most k(1 + Δ). It implies 2k v + Δk e ≥ δ + (p − 1).1↔ k + Δk ≥ δ + (p− 1)↔k ≥ δ + p − 1/Δ + 1. Since rvs(G) should be integer and we need a sharpest lower bound, it implies It completes the proof.
Proof. Given that a graph G of order n is with l pendant vertices. e labeling of graph G is with respect to two components, namely, the pendant vertices and otherwise vertices. us, we will split our proof into two cases.
Define an injection f as the labels. Since the weight l is contributed by one vertex and one edge labels, it will give four possibilities.
us, we will have a different weight for every pendant. us, the labels of vertex and edge of the pendant are the following.
Case 2. e vertex set apart from pendant vertices V(G) − V l must have a degree of at least two. e cardinality of V(G) − V l is less than or equal to the cardinality of V l . It implies that the vertex or edge labels of pendant vertices can be re-used on labels of V(G) − V l . us, the vertex weight of V(G) − V l will be different with the vertices of V l since it has more combination, namely, 2k + 1, 2k + 2, . . . , n.
Based on Case 1 and Case 2, the reflexive vertex strength of graph G is rvs(G) � l 2 + 1, for l even and l 2 odd, It concludes the proof.
Proof. Moreover, to determine the label of vertices V(S n ) � u i , v i ; 1 ≤ i ≤ n and edge set E(S n ) � u i v i , 1 ≤ i ≤ n ∪ u i u i+1 , 1 ≤ i ≤ n − 1 ∪ u 1 u n , we will use (Algorithm 1) k � n 2 + 1, if n even and n 2 odd, It concludes the proof.
For an illustration, see Figure 1.

Theorem 2.
Let H n be a helm graph, and for every n ≥ 3, rvs H n � n 2 + 1, if n even and n 2 odd, Proof. Let H n be a helm graph with vertex set Furthermore, we will show the upper bound of vertex irregular reflexive k-labeling by defining the injection f and g in the following: Based on the above injection, the overall vertex weight sets are It is easy to see that the above elements of set are all different. It concludes the proof. Theorem 3. SS n be a subdivided star graph, and for every n ≥ 3, Proof. Let SS n be a subdivided star graph with vertex set V(SS n ) � A, x i , y i ; 1 ≤ i ≤ n , |V(SS n )| � 2n + 1 and edge set E(SS n ) � Ay i , x i y i , 1 ≤ i ≤ n , |E(H n )| � 2n. e maximum degree of SS n is n. e graph SS n has one central vertex of degree n. Since the central vertex has degree of much greater than the other vertices, it must have a different vertex weight than the others. Based on Lemma 1, we have the following lower bound: (1) Define v ∈ V(G), e ∈ E(G) and injecton f for labeling of the graph elements (2) Assign the labels of vertices and edges of pendants v i according to eorem 1.
For the illustration of the vertex irregular reflexive, k-labeling of SS 3 and SS 4 can be depicted in Figure 2.
Furthermore, we will show the upper bound of vertex irregular reflexive k-labeling by defining the injection f and g. For n ≥ 5, we have the following: For n ≡ 3, 4(mod6), we have the following function for y i and Ay i : For otherwise n, we have International Journal of Mathematics and Mathematical Sciences k − 1, if k + 1 ≤ i ≤ n, (i odd for k even)and(i even for k odd), k, if k + 1 ≤ i ≤ n, (i even for k even)and(i odd for k odd), Based on the above injection, the overall vertex weight sets of the subdivided star SS n are w x i � i,