The Distance Irregular Reflexive k-Labeling of Graphs CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(4) (2023), Pages 622-629 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: January 18, 2023 Reviewed: February 28, 2023 Accepted: March 16, 2023 DOI: http://dx.doi.org/10.18860/ca.v7i4.19747 The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin1,2, Dafik1,3, N. Mohanapriya 4 , Marsidi 5 , Ismail Naci Cangul 6 1PUI-PT Combinatorics and Graph, University of Jember, Indonesia 2 Department of Mathematics, University of Jember, Indonesia 3 Department of Mathematics Education, University of Jember, Indonesia 4Department of Mathematics, Kongunadu Arts and Science College, India 5Department of Mathematics Education, University of PGRI Argopuro Jember, Indonesia 6Uludag University, Mathematics Department, Gorukle 16059 Bursa, Turkey Email: ikahesti.fmipa@unej.ac.id ABSTRACT A total π‘˜-labeling is a function 𝑓𝑒 from the edge set to the set {1,2,…,π‘˜π‘’} and a function 𝑓𝑣 from the vertex set to the set {0,2,4,…,2π‘˜π‘£}, where π‘˜ = max{ ke,2kv}. A distance irregular reflexive k-labeling of graph G is total k-labeling. If for every two different vertices 𝑒 and 𝑒′ of 𝐺, 𝑀(𝑒) β‰  w(uβ€²), where w(u) = Ξ£ui∈ N(u) fv(ui) + Ξ£uv∈ E(G) fe(uv). The minimum k for graph 𝐺 which has a distance irregular reflexive k- labeling is called distance reflexive strength of the graph 𝐺, denoted by π·π‘Ÿπ‘’π‘“(G). In this paper, we determine the lower bound of distance reflexive strength of any graph and the exact value of distance reflexive strength of path, star, and friendship graph. Copyright Β© 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC BY-SA License (https://creativecommons.org/licenses/by-sa/4.0/) Keywords: distance irregular reflexive k-labeling; distance reflexive strength; path; star; friendship. INTRODUCTION A simple graph 𝐺 is a pair (𝑉(𝐺),𝐸(𝐺)), where 𝑉(𝐺) denotes a non-empty vertex set, whereas 𝐸(𝐺) denotes an unordered pair of two distinct vertices 𝑒,𝑣 ∈ 𝑉(𝐺). If there is a path between every pair of distinct vertices in 𝐺, the graph is said to be connected. Among the various types of problems that arise while studying graph theory, one that has grown in popularity in recent years is graph labeling. Throughout this paper all graphs are simple, connected and undirected 𝐺(𝑉,𝐸). The vertex set and edge set are denoted by V(G) and E(G), respectively [1]. A graph labeling is a map that connects graph elements to numbers (usually to the positive or non-negative integers). Gallian [2] describes a recent concept of graph labeling. The most common choices of domain are the set of labels to the vertex set (called vertex labeling), the edge set (called edge labeling), or to both vertices and edges (called total labeling). In many cases it is quite interesting to consider the sum of all labels associated with a graph element called the weight of the element which is denoted by 𝑀. In this research, we concentrate on studying a novel kind of labeling called irregular labeling. If each vertex has a different weight (label sum), when we assign positive integer labels to the edges or vertices of a connected graph with at least three vertices, http://dx.doi.org/10.18860/ca.v7i4.19747 https://creativecommons.org/licenses/by-sa/4.0/ The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 623 they become irregular. The irregularity strength of a graph G is the minimum of all such irregular assignments' maximum weights. On [3-13], you can view the results of a graph's irregularity strength. The overall vertex irregularity strength on graph G was evaluated by Baca et al.; for more information, see [15]. We define a π‘˜-labeling for a graph 𝐺 and assign the numbers {1,2,…,π‘˜} to the graph's elements by using a vertex irregular reflexive labeling. Let π‘˜ be a natural number; total π‘˜-irregular labeling is a function𝑓 ∢ 𝑉(𝐺) βˆͺ 𝐸(𝐺) β†’ { 1,2,3, . . . ,π‘˜}. A vertex irregular reflexive total π‘˜-labeling is one of the many types of irregular π‘˜-labeling. Vertex irregular reflexive labeling is described in detail by Tanna et al. in their article [14]. Total π‘˜-labeling is a function 𝑓𝑒 from 𝐸(𝐺) to the first natural number up to π‘˜π‘’ and a function 𝑓𝑣 from 𝑉(𝐺) to the non-negative even number up to 2π‘˜π‘£, where π‘˜ = π‘šπ‘Žπ‘₯{π‘˜π‘’,2π‘˜π‘£}. A vertex irregular reflexive π‘˜-labeling of the graph 𝐺 is the total π‘˜-labeling, if for every two different vertices π‘₯ and π‘₯β€² of 𝐺, 𝑀𝑑(π‘₯) β‰  𝑀𝑑(π‘₯β€²), where 𝑀𝑑(π‘₯) = 𝑓𝑣(π‘₯) + Ξ£π‘₯π‘¦βˆˆ 𝐸(𝐺) 𝑓𝑒(π‘₯𝑦). The reflexive vertex strength of the graph G, denoted by π‘Ÿπ‘£π‘ (𝐺), is the minimum π‘˜ for a graph 𝐺 with a vertex irregular reflexive π‘˜-labeling. The minimum number of edges between two vertices 𝑒 and 𝑣 is called the distance between these vertices and is denoted as 𝑑(𝑒,𝑣). Motivated by the definition of distance irregular labeling [16] and irregular reflexive labeling of graphs, we develop a new concept namely distance irregular reflexive labeling. We have given a formal definition as follows: A total π‘˜-labeling is a function 𝑓𝑒 from 𝐸(𝐺) to the set {1,2,…,π‘˜π‘’} and a function 𝑓𝑣 from 𝑉(𝐺) to the set {0,2,4,…,2π‘˜π‘£}, where π‘˜ = π‘šπ‘Žπ‘₯{π‘˜π‘’,2π‘˜π‘£}. A distance irregular reflexive k-labeling of the graph 𝐺 is the total π‘˜-labeling, if for every 𝑒,𝑒′ ∈ 𝑉(𝐺),𝑒 β‰  𝑒′, 𝑀(𝑒) β‰  𝑀(𝑒′), where 𝑀(𝑒) = Ξ£π‘’π‘–βˆˆ 𝑁(𝑒)𝑓𝑣(𝑒𝑖) + Ξ£π‘’π‘£βˆˆ 𝐸(𝐺)𝑓𝑒(𝑒𝑣) and 𝑁(𝑒) is the neighbourhood of distance 1. The minimum π‘˜ for graph 𝐺 which has a distance irregular reflexive π‘˜- labeling is called distance reflexive strength of the graph 𝐺, denoted by β±°π‘Ÿπ‘’π‘“(𝐺). In this paper, we determine the lower bound of distance reflexive strength of any graph and the exact value of distance reflexive strength for three types of connected graphs: path, star, and friendship graph. METHOD The method used in determining the distance reflexive strength of a graph is as follows: 1. Defines a graph as data or a research object. 2. Determine the vertex set and edge set of the graph. 3. Determine the lower bound distance reflexive strength of a graph with the following Lemma and Corollary. Lemma: β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ and Corollary: β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ + 1, if 𝑝 + 𝛿 βˆ’ 1 > Ξ”(2𝑑 βˆ’ 1). 4. Construct vertex and edge labels based on the definition of distance irregular reflexive labeling. 5. Determine the upper bound of the distance reflexive strength of a graph with the obtained function in point 4. 6. If the upper bound of distance reflexive strength of a graph is the same as the lower bound of distance reflexive strength, then the value of distance reflexive strength can be determined from the graph. The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 624 7. If the upper bound of distance reflexive strength of the graph is not the same as the lower bound of distance reflexive strength, then point 4 is repeated until the upper bound of distance reflexive strength is equal to the lower bound of distance reflexive strength. RESULTS AND DISCUSSION Before determining the distance reflexive strength of a graph, we need to determine the lower bound which is used as a basic reference and proof in the theorem. In addition, the lower bound is used to guarantee the obtained distance reflexive strength is the smallest or minimum value. The lower bound of distance reflexive strength is presented in the following Lemma. Lemma 1. Let 𝐺 be a graph which has 𝑝,𝛿, and Ξ”. For any graph G, β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝 + 𝛿 βˆ’ 1 2Ξ” βŒ‰ where 𝑝,𝛿, and Ξ” are order, minimum degree, and maximum degree, respectively. Proof. Assume that 𝐺 is a graph of order 𝑝, the minimum degree 𝛿, and the maximum degree Ξ”. The greatest label on the graph for every distance irregular reflexive π‘˜-labeling is π‘˜β€² = β±°π‘Ÿπ‘’π‘“(𝐺). We choose 𝑒 as a vertex of degree 𝛿, in such a case that the vertex weight of 𝑒 is at least: 𝑀(𝑒) = Ξ£π‘’π‘–βˆˆ 𝑁(𝑒) 𝑓𝑣(𝑒𝑖) + Ξ£π‘’π‘£βˆˆ 𝐸(𝐺) 𝑓𝑒(𝑒𝑣) = 0 + 1(𝛿) = 𝛿. Moreover, to get the minimum π‘˜β€² the vertex weight must form an arithmetic sequence, namely 𝛿,𝛿 + 1,𝛿 + 2,…,𝛿 + 𝑝 βˆ’ 1. In other hand, we choose 𝑒′ as a vertex of degree Ξ”, such that 𝑀(𝑒′) = Σ𝑒𝑖 β€²βˆˆ 𝑁(𝑒′) 𝑓𝑣(𝑒𝑖 β€²) + Ξ£π‘’β€²π‘£β€²βˆˆ 𝐸(𝐺) 𝑓𝑒(𝑒′𝑣′) = 2π‘˜π‘£(Ξ”)+ π‘˜π‘’(Ξ”). Since π‘˜β€² = π‘šπ‘Žπ‘₯{π‘˜π‘’,2π‘˜π‘£}, then 𝑀(𝑒′) = 2π‘˜π‘£(Ξ”)+ π‘˜π‘’(Ξ”) ≀ π‘˜β€²(2Ξ”). It is obtained the largest vertex weight is 𝑝 + 𝛿 βˆ’ 1. Since 𝑝 + 𝛿 βˆ’ 1 ≀ 𝑀𝑑(𝑒′), then π‘˜β€²(2Ξ”) β‰₯ 𝑝 + 𝛿 βˆ’ 1, such that π‘˜β€² β‰₯ 𝑝+π›Ώβˆ’1 2Ξ” . Since β±°π‘Ÿπ‘’π‘“(𝐺) should be integer and we need a sharpest lower bound, it implies π‘˜β€² β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2π›₯ βŒ‰ or β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰. ∎ From the Lemma 1, we establish the corollary that can be used to calculate the exact value of distance reflexive strength as follows. Corollary 1. Let 𝐺 be a graph which has 𝑝,𝛿, and Ξ”. If 𝑝 + 𝛿 βˆ’ 1 > Ξ”(2𝑑 βˆ’ 1), then β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰+ 1, where 𝑝,𝛿,Ξ”, and 𝑑 are order, minimum degree, maximum degree, and an odd integer, respectively. Proof. Let 𝐺 be a graph which has 𝑝,𝛿, and Ξ”. Let 𝑝 + 𝛿 βˆ’ 1 = 𝑑(2Ξ”). According to Lemma 1, we have β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ > ⌈ Ξ”(2π‘‘βˆ’1) 2Ξ” βŒ‰ = ⌈ 2π‘‘βˆ’1 2 βŒ‰ = 𝑑. Moreover, suppose 𝑑 is β±°π‘Ÿπ‘’π‘“(𝐺), 𝑑 will be the label with the largest size under any distance irregular reflexive k-labeling. On the vertex with the maximum degree, the greatest weight can be attained. Let 𝑒′ be the vertex of greatest degree, with uβ€² is vertex weight being The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 625 𝑀(𝑒′) = Σ𝑒𝑖 β€²βˆˆ 𝑁(𝑒′) 𝑓𝑣(𝑒𝑖 β€²) + Ξ£{π‘’β€²π‘£β€²βˆˆ 𝐸(𝐺)} ≀ (𝑑 βˆ’ 1)(Ξ”) + 𝑑(Ξ”) = Ξ”(2𝑑 βˆ’ 1). Based on Lemma 1, we obtain 𝑝 + 𝛿 βˆ’ 1 ≀ Ξ”(2𝑑 βˆ’ 1). It contradicts with 𝑝 + 𝛿 βˆ’ 1 > Ξ”(2𝑑 βˆ’ 1). Hence 𝑑 β‰  β±°π‘Ÿπ‘’π‘“(𝐺), such that β±°π‘Ÿπ‘’π‘“(𝐺) > ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ > ⌈ Ξ”(2π‘‘βˆ’1) 2Ξ” βŒ‰ = ⌈ 2π‘‘βˆ’1 2 βŒ‰ = 𝑑. Thus, it concludes that β±°π‘Ÿπ‘’π‘“(𝐺) β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ + 1. ∎ We use those lemma and corollary above to determine the distance reflexive strength of path, star, and friendship graph which presented in the following theorems. Theorem 1. Let 𝐺𝑛 be a graph isomorphic with a path graph. For βˆ€π‘› β‰₯ 3, β±°π‘Ÿπ‘’π‘“(𝐺𝑛) = { ⌈ 𝑛 4 βŒ‰ + 1, if 𝑛 ≑ 3,4(π‘šπ‘œπ‘‘ 8) ⌈ 𝑛 4 βŒ‰, if 𝑛 ≑ 0,5,6,7(π‘šπ‘œπ‘‘ 8) where 𝑛 is natural number. Proof. The graph 𝐺𝑛, 𝑛 β‰₯ 3 has a vertex set is 𝑉(G𝑛) = {π‘Žπ‘–; 1 ≀ 𝑖 ≀ 𝑛 } and an edge set 𝐸(G𝑛) = { π‘Žπ‘–π‘Žπ‘–+1; 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1}. As such 𝛿(𝑃𝑛) = 1 and Ξ”(𝑃𝑛) = 2, for 𝑛 ≑ 0,5,6,7 (π‘šπ‘œπ‘‘ 8) base on Lemma 1 we have β±°π‘Ÿπ‘’π‘“(G𝑛) β‰₯ ⌈ 𝑝 + 𝛿 βˆ’ 1 2Ξ” βŒ‰ = ⌈ 𝑛 + 1 βˆ’ 1 2(2) βŒ‰ = ⌈ 𝑛 4 βŒ‰. For 𝑛 ≑ 3 (π‘šπ‘œπ‘‘ 8), it gives β±°π‘Ÿπ‘’π‘“(G𝑛) β‰₯ ⌈ 𝑛 4 βŒ‰ = ⌈ 8𝑏+3 4 βŒ‰ = 2𝑏 + 1 and for 𝑛 ≑ 4 (π‘šπ‘œπ‘‘ 8), it gives β±°π‘Ÿπ‘’π‘“(G𝑛) β‰₯ ⌈ 𝑛 4 βŒ‰ = ⌈ 8𝑏+4 4 βŒ‰ = 2𝑏 + 1, where 𝑏 is natural number. Since ⌈ 𝑛 4 βŒ‰ = 2𝑏 + 1 is odd number, 2(2⌈ 𝑛 4 βŒ‰ βˆ’ 1) will be the biggest vertex label on path. In other hand, 𝑛 > 2(2⌈ 𝑛 4 βŒ‰ βˆ’ 1), this condition satisfies with Corollary 1, such that we have β±°π‘Ÿπ‘’π‘“(G𝑛) β‰₯ ⌈ 𝑛 4 βŒ‰ + 1. Furthermore, we show the upper bound of β±°π‘Ÿπ‘’π‘“(G𝑛) by constructing and defining the function as follows. We give an illustration in Figure 1 to show the upper bound of 𝑃3,𝑃4,𝑃5,𝑃6, and 𝑃7. Now we give the construction of vertex and edge labels in the following steps: (i) Give the label on vertices with the function as follows. For 1 ≀ 𝑖 ≀ 2π‘˜ + 1, 𝑓𝑣(π‘Žπ‘–) = π‘˜ where π‘˜ = { ⌈ 𝑛 4 βŒ‰ + 1, if 𝑛 ≑ 3,4 (π‘šπ‘œπ‘‘ 8) ⌈ 𝑛 4 βŒ‰ , if 𝑛 ≑ 0,5,6,7 (π‘šπ‘œπ‘‘ 8) The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 626 Figure 1. The construction of vertex and edge labels on 𝑃3,𝑃4,𝑃5,𝑃6, and 𝑃7. (ii) Give the label on edges with the function as follows. For 1 ≀ 𝑖 ≀ 2π‘˜, 𝑓𝑒(π‘Žπ‘–π‘Žπ‘–+1) = π‘˜ + 1 βˆ’ ⌈ 𝑖 2 βŒ‰. (iii) The vertex weight, 𝑀(π‘Ž1) = 2π‘˜ and give the vertex weight 𝑀(π‘Žπ‘–) = 4π‘˜ + 2 βˆ’ 𝑖, for 2 ≀ 𝑖 ≀ 2π‘˜. (iv) Determine 𝑀(π‘Ž2π‘˜+1) = 2π‘˜ + 1, and for 2π‘˜ + 2 ≀ 𝑖 ≀ 𝑛 βˆ’ 1,𝑀(π‘Žπ‘–) = 4π‘˜ + 1 βˆ’ 𝑖. (v) For 2π‘˜ + 2 ≀ 𝑖 ≀ 𝑛, label each vertex π‘Žπ‘– with 𝑓𝑣(π‘Žπ‘–βˆ’1) and label edge π‘Žπ‘–βˆ’1π‘Žπ‘– with 𝑀(π‘Žπ‘–βˆ’1) βˆ’ 𝑓𝑣(π‘Žπ‘–βˆ’2) βˆ’ 𝑓𝑒(π‘Žπ‘–βˆ’2π‘Žπ‘–βˆ’1) βˆ’ 𝑓𝑣(π‘Žπ‘–). (vi) If the edge label of point [4] is 0, then label each vertex π‘Žπ‘– with 𝑓𝑣(π‘Žπ‘–βˆ’1) βˆ’ 2 and label edge π‘Žπ‘–βˆ’1π‘Žπ‘– with 𝑀(π‘Žπ‘–βˆ’1) βˆ’ 𝑓𝑣(π‘Žπ‘–βˆ’2) βˆ’ 𝑓𝑒(π‘Žπ‘–βˆ’2π‘Žπ‘–βˆ’1) βˆ’ 𝑓𝑣(π‘Žπ‘–). (vii) The vertex weight of π‘Žπ‘› as follows. 𝑀(π‘Žπ‘›) = 𝑓𝑣(π‘Žπ‘›βˆ’1) + 𝑓𝑒(π‘Žπ‘›βˆ’1π‘Žπ‘›), since 𝑀(π‘Žπ‘›) is the sum of two labels (vertex and edge), then 𝑀(π‘Žπ‘›) must be different from the others. Since the all vertices are distinct, β±°π‘Ÿπ‘’π‘“(G𝑛) β‰₯ π‘˜ and β±°π‘Ÿπ‘’π‘“(G𝑛) ≀ π‘˜, it concludes that β±°π‘Ÿπ‘’π‘“(G𝑛) = π‘˜ for 𝑛 β‰₯ 3. ∎ The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 627 Theorem 2. Let H𝑛 be a graph isomorphic with a star graph. For βˆ€π‘› β‰₯ 3, β±°π‘Ÿπ‘’π‘“(H𝑛) = 𝑛. Proof. The graph 𝐻𝑛,𝑛 β‰₯ 3 has a vertex set 𝑉(H𝑛) = {𝛼,π‘₯𝑖; 1 ≀ 𝑖 ≀ 𝑛} and an edge set 𝐸(H𝑛) = {𝛼 π‘₯𝑖; 1 ≀ 𝑖 ≀ 𝑛}. The star graph has n vertices of degree 1 and one dominant vertex. Since 𝛿(H𝑛) = 1 and the dominant vertex always contributes the constant label to other vertices, the smallest vertex weight is at least 1 and the largest vertex weight on vertices of degree 1 is at least n. Since 𝛼 is the dominant vertex and we can label it with 0, the vertex weight on vertices of degree 1 depends on 1 label (only edge), such that β±°π‘Ÿπ‘’π‘“ (H𝑛) β‰₯ 𝑛. Since Ξ”(H𝑛) = 𝑛, then β±°π‘Ÿπ‘’π‘“(H𝑛) β‰₯ 𝑛 β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ and satisfies the Lemma 1. Furthermore, we show the upper bound of β±°π‘Ÿπ‘’π‘“(H𝑛) by defining the following function: 𝑓𝑣(𝛼) = 𝑓𝑣(π‘₯𝑖) = 0 𝑓𝑒(𝛼 π‘₯𝑖) = 𝑖: 1 ≀ 𝑖 ≀ 𝑛 By those vertex and edge labels, we have the vertex weight as follows. 𝑀(𝛼) = βˆ‘[𝑓𝑣(π‘₯𝑖) + 𝑓𝑒(𝛼 π‘₯𝑖)] 𝑛 𝑖=1 = 𝑛(𝑛 + 1) 2 𝑀(π‘₯𝑖) = 𝑖: 1 ≀ 𝑖 ≀ 𝑛 We know that all vertex weights of graph H𝑛 are distinct and it gives a distance reflexive strength on the star graph. Since β±°π‘Ÿπ‘’π‘“(H𝑛) ≀ 𝑛 and β±°π‘Ÿπ‘’π‘“(H𝑛) β‰₯ 𝑛, it concludes that β±°π‘Ÿπ‘’π‘“(H𝑛) = 𝑛 for 𝑛 β‰₯ 3. ∎ Theorem 3. Let M𝑛 be a graph isomorphic with friendship graph (𝐹𝑛). For βˆ€π‘› β‰₯ 3, β±°π‘Ÿπ‘’π‘“(M𝑛) = { ⌈ 2n + 1 3 βŒ‰ + 1, if 𝑛 ≑ 1(π‘šπ‘œπ‘‘ 3) ⌈ 2𝑛 + 1 3 βŒ‰ , otherwise where 𝑛 is natural number. Proof. The graph M𝑛,𝑛 β‰₯ 3 has a vertex set 𝑉(M𝑛) = {π‘Ž,𝑒𝑖,𝑣𝑖; 1 ≀ 𝑖 ≀ 𝑛} and an edge set 𝐸(M𝑛) = {π‘Žπ‘’π‘–; 1 ≀ 𝑖 ≀ 𝑛} βˆͺ {π‘Žπ‘£π‘–; 1 ≀ 𝑖 ≀ 𝑛} βˆͺ {𝑒𝑖𝑣𝑖; 1 ≀ 𝑖 ≀ 𝑛}. The graph M𝑛 has 2𝑛 vertices of degree 2 and one dominant vertex. Since 𝛿(M𝑛) = 2 and the dominant vertex always contributes the constant label to other vertices, the smallest vertex weight is at least 2 and the biggest vertex weight on vertices of degree 2 is at least 2𝑛 + 1. Since a is the dominant vertex and we can label it with 0, the vertex weight on vertices of degree 2 depends on 3 labels (2 edges and 1 adjacent vertex). Since label of vertex must be an even number, in such a case that β±°π‘Ÿπ‘’π‘“(M𝑛) β‰₯ ⌈ 2𝑛+1 3 βŒ‰. Since Ξ”(M𝑛) = 2𝑛, then β±°π‘Ÿπ‘’π‘“(M𝑛) β‰₯ ⌈ 2𝑛+1 3 βŒ‰ β‰₯ ⌈ 𝑝+π›Ώβˆ’1 2Ξ” βŒ‰ and satisfies the Lemma 1. For 𝑛 ≑ 1(π‘šπ‘œπ‘‘ 3), there is an integer s such that 2𝑛 = 2(3𝑠 + 1) = 6𝑠 + 2, then ⌈ 2𝑛+1 3 βŒ‰ = ⌈ 6𝑠+3 3 βŒ‰ = 2𝑠 + 1. The vertex weight 2𝑛 + 1 = 6𝑠 + 3 is the sum of 3 labels where each label is not more than 2𝑠 + 1, such that 2𝑛 + 1 = 6𝑠 + 3 = (2𝑠 + 1) + (2𝑠 + 1) + (2𝑠 + 1). As such the vertex label should be an even number, for 𝑛 ≑ 1(π‘šπ‘œπ‘‘ 3) we get β±°π‘Ÿπ‘’π‘“(M𝑛) β‰₯ ⌈ 2𝑛+1 3 βŒ‰ + 1. Furthermore, we show the upper bound of β±°π‘Ÿπ‘’π‘“(M𝑛) by defining the following function: The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 628 𝑓𝑣(π‘Ž) = 0 𝑓(𝑒𝑖) = { 0, if 𝑖 = 1 2⌈ 𝑖 βˆ’ 1 3 βŒ‰, if 2 ≀ 𝑖 ≀ 𝑛 𝑓𝑣(𝑒𝑖) = 𝑓𝑣(𝑣𝑖) 𝑓𝑒(𝑒𝑖𝑣𝑖) = { 2 3 𝑖, if 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 3) 1 3 (2𝑖 + 1), if 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 3) 1 3 (2𝑖 βˆ’ 1), if 𝑖 ≑ 2 (π‘šπ‘œπ‘‘ 3) 𝑓𝑒(π‘Žπ‘’π‘–) = 𝑓𝑒(𝑒𝑖𝑣𝑖) 𝑓𝑒(π‘Žπ‘£π‘–) = { 2 3 𝑖 + 1, if 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 3) 1 3 (2𝑖 + 1) + 1, if 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 3) 1 3 (2𝑖 βˆ’ 1) + 1, if 𝑖 ≑ 2 (π‘šπ‘œπ‘‘ 3) By those vertex and edge labels, we have the vertex weight as follows. 𝑀(π‘Ž) = βˆ‘[𝑓𝑣(𝑒𝑖) + 𝑓𝑣(𝑣𝑖) + 𝑓𝑒(π‘Žπ‘’π‘–) + 𝑓𝑒(π‘Žπ‘£π‘–)] 𝑛 𝑖=1 𝑀(𝑒𝑖) = 2𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 𝑀(𝑣𝑖) = 2𝑖 + 1 ∢ 1 ≀ 𝑖 ≀ 𝑛 We know that the all vertex weights of graph M𝑛 are distinct. Thus, it gives a distance reflexive strength on M𝑛 for 𝑛 β‰₯ 3. ∎ CONCLUSIONS We determined the lower bound and β±°π‘Ÿπ‘’π‘“(𝐺), where 𝐺 be several graphs which isomorphic with path, star, and friendship graphs. Since every graph has different characteristics, the distance reflexive strength varies. Based on the characteristics of these graphs, we can characterize the distance reflexive strength. Characterizing the distance reflexive strength is a difficult problem that requires extensive research. As a result, we present the following open problems. Open problem 1. Find out the exact value of β±°π‘Ÿπ‘’π‘“(𝑃𝑛) for 𝑛 ≑ 1,2 (π‘šπ‘œπ‘‘ 8) and any graphs apart from those families. Open problem 2. Characterize the distance reflexive strength of graph. The Distance Irregular Reflexive k-Labeling of Graphs Ika Hesti Agustin 629 REFERENCES [1] G. Chartrand, L. Lesniak, P. Zhang, Graphs & Digraphs, sixth ed., Taylor & Francis Group, Boca Raton, New York, 2016. [2] Joseph A. Gallian. 2017. A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatorics, 20:1-432. [3] Ali Ahmad and M. Baca. 2013. On Vertex Irregular Total Labelings. {Ars Combinatoria}, 112:129-139. [4] Haryanti, A., D. Indriati, dan T. S. Martini. 2019. On the Total Vertex Irregularity Strength of Firecracker Graph. Journal of Physics: Conference Series, 1211:1-9. [5] Nurdin, E. T. Baskoro, A. N. M. Salman, dan N. N. Gaos. 2010. On the Total Vertex Irregularity Strength of Trees. Discrete Mathematics, 310:3043-3048. [6] Ramdani, R., A. N. M. salman, dan H. Assiyatun. 2015. On the Total Irregularity Strength of Regular Graphs. Journal Mathematics Fundamental Science, 47 (3):281- 295. [7] Slamin, Dafik, and W. Winnona. 2011. Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs. The Electronic Journal of Combinatorics, 2012:1-9. [8] Susilawati, E. T. Baskoro, dan R. Simanjuntak. 2015. Total Vertex-Irregularity Labelings for Subdivision of Several Classes of Trees. Procedia Computer Science, 74:112-117. [9] Susilawati, S., E. T. Baskoro, dan R. Simanjuntak. 2018. Total Vertex Irregularity Strength of Trees with Maximum Degree Five. Electronic Journal Graph Theory Apply, 6 (2):250-257. [10] Agustin I.H, Imam Utoyo, Dafik, N. Mohanapriya, Ismail Naci Cangul, 2020. On Vertex irregular reflexive labeling of cycle and friendship graph. ASCM Published By JANGJEON Mathematical Society. (submitted) [11] Agustin I H, Dafik, Marsidi , Albirri E R. On the total H-irregularity strength of graphs: A new notion. Journal of Physics: Conf. Series, 855 (2017) 012004. [12] Agustin I.H, Imam Utoyo, Dafik, M.Venkatachalam, and Surahmat. 2020. On the construction of the reflexive vertex k-labeling of any graph with pendant vertex. (submitted) [13] R Nisviasari, Dafik and I H Agustin. The total H-irregularity strength of triangular ladder and grid graphs. Journal of Physics: Conf. Series, 1211 (2019) 012005. [14] Tanna, D., J. Ryan, A. Semanǐcov́a-Fěnov̌ćikov́a dan M.Baca. 2018. Vertex Irregular Reflexive Labeling of Prisms and Wheels. AKCE International Journal of Graphs and Combinatorics. [15] Ba\u{c}a, M., S. Jendrol, M. Miller, dan J. Ryan. 2007. On Irregular Total Labelings. Discrete Mathematics, 307:1378-1388.. [16] Slamin. 2017. ON DISTANCE IRREGULAR LABELING OF GRAPHS. Far East Journal of Mathematical Sciences (FJMS). Volume 102, Number 5, 2017, Pages 919-932. http://dx.doi.org/10.17654/MS102050919.