Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1, 481-492 2025 Publisher: Learning Gate DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate Β© 2025 by the authors; licensee Learning Gate History: Received: 28 November 2024; Revised: 26 December 2024; Accepted: 3 January 2025; Published: 9 January 2025 * Correspondence: nurdin1701@unhas.ac.id Development of navigation systems in optimal utilization of transportation routes in topology helm graphs using graph labeling Nurdin Hinding1*, Nurtiti Sunusi2, Yuni Syahrani3, Harry Maulana Buhari4 1Graph Theory Research Group, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Indonesia, 90245; nurdin1701@unhas.ac.id (N.H.). 2Stochastic Modeling Research Group Department of Statistics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Indonesia, 90245. 3,4Undergraduate’s Program in Mathematics, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Indonesia, 90245. Abstract: Graph theory plays a vital role in many fields. The graph concept models many relationships and processes in physical, biological, social, transportation, and information systems. One of the essential fields in graph theory is graph labeling. Graph labeling is widely used in various applications such as coding theory, x-ray crystallography, radar, astronomy, circuit design, addressing of communication networks, database management, etc. In this research, we will examine a form of topology of transportation routes based on graph labeling. The topology of the transportation routes that will be studied is a modified of helm graph. This research aims to determine the total vertex irregularity strength of the modified helm graph denoted by 𝐻𝑛 for 𝑛 β‰₯ 3. The lower bound is obtained based on the properties of the graph 𝐻𝑛 using the existing supporting theorem. The upper bound is obtained by labeling the vertices and edges of the graph 𝐻𝑛 in some simple cases. From this labeling, total vertex irregular labeling will be constructed for any 𝑛. Based on the results of this research, the total vertex irregularity strength of the graph 𝐻𝑛 is 𝑑𝑣𝑠(𝐻𝑛) = ⌈ 3+3𝑛 4 βŒ‰ with 𝑛 β‰₯ 3. Keywords: Helm graph, Irregular labeling, Irregular strength, Transportation system. 1. Introduction One of the problems encountered in big cities, such as Jakarta, Surabaya, and Makassar, is the increase in the number of vehicles, which needs to be balanced with road infrastructure development. This will lead to the appearance of congestion points on the highway. Various attempts have been made to resolve this problem. One way is to theoretically study transportation networks using graph models, especially labeled graphs. A labeling of a graph is generally defined as a function of the subsets of elements from G to a set of numbers, typically a set of positive or non-negative integers, which fulfill certain conditions. Several types of graph labeling have been studied, including graceful labeling, magic labeling, anti-magic labeling, and irregular labeling. One good source of information about graph labeling is "A Dynamic Survey of Graph Labeling" [1]. Our research topic is focused on irregular labeling. The concept of irregular labeling on a graph was first introduced by Chartrand, et al. [2]. In Bac Μ†a, et al. [3] introduced another irregular labeling based on total labeling, namely edge irregular total labeling and vertex total irregular labeling. Determination of the vertex irregularity strength, the total edge irregularity strength, and the total vertex irregularity strength of a graph can only be done partially for some land transportation network 482 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate models. Several models of transportation networks are still open issues that that need to be resolved entirely. Our previous research on labeling total irregularities of vertices has been studied for graphs in general. This research produces a lower bound on the total vertex irregularity strength, but the exact value has yet to be obtained. In the same paper, a conjecture is given, which has yet to be resolved [4]. 2. Total Vertex Irregular Graph Historically, Leonhard Euler used graphs to solve the Konigsberg bridge problem in 1736. One of the main points of discussion in graphs is graph labeling. Graph labeling is a function that maps the elements of a graph (vertices or edges) to positive or non-negative integers. If the labeling domain is a set of vertices, it is called point labeling. If the labeling domain is an edge set, then it is called an edge label, and if the labeling domain is a set of vertices and edges, then it is called total labeling [4]. Definition 1. Bac Μ†a, et al. [3] The vertex weight 𝑣 on the total labeling 𝑓 is the vertex label 𝑣 added to the sum of all the edges labels incident to 𝑣, i.e 𝑀𝑑(𝑣) = 𝑓(𝑣) +βˆ‘ 𝑓(𝑒𝑣) π‘’π‘£βˆˆπΈ . Definition 2. Bac Μ†a, et al. [3]Let 𝐺(𝑉, 𝐸) be a simple graph. A total labeling 𝑓: 𝑉 βˆͺ 𝐸 β†’ {1,2,… , π‘˜} is a total vertex irregular k-labeling on 𝐺 if for every two different vertices in 𝑉 satisfy 𝑀𝑑(π‘₯) β‰  𝑀𝑑(𝑦), where 𝑀𝑑(π‘₯) = 𝑓(π‘₯) + βˆ‘ 𝑓(π‘₯𝑒)π‘₯π‘’βˆˆπΈ . Definition 3. Bac Μ†a, et al. [3] Let 𝐺(𝑉, 𝐸) be a simple graph. The total vertex irregularity strength of 𝐺, dinoted by 𝑑𝑣𝑠(𝐺) is a minimum positive integer number π‘˜ such that 𝐺 have a total irregular π‘˜ βˆ’ π‘™π‘Žπ‘π‘’π‘™π‘™π‘–π‘›π‘”. Theorem 1. Nurdin [4] Let 𝐺(𝑉, 𝐸) be a connected simple graph on 𝑛𝑖 vertices of degree 𝑖 (𝑖 = 𝛿, 𝛿 + 1, 𝛿 + 2,… , βˆ†), where 𝛿 and βˆ† are minimum and maximum degree on 𝐺, respectively. Then 𝑑𝑣𝑠(𝐺) β‰₯ π‘šπ‘Žπ‘₯ {⌈ 𝛿 + 𝑛𝛿 𝛿 + 1 βŒ‰ , ⌈ 𝛿 + 𝑛𝛿 + 𝑛𝛿+1 𝛿 + 2 βŒ‰ , … , ⌈ 𝛿 + βˆ‘ 𝑛𝑖 βˆ† 𝑖=𝛿 βˆ† + 1 βŒ‰}. Several studies have been carried out using total irregular labeling. Aarthi [5], reminded the total vertex irregularity strength in the helm graph Aarthi [5]. Ahmad, et al. [6] determined the total vertex irregularity strength in the composite isomorphic helm graph Ahmad, et al. [6]. Indriati Widodo, et al. [7] determined generalized helm graphs' total vertex irregularity strength [7]. Previously, Nurdin [4] had determined a caterpillar graph’s total vertex irregularity strength Nurdin [4]. Likewise, in Hinding, et al. [8] determined the irregularity strength of a diamond graph [8]. In this regard, Siddiqui Nurdin and Baskoro [9] examined the total edge irregularity strength of the disjoint union of the helm graph [9]. However, researchers have yet to determine the total irregularity strength of the modified helm graph vertices. Therefore, this study evaluates the total irregularity strength of the modified helm graph. 3. Results The graph for which the total vertex irregularity strength will be determined in this paper is the helm g, which is modified so that the vertices are only degrees 3 or 5. Formally, the graph in question is as follows. Definition 4. Let 𝐻𝑛 be a helm graph. The ℋ𝑛 is a graph constructed from helm graph 𝐻𝑛 by removing the central vertex and adding 2𝑛 vertices, namely 𝑀1, 𝑣1, 𝑀2, 𝑣2, … , 𝑀𝑛, 𝑣𝑛 such that 1. π‘₯1𝑀𝑛, π‘₯𝑖+1𝑀𝑖 for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1; 2. π‘₯𝑛𝑣1, π‘₯π‘–βˆ’1𝑣𝑖 for𝑖 = 2, 3, … , 𝑛; 483 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate 3. 𝑦𝑖𝑀𝑖, 𝑦𝑖𝑣𝑖, 𝑀𝑖𝑣𝑖 for 𝑖 = 2, 3, … , 𝑛 The set of vertices and set of edges of the graph 𝐻𝑛 are as follows 𝑉(ℋ𝑛) = {π‘₯𝑖, 𝑦𝑖 , 𝑀𝑖 , 𝑣𝑖| 𝑖 = 1, 2,… , 𝑛} and 𝐸(ℋ𝑛) = {π‘₯1𝑀𝑛, π‘₯𝑛𝑣𝑖} βˆͺ {π‘₯𝑖+1𝑀𝑖| 𝑖 = 1, 2, … , 𝑛 βˆ’ 1} βˆͺ {π‘₯π‘–βˆ’1𝑣𝑖| 𝑖 = 2, 3, … , 𝑛} βˆͺ {π‘₯𝑖𝑦𝑖, 𝑦𝑖𝑀𝑖 , 𝑦𝑖𝑣𝑖, 𝑀𝑖𝑣𝑖| 𝑖 = 1, 2, … , 𝑛} βˆͺ {π‘₯1π‘₯2, π‘₯2π‘₯3, … , π‘₯π‘›βˆ’1π‘₯𝑛, π‘₯𝑛π‘₯1}. In determining the total vertex irregularity strength of the modified helm graph, the lower bound will be analyzed first based on the properties of the graph 𝓗𝒏 and use Theorem 1. Meanwhile, to determine the upper bound, a total vertex irregular labeling will be constructed Theorem 2. Let 𝓗𝒏 be modified helm graph. For 𝒏 β‰₯ πŸ‘ and 𝒏 is a positive integer number, then 𝒕𝒗𝒔(𝓗𝒏) β‰₯ ⌈ πŸ‘ + πŸ‘π’ πŸ’ βŒ‰. Next, to prove that 𝒕𝒗𝒔(𝓗𝒏) ≀ ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰ , a total vertex irregular labeling is constructed as follows. There are four cases in the construction of a total vertex labeling on 𝓗𝒏. Case 1. For 𝑛 = 4𝑑 βˆ’ 1, where 𝑑 is some positive integer. The total vertex labeling 𝑓 is 𝑓(π‘₯1) = 1. 𝑓(π‘₯2) = 𝑠 + 1. 𝑓(π‘₯𝑖) = { 3𝑖+3 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+3 4 , 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 4. 𝑓(π‘₯𝑖) = { 3𝑖+4 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+4 4 , 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 3. 𝑓(π‘₯𝑖) = { 3𝑖+5 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+5 4 , 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 2. 𝑓(π‘₯𝑖) = { 3𝑖+6 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+6 4 , 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 1. 𝑓(π‘₯𝑛) = 3𝑛+3 4 . 𝑓(𝑦𝑖) = 1, 𝑖 = 1, 2,… , 𝑛. 𝑓(𝑀1) = 1. 𝑓(𝑀2) = { 2, 𝑠 = 1 1, 𝑠 = 2, 3,… 𝑓(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+3 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 4. 484 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate 𝑓(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+4 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 3. 𝑓(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+5 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 2. 𝑓(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+6 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 1. 𝑓(𝑀𝑛) = 2𝑠 + 1. 𝑓(𝑣𝑖) = 3𝑠, 𝑖 = 1, 2, … , 𝑛. 𝑓(π‘₯1π‘₯2) = 𝑓(π‘₯2π‘₯3) = 𝑓(π‘₯π‘–βˆ’1π‘₯𝑖) = 𝑓(π‘₯𝑖π‘₯𝑖+1) = 𝑓(π‘₯π‘›βˆ’1π‘₯𝑛) = 𝑓(π‘₯𝑛π‘₯1) = 3𝑠. 𝑓(π‘₯1𝑦1) = 𝑓(π‘₯2𝑦2) = 1. 𝑓(π‘₯𝑖𝑦𝑖) = { 𝑖+1 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 4. 𝑖 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 3. π‘–βˆ’1 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 2. π‘–βˆ’2 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 1. 𝑓(π‘₯𝑛𝑦𝑛) = 𝑛+1 4 . 𝑓(π‘₯2𝑀1) = 2𝑠. 𝑓(π‘₯3𝑀2) = { 3𝑠 βˆ’ 1, 𝑠 = 1 2𝑠 + 1, 𝑠 = 2, 3, … 𝑓(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+3 4 βˆ’ 1, 𝑖 ≀ 𝑠 3𝑠 βˆ’ 1, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 4. 𝑓(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+4 4 βˆ’ 1, 𝑖 ≀ 𝑠 3𝑠 βˆ’ 1, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,…𝑛 βˆ’ 3. 𝑓(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+5 4 βˆ’ 1, 𝑖 ≀ 𝑠 3𝑠 βˆ’ 1, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 2. 𝑓(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+6 4 βˆ’ 1, 𝑖 ≀ 𝑠 3𝑠 βˆ’ 1, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 1. 𝑓(π‘₯1𝑀𝑛) = 𝑓(π‘₯π‘›π‘€π‘›βˆ’1) = 3𝑛+3 4 βˆ’ 1 = 3𝑠 βˆ’ 1. 𝑓(π‘₯π‘–π‘€π‘–βˆ’1) = { 2𝑠 + 𝑖 βˆ’ 2, 𝑖 ≀ 𝑠 3𝑠 βˆ’ 1, 𝑖 > 𝑠 𝑓(π‘₯1𝑣2) = 𝑓(π‘₯𝑖𝑣𝑖+1) = 𝑓(π‘₯π‘›βˆ’1𝑣𝑛) = 𝑓(π‘₯𝑛𝑣1) = 3𝑠. 𝑓(𝑦𝑖𝑀𝑖) = 1, 𝑖 = 1, 2,… , 𝑛. 𝑓(𝑦1𝑣1) = 1. 𝑓(𝑦2𝑣2) = 2. 485 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate 𝑓(𝑦𝑖𝑣𝑖) = { 3𝑖+3 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 4, 3𝑖+4 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 3, 3𝑖+5 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 2, 3𝑖+6 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 1, 𝑓(𝑦𝑛𝑣𝑛) = 3𝑛+3 4 . 𝑓(𝑀1𝑣1) = 𝑓(𝑀2𝑣2) = 2𝑠 + 1. 𝑓(𝑀𝑖𝑣𝑖) = { 𝑖+1 4 + 2𝑠, 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 4, 𝑖 4 + 2𝑠, 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 3, π‘–βˆ’1 4 + 2𝑠, 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 2, π‘–βˆ’2 4 + 2𝑠, 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 1, 𝑓(𝑀𝑛𝑣𝑛) = 𝑛 + 1 4 + 2𝑠. Based on this function, it can be shown that all vertices weights are different and the maximum value of the function is ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. This shows that 𝑓: 𝑉 βˆͺ 𝐸 β†’ {1, 2, 3,β‹― , ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰} is a function a total vertex irregular π‘˜ βˆ’ π‘™π‘Žπ‘π‘’π‘™π‘™π‘–π‘›π‘” where π‘˜ = ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. Thus it is obtained that 𝒕𝒗𝒔(𝓗𝒏) ≀ ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰ for 𝑛 = 4𝑑 βˆ’ 1, where 𝑑 is some positive integer. Case 2. For 𝑛 = 4𝑑, where 𝑑 is some positive integer. The total vertex labeling 𝑔 is 𝑔(π‘₯1) = 1. 𝑔(π‘₯2) = 𝑠 + 1. 𝑔(π‘₯𝑖) = { 3𝑖+3 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+3 4 , 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1. 𝑔(π‘₯𝑖) = { 3𝑖+4 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+4 4 , 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4. 𝑔(π‘₯𝑖) = { 3𝑖+5 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+5 4 , 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 3. 𝑔(π‘₯𝑖) = { 3𝑖+6 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+6 4 , 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2. 𝑔(π‘₯𝑛) = 3𝑛+4 4 . 𝑔(𝑦𝑖) = 1, 𝑖 = 1, 2, … , 𝑛. 𝑔(𝑀1) = 1. 𝑔(𝑀2) = { 2, 𝑠 = 1 1, 𝑠 = 2, 3, … 486 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate 𝑔(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+3 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1. 𝑔(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+4 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4. 𝑔(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+5 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 3. 𝑔(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+6 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2. 𝑔(𝑀𝑛) = 2𝑠 + 2. 𝑔(𝑣𝑖) = 3𝑠 + 1, 𝑖 = 1, 2, … , 𝑛. 𝑔(π‘₯1π‘₯2) = 𝑔(π‘₯π‘›βˆ’1π‘₯𝑛) = 3𝑠. 𝑔(π‘₯2π‘₯3) = 𝑔(π‘₯𝑛π‘₯1) = 3𝑠 + 1. 𝑔(π‘₯𝑖π‘₯𝑖+1) = { 3𝑠, { 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1. 𝑖 = 5, 9, 13,…𝑛 βˆ’ 3. 3𝑠 + 1, { 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4. 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2. 𝑔(π‘₯π‘–βˆ’1π‘₯𝑖) = { 3𝑠 + 1, { 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1. 𝑖 = 5, 9, 13,…𝑛 βˆ’ 3. 3𝑠, { 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4. 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2. 𝑔(π‘₯1𝑦1) = 𝑓(π‘₯2𝑦2) = 1. 𝑔(π‘₯𝑖𝑦𝑖) = { 𝑖+1 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1. 𝑖 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4. π‘–βˆ’1 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 3. π‘–βˆ’2 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2. 𝑔(π‘₯𝑛𝑦𝑛) = 𝑛 4 . 𝑔(π‘₯2𝑀1) = 2𝑠 + 1. 𝑔(π‘₯3𝑀2) = { 3𝑠, 𝑠 = 1 2𝑠 + 2, 𝑠 = 2, 3,… 𝑔(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+3 4 , 𝑖 ≀ 𝑠 3𝑠, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1. 𝑔(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+4 4 , 𝑖 ≀ 𝑠 3𝑠, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4. 𝑔(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+5 4 , 𝑖 ≀ 𝑠 3𝑠, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 3. 𝑔(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+6 4 , 𝑖 ≀ 𝑠 3𝑠, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2. 𝑔(π‘₯1𝑀𝑛) = 𝑔(π‘₯π‘›π‘€π‘›βˆ’1) = 3𝑛+4 4 βˆ’ 1 = 3𝑠. 487 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate 𝑔(π‘₯π‘–π‘€π‘–βˆ’1) = { 2𝑠 + 𝑖 βˆ’ 1, 𝑖 ≀ 𝑠 3𝑠, 𝑖 > 𝑠 𝑔(π‘₯1𝑣2) = 𝑔(π‘₯𝑖𝑣𝑖+1) = 𝑔(π‘₯π‘›βˆ’1𝑣𝑛) = 𝑔(π‘₯𝑛𝑣1) = 3𝑠 + 1 𝑔(𝑦𝑖𝑀𝑖) = 1, 𝑖 = 1, 2, … , 𝑛. 𝑔(𝑦1𝑣1) = 1. 𝑔(𝑦2𝑣2) = 2. 𝑔(𝑦𝑖𝑣𝑖) = { 3𝑖+3 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1, 3𝑖+4 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4, 3𝑖+5 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 3, 3𝑖+6 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2, 𝑔(𝑦𝑛𝑣𝑛) = 3𝑛+4 4 . 𝑔(𝑀1𝑣1) = 𝑔(𝑀2𝑣2) = 2𝑠 + 1. 𝑔(𝑀𝑖𝑣𝑖) = { 𝑖+1 4 + 2𝑠, 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 1, 𝑖 4 + 2𝑠, 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 4, π‘–βˆ’1 4 + 2𝑠, 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 3, π‘–βˆ’2 4 + 2𝑠, 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 2, 𝑔(𝑀𝑛𝑣𝑛) = 𝑛 4 + 2𝑠. Based on this function, it can be shown that all vertices weights are different and the maximum value of the function is ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. This shows that 𝑔: 𝑉 βˆͺ 𝐸 β†’ {1, 2, 3,β‹― , ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰} is a function a total vertex irregular π‘˜ βˆ’ π‘™π‘Žπ‘π‘’π‘™π‘™π‘–π‘›π‘” where π‘˜ = ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. Thus it is obtained that 𝒕𝒗𝒔(𝓗𝒏) ≀ ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰ for 𝒏 = πŸ’π’•, where 𝒕 is some positive integer. Case 3. For 𝑛 = 4𝑑 + 1, where 𝑑 is some positive integer. The total vertex labeling β„Ž is β„Ž(π‘₯1) = 1. β„Ž(π‘₯2) = 𝑠 + 1. β„Ž(π‘₯𝑖) = { 3𝑖+3 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+3 4 , 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 2. β„Ž(π‘₯𝑖) = { 3𝑖+4 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+4 4 , 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 1. β„Ž(π‘₯𝑖) = { 3𝑖+5 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+5 4 , 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 4. β„Ž(π‘₯𝑖) = { 3𝑖+6 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+6 4 , 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 3. β„Ž(π‘₯𝑛) = 3𝑛+5 4 . 488 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate β„Ž(𝑦𝑖) = 1, 𝑖 = 1, 2, … , 𝑛. β„Ž(𝑀1) = 1. β„Ž(𝑀2) = { 2, 𝑠 = 1 1, 𝑠 = 2, 3, … β„Ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+3 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 2. β„Ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+4 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 1. β„Ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+5 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 4. β„Ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+6 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 3. β„Ž(𝑀𝑛) = 2𝑠 + 3. β„Ž(𝑣𝑖) = 3𝑠 + 2, 𝑖 = 1, 2,… , 𝑛. β„Ž(π‘₯1π‘₯2) = β„Ž(π‘₯2π‘₯3) = β„Ž(π‘₯π‘–βˆ’1π‘₯𝑖) = β„Ž(π‘₯𝑖π‘₯𝑖+1) = β„Ž(π‘₯π‘›βˆ’1π‘₯𝑛) = β„Ž(π‘₯𝑛π‘₯1) = 3𝑠 + 1. β„Ž(π‘₯1𝑦1) = β„Ž(π‘₯2𝑦2) = 1. β„Ž(π‘₯𝑖𝑦𝑖) = { 𝑖+1 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 2. 𝑖 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 1. π‘–βˆ’1 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 4. π‘–βˆ’2 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 3. β„Ž(π‘₯𝑛𝑦𝑛) = π‘›βˆ’1 4 . β„Ž(π‘₯2𝑀1) = 2𝑠 + 2. β„Ž(π‘₯3𝑀2) = { 3𝑠 + 1, 𝑠 = 1 2𝑠 + 3, 𝑠 = 2, 3, … β„Ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+3 4 + 1, 𝑖 ≀ 𝑠 3𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 2. β„Ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+4 4 + 1, 𝑖 ≀ 𝑠 3𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 1. β„Ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+5 4 + 1, 𝑖 ≀ 𝑠 3𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 4. β„Ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+6 4 + 1, 𝑖 ≀ 𝑠 3𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 3. β„Ž(π‘₯1𝑀𝑛) = β„Ž(π‘₯π‘›π‘€π‘›βˆ’1) = 3𝑛+5 4 βˆ’ 1 = 3𝑠 + 1. β„Ž(π‘₯π‘–π‘€π‘–βˆ’1) = { 2𝑠 + 𝑖, 𝑖 ≀ 𝑠 3𝑠 + 1, 𝑖 > 𝑠 β„Ž(π‘₯1𝑣2) = β„Ž(π‘₯𝑖𝑣𝑖+1) = β„Ž(π‘₯π‘›βˆ’1𝑣𝑛) = β„Ž(π‘₯𝑛𝑣1) = 3𝑠 + 2. β„Ž(𝑦𝑖𝑀𝑖) = 1, 𝑖 = 1, 2, … , 𝑛. β„Ž(𝑦1𝑣1) = 1. 489 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate β„Ž(𝑦2𝑣2) = 2. β„Ž(𝑦𝑖𝑣𝑖) = { 3𝑖+3 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 2, 3𝑖+4 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 1, 3𝑖+5 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 4, 3𝑖+6 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 3, β„Ž(𝑦𝑛𝑣𝑛) = 3𝑛+5 4 . β„Ž(𝑀1𝑣1) = β„Ž(𝑀2𝑣2) = 2𝑠 + 1. β„Ž(𝑀𝑖𝑣𝑖) = { 𝑖+1 4 + 2𝑠, 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 2, 𝑖 4 + 2𝑠, 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 1, π‘–βˆ’1 4 + 2𝑠, 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 4, π‘–βˆ’2 4 + 2𝑠, 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 3, β„Ž(𝑀𝑛𝑣𝑛) = 𝑛 βˆ’ 1 4 + 2𝑠 Based on this function, it can be shown that all vertices weights are different and the maximum value of the function is ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. This shows that β„Ž: 𝑉 βˆͺ 𝐸 β†’ {1, 2, 3,β‹― , ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰} is a function a total vertex irregular π‘˜ βˆ’ π‘™π‘Žπ‘π‘’π‘™π‘™π‘–π‘›π‘” where π‘˜ = ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. Thus it is obtained that 𝒕𝒗𝒔(𝓗𝒏) ≀ ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰ for 𝑛 = 4𝑑 + 1, where 𝑑 is some positive integer. Case 4. For 𝑛 = 4𝑑 + 2, where 𝑑 is some positive integer. The total vertex labeling π‘ž is π‘ž(π‘₯1) = 1. π‘ž(π‘₯2) = 𝑠 + 1. π‘ž(π‘₯𝑖) = { 3𝑖+3 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+3 4 , 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3. π‘ž(π‘₯𝑖) = { 3𝑖+4 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+4 4 , 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2. π‘ž(π‘₯𝑖) = { 3𝑖+5 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+5 4 , 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 1. π‘ž(π‘₯𝑖) = { 3𝑖+6 4 + 𝑠 βˆ’ 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑖+6 4 , 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4. π‘ž(π‘₯𝑛) = 3𝑛+6 4 . π‘ž(𝑦𝑖) = 1, 𝑖 = 1, 2, … , 𝑛. π‘ž(𝑀1) = 1. π‘ž(𝑀2) = { 2, 𝑠 = 1 1, 𝑠 = 2, 3,… 490 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate π‘ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+3 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3. π‘ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+4 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2. π‘ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+5 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 1. π‘ž(𝑀𝑖) = { 1, 𝑖 ≀ 𝑠 3𝑖+6 4 βˆ’ 𝑠 + 1, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4. π‘ž(𝑀𝑛) = 2𝑠 + 4. π‘ž(𝑣𝑖) = 3𝑠 + 3, 𝑖 = 1, 2, … , 𝑛. π‘ž(π‘₯1π‘₯2) = π‘ž(π‘₯π‘›βˆ’1π‘₯𝑛) = 3𝑠 + 1. π‘ž(π‘₯2π‘₯3) = π‘ž(π‘₯𝑛π‘₯1) = 3𝑠 + 2. π‘ž(π‘₯𝑖π‘₯𝑖+1) = { 3𝑠 + 1, { 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3. 𝑖 = 5, 9, 13,…𝑛 βˆ’ 1. 3𝑠 + 2, { 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2. 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4. π‘ž(π‘₯π‘–βˆ’1π‘₯𝑖) = { 3𝑠 + 2, { 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3. 𝑖 = 5, 9, 13,…𝑛 βˆ’ 1. 3𝑠 + 1, { 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2. 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4. π‘ž(π‘₯1𝑦1) = π‘ž(π‘₯2𝑦2) = 1. π‘ž(π‘₯𝑖𝑦𝑖) = { 𝑖+1 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3. 𝑖 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2. π‘–βˆ’1 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 1. π‘–βˆ’2 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4. π‘ž(π‘₯𝑛𝑦𝑛) = π‘›βˆ’2 4 . π‘ž(π‘₯2𝑀1) = 2𝑠 + 3. π‘ž(π‘₯3𝑀2) = { 3𝑠 + 2, 𝑠 = 1 2𝑠 + 4, 𝑠 = 2, 3, … π‘ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+3 4 + 2, 𝑖 ≀ 𝑠 3𝑠 + 2, 𝑖 > 𝑠 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3. π‘ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+4 4 + 2, 𝑖 ≀ 𝑠 3𝑠 + 2, 𝑖 > 𝑠 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2. π‘ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+5 4 + 2, 𝑖 ≀ 𝑠 3𝑠 + 2, 𝑖 > 𝑠 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 1. π‘ž(π‘₯𝑖+1𝑀𝑖) = { 2𝑠 + 3𝑖+6 4 + 2, 𝑖 ≀ 𝑠 3𝑠 + 2, 𝑖 > 𝑠 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4. π‘ž(π‘₯1𝑀𝑛) = π‘ž(π‘₯π‘›π‘€π‘›βˆ’1) = 3𝑛+6 4 βˆ’ 1 = 3𝑠 + 2. 491 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 1: 481-492, 2025 DOI: 10.55214/25768484.v9i1.4161 Β© 2025 by the authors; licensee Learning Gate π‘ž(π‘₯π‘–π‘€π‘–βˆ’1) = { 2𝑠 + 𝑖 + 1, 𝑖 ≀ 𝑠 3𝑠 + 2, 𝑖 > 𝑠 π‘ž(π‘₯1𝑣2) = π‘ž(π‘₯𝑖𝑣𝑖+1) = π‘ž(π‘₯π‘›βˆ’1𝑣𝑛) = π‘ž(π‘₯𝑛𝑣1) = 3𝑠 + 3. π‘ž(𝑦𝑖𝑀𝑖) = 1, 𝑖 = 1, 2,… , 𝑛. π‘ž(𝑦1𝑣1) = 1. π‘ž(𝑦2𝑣2) = 2. π‘ž(𝑦𝑖𝑣𝑖) = { 3𝑖+3 4 , 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3, 3𝑖+4 4 , 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2, 3𝑖+5 4 , 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 1, 3𝑖+6 4 , 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4, π‘ž(𝑦𝑛𝑣𝑛) = 3𝑛+6 4 . π‘ž(𝑀1𝑣1) = π‘ž(𝑀2𝑣2) = 2𝑠 + 1. π‘ž(𝑀𝑖𝑣𝑖) = { 𝑖+1 4 + 2𝑠, 𝑖 = 3, 7, 11,… , 𝑛 βˆ’ 3, 𝑖 4 + 2𝑠, 𝑖 = 4, 8, 12,… , 𝑛 βˆ’ 2, π‘–βˆ’1 4 + 2𝑠, 𝑖 = 5, 9, 13,… , 𝑛 βˆ’ 1, π‘–βˆ’2 4 + 2𝑠, 𝑖 = 6, 10, 14,… , 𝑛 βˆ’ 4, π‘ž(𝑀𝑛𝑣𝑛) = 𝑛 βˆ’ 2 4 + 2𝑠. Based on this function, it can be shown that all vertices weights are different and the maximum value of the function is ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. This shows that π‘ž: 𝑉 βˆͺ 𝐸 β†’ {1, 2, 3,β‹― , ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰} is a function a total vertex irregular π‘˜ βˆ’ π‘™π‘Žπ‘π‘’π‘™π‘™π‘–π‘›π‘” where π‘˜ = ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰. Thus it is obtained that 𝒕𝒗𝒔(𝓗𝒏) ≀ ⌈ πŸ‘+πŸ‘π’ πŸ’ βŒ‰ for 𝑛 = 4𝑑 + 2, where 𝑑 is some positive integer. Based on these four cases, it is concluded that 𝑑𝑣𝑠(ℋ𝑛) = ⌈ 3+3𝑛 4 βŒ‰ for 𝑛 β‰₯ 3 and n is a positive integer number. 4. Conclusion This research contributes significantly to understanding transportation route optimization through the lens of graph theory, specifically focusing on graph labeling within modified helm graph topologies. By applying irregular labeling, this study explores the vertex irregularity strength required for efficient transportation network configurations, which is critical in addressing the increasing congestion in urban transportation systems. The findings establish the lower and upper bounds of vertex irregularity strength within these graph structures, providing foundational parameters for further modeling and practical application in real-world transportation networks. These insights enhance theoretical perspectives in graph theory and offer a pathway for developing algorithms that can be used to optimize route navigation in complex transportation systems. Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. 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