IHJPAS. 36(1)2023 284 This work is licensed under a Creative Commons Attribution 4.0 International License On Antimagic Labeling for Some Families of Graphs Abstract Antimagic labeling of a graph 𝐺 with 𝑝 vertices and π‘ž edges is assigned the labels for its edges by some integers from the set {1,2, … , π‘ž}, such that no two edges received the same label, and the weights of vertices of a graph 𝐺 are pairwise distinct. Where the vertex-weights of a vertex 𝑣 under this labeling is the sum of labels of all edges incident to this vertex, in this paper, we deal with the problem of finding vertex antimagic edge labeling for some special families of graphs called strong face graphs. We prove that vertex antimagic, edge labeling for strong face ladder graph 𝐿𝑛 βˆ— , strong face wheel graph π‘Šπ‘› βˆ—, strong face fan graph 𝐹𝑛 βˆ—, strong face prism graph (𝐢𝑛 Γ— 𝑃2) βˆ— and finally strong face friendship graph (𝑇𝑛) βˆ—. Keywords: Antimagic graph, vertex antimagic graph, edge labeling, strong face graph. 1. Introduction Let a graph G = (V, E) be a finite, simple and undirected graph, where V(G) and E(G) are the vertex set and edge set respectively. An antimagic labeling graphs had first been introduced by [1]. If a graph G with p vertices and q edges can have its edges labeled without repetition and the sums of the labels of the edges incident to each vertex are pairwise distinct, the graph is said to be antimagic [2]. In addition [3] show that If all vertex weights are distinct, an edge labeling of a graph G is said to have a vertex antimagic edge labeling (VAE labeling). Hartsfield and Ringel proved that the path graphs 𝑃𝑛, complete graph 𝐾𝑛, 𝑛 β‰₯ 3, wheels and cycles graphs are antimagic, moreover, they conjectured that every tree except 𝑃2 is antimagic, and every connected graph except 𝑃2 is antimagic, both these conjectures are still open. In this paper, we will prove several graphs derived from a plane graph are edge labeling vertex antinagic, these graphs are called strong face graphs. The strong face graph, first introduced by [4]. Where they proved that face antimagic total labeling for some families of these graphs. In our study we address the problem of finding vertex antimagic edge labeling for this family of graphs. doi.org/10.30526/36.1.3209 Article history: Received 15 September 2022, Accepted 9 October 2022, Published in January 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepagejih.uobaghdad.edu.iq Noor Khalil Shawkat Department of Mathematics , College of Education for Pure Sciences,Ibn Al –Haitham/ University of Baghdad, Baghdad, Iraq. nour.khaleel1203a@ihcoedu.uobaghdad.edu.iq Mohammed Ali Ahmed Department of Mathematics , College of Education for Pure Sciences,Ibn Al –Haitham/ University of Baghdad, Baghdad, Iraq. mohammad.a.a@ihcoedu.uobaghdad.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:nour.khaleel1203a@ihcoedu.uobaghdad.edu.iq mailto:mohammad.a.a@ihcoedu.uobaghdad.edu.iq IHJPAS. 36(1)2023 285 The strong face plane graphs are obtained from a plane graph, by adding a new vertex to every face, in such way that, the results graphs are all three-sided faces, moreover if the faces of original plane graphs are three sided faces, then the number of faces will be increasing. We address the problem of finding vertex antimagic edge labeling, for short (VAEL), for the strong face ladder graph 𝐿𝑛 βˆ— , the strong face wheel graph π‘Šπ‘› βˆ—, the strong face fan graph 𝐹𝑛 βˆ—, The strong face prism graph (𝐢𝑛 Γ— 𝑃2) βˆ—, and finally the strong face friendship graph (𝑇𝑛) βˆ—. 2. Main Results Theorem1 For every 𝑛 β‰₯ 3, 𝑛 β‰’ 2, (mod 4) the strong face ladder graph 𝐿𝑛 βˆ— is VAEL. Proof: we define the vertex and edge sets of 𝐿𝑛 βˆ— graph as follows: 𝑉( 𝐿𝑛 βˆ— ) = {𝑣𝑖 : 𝑖 = 1, 2, … ,3𝑛 βˆ’ 1}, 𝐸( 𝐿𝑛 βˆ— ) = {𝑣𝑖 𝑣𝑖+1, 𝑣𝑛+𝑖 𝑣𝑛+𝑖+1, 𝑣𝑖 𝑣2𝑛+𝑖 , 𝑣𝑖+1𝑣2𝑛+𝑖, 𝑣𝑛+𝑖 𝑣2𝑛+𝑖, 𝑣𝑛+𝑖+1𝑣2𝑛+𝑖 ∢ 𝑖 = 1, 2, … , 𝑛 βˆ’ 1} βˆͺ {𝑣𝑖 𝑣𝑛+𝑖 : 𝑖 = 1, 2 … , 𝑛}. For 𝑛 β‰₯ 3, 𝑛 β‰’ 2 (mod 4) we define the labeling of 𝐿𝑛 βˆ— as: πœ‡1: 𝐸( 𝐿𝑛 βˆ— ) β†’ {1, 2, … , 7𝑛 βˆ’ 6}. Such that; πœ‡1(𝑣𝑖 𝑣𝑛+𝑖 ) = 7𝑖 βˆ’ 6 for 𝑖 = 1, 2, … , 𝑛. For 𝑖 = 1, 2, … , 𝑛 βˆ’ ,1, we have: πœ‡1(𝑣𝑖 𝑣𝑖+1) = 7𝑖 βˆ’ 5, πœ‡1(𝑣𝑛+𝑖 𝑣𝑛+𝑖+1) = 7𝑖 βˆ’ 2, πœ‡1(𝑣𝑖 𝑣2𝑛+𝑖 ) = 7𝑖 βˆ’ 4, πœ‡1(𝑣𝑖+1𝑣2𝑛+𝑖 ) = 7𝑖 βˆ’ 1, πœ‡1(𝑣𝑛+𝑖 𝑣2𝑛+𝑖 ) = 7𝑖 βˆ’ 3, πœ‡1(𝑣𝑛+𝑖+1𝑣2𝑛+𝑖 ) = 7𝑖. For the vertex-weights we get: π‘€π‘‘πœ‡1 (𝑣1) = πœ‡1(𝑣1𝑣2) + πœ‡1(𝑣1𝑣𝑛+1) + πœ‡1(𝑣1𝑣2𝑛+1) = 6, π‘€π‘‘πœ‡1 (𝑣𝑖 ) = πœ‡1(𝑣𝑖 𝑣𝑖+1) + πœ‡1(𝑣𝑖 π‘£π‘–βˆ’1) + πœ‡1(𝑣1𝑣2𝑛+𝑖 ) + πœ‡1(𝑣𝑖 𝑣2𝑛+π‘–βˆ’1) + πœ‡1(𝑣𝑖 𝑣𝑛+𝑖 ) for 𝑖 = 2, 3 … , 𝑛 βˆ’ 1 = (7𝑖 βˆ’ 5) + (7𝑖 βˆ’ 12) + (7𝑖 βˆ’ 4) + (7𝑖 βˆ’ 8) + (7𝑖 βˆ’ 6) = 35𝑖 βˆ’ 35 for𝑖 = 2, 3 … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡1 (𝑣𝑛) = πœ‡1(π‘£π‘›π‘£π‘›βˆ’1) + πœ‡1(𝑣𝑛𝑣2𝑛 ) + πœ‡1(𝑣𝑛 𝑣3π‘›βˆ’1) = (7𝑛 βˆ’ 12) + (7𝑛 βˆ’ 6) + (7𝑛 βˆ’ 8) = 21𝑛 βˆ’ 26, π‘€π‘‘πœ‡1 (𝑣𝑛+1) = πœ‡1(𝑣𝑛+1𝑣1) + πœ‡1(𝑣𝑛+1𝑣𝑛+2) + πœ‡1(𝑣𝑛+1𝑣2𝑛+1) = 1 + 5 + 4 = 10, π‘€π‘‘πœ‡1 (𝑣𝑛+𝑖 ) = πœ‡1(𝑣𝑛+𝑖 𝑣𝑛+𝑖+1) + πœ‡1(𝑣𝑛+𝑖 𝑣𝑛+π‘–βˆ’1) + πœ‡1(𝑣𝑛+𝑖 𝑣2𝑛+𝑖 ) + πœ‡1(𝑣𝑛+𝑖 𝑣2𝑛+π‘–βˆ’1) +πœ‡1(𝑣𝑛+𝑖 𝑣𝑖 ) for 𝑖 = 2, 3 … 𝑛, βˆ’1, = (7𝑖 βˆ’ 2) + (7𝑖 βˆ’ 9) + (7𝑖 βˆ’ 3) + (7𝑖 βˆ’ 7) + (7𝑖 βˆ’ 6) for𝑖 = 2, 3 … , 𝑛 βˆ’ 1 = 35𝑖 βˆ’ 27 for 𝑖 = 2, 3 … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡1 (𝑣2𝑛) = πœ‡1(𝑣2𝑛𝑣2π‘›βˆ’1) + πœ‡1(𝑣2𝑛𝑣𝑛) + πœ‡1(𝑣2𝑛𝑣3π‘›βˆ’1) = (7𝑛 βˆ’ 9) + (7𝑛 βˆ’ 6) + (7𝑛 βˆ’ 7) = 21𝑛 βˆ’ 22. Finally: π‘€π‘‘πœ‡1 (𝑣2𝑛+𝑖 ) = πœ‡1(𝑣2𝑛+𝑖 𝑣𝑖 ) + πœ‡1(𝑣2𝑛+𝑖 𝑣𝑖+1) + πœ‡1(𝑣2𝑛+𝑖 𝑣𝑛+𝑖 ) + πœ‡1(𝑣2𝑛+𝑖 𝑣𝑛+𝑖+1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, = (7𝑖 βˆ’ 4) + (7𝑖 βˆ’ 1) + (7𝑖 βˆ’ 3) + (7𝑖) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, = 28𝑖 βˆ’ 8 for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1. IHJPAS. 36(1)2023 286 Based on the vertex weight of the previous we conclude that they are all distinct and the following notes are observed: 1- π‘€π‘‘πœ‡1 (𝑣1) < π‘€π‘‘πœ‡1 (𝑣𝑛+1) < π‘€π‘‘πœ‡1 (𝑣𝑛 ) < π‘€π‘‘πœ‡1 (𝑣2𝑛), 2- π‘€π‘‘πœ‡1 (𝑣2𝑛+𝑖 ) < π‘€π‘‘πœ‡1 (𝑣2𝑛+𝑖+1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 2, 3- π‘€π‘‘πœ‡1 (𝑣𝑖 ) < π‘€π‘‘πœ‡1 (𝑣𝑛+𝑖 ) < π‘€π‘‘πœ‡1 (𝑣𝑖+1) < π‘€π‘‘πœ‡1 (𝑣𝑛+𝑖+1) for 𝑖 = 2,3, … , 𝑛 βˆ’ 2. Thus, the strong face ladder graph 𝐿𝑛 βˆ— is vertex antimagic edge labeling for every 𝑛 β‰₯ 3, 𝑛 β‰’ 2 (mod 4). However, when 𝑛 = 6, we can easily show that π‘€π‘‘πœ‡1 (𝑣12) = π‘€π‘‘πœ‡1 (𝑣16) and similarly when 𝑛 = 10, then π‘€π‘‘πœ‡1 (𝑣20) = π‘€π‘‘πœ‡1 (𝑣27) and so on. Theorem 2 The strong face wheel graph π‘Šπ‘› βˆ— is VAEL, for every 𝑛 β‰₯ 3, 𝑛 β‰’ 0 (mod 3). Proof: we define the vertex and edge sets of π‘Šπ‘› graph as follows: 𝑉( π‘Šπ‘› βˆ—) = {𝑣𝑖 : 𝑖 = 1, 2, … ,2𝑛 + 1}, 𝐸( π‘Šπ‘› βˆ—) = {𝑣𝑖 𝑣2𝑛+1, 𝑣2𝑛+1𝑣𝑛+𝑖 , 𝑣𝑛+𝑖 𝑣𝑖 ∢ 𝑖 = 1, 2, … , 𝑛} βˆͺ {𝑣𝑛+𝑖 𝑣𝑖+1,𝑣𝑖 𝑣𝑖+1 ∢ 𝑖 = 1, 2 … 𝑛 βˆ’ 1} βˆͺ {𝑣1𝑣𝑛, 𝑣1𝑣2𝑛 }. For 𝑛 β‰₯ 3, 𝑛 β‰’ 0 (mod 3) we define the labeling of π‘Šπ‘› βˆ— as: πœ‡2: 𝐸( π‘Šπ‘› βˆ—) β†’ {1, 2, … , 5𝑛}. Such that: For 𝑖 = 1,2, … , 𝑛, πœ‡2(𝑣2𝑛+1𝑣𝑛+𝑖 ) = 4𝑖, πœ‡2(𝑣2𝑛+1𝑣𝑖 ) = 4𝑖 βˆ’ 2, πœ‡2(𝑣𝑖 𝑣𝑛+𝑖 ) = 4𝑖 βˆ’ 1. And for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, πœ‡2(𝑣𝑖 𝑣𝑖+1) = 4𝑛 + 𝑖, πœ‡2(𝑣𝑛+𝑖 𝑣𝑖+1) = 4𝑖 + 1. Finally, πœ‡2(𝑣𝑛 𝑣1) = 5𝑛, πœ‡2(𝑣2𝑛 𝑣1) = 1. For the vertex-weights we get: π‘€π‘‘πœ‡2 (𝑣1) = πœ‡2(𝑣1𝑣2) + πœ‡2(𝑣1𝑣𝑛 ) + πœ‡2(𝑣1𝑣𝑛+1) + πœ‡2(𝑣1𝑣2𝑛+1) + πœ‡2(𝑣1𝑣2𝑛 ) = (4𝑛 + 1) + (5𝑛) + 3 + 2 + 1 = 9𝑛 + 7, π‘€π‘‘πœ‡2 (𝑣𝑖 ) = πœ‡2(𝑣𝑖 𝑣𝑖+1) + πœ‡2(𝑣𝑖 π‘£π‘–βˆ’1) + πœ‡2(𝑣𝑖 𝑣𝑛+𝑖 ) + πœ‡2(𝑣𝑖 𝑣2𝑛+1) + πœ‡2(𝑣𝑖 𝑣𝑛+π‘–βˆ’1) for 𝑖 = 2, 3 … 𝑛 βˆ’ 1, = (4𝑛 + 𝑖) + (4𝑛 βˆ’ 1 + 𝑖) + (4𝑖 βˆ’ 1) + (4𝑖 βˆ’ 2) + (4𝑖 βˆ’ 3) = 8𝑛 + 14𝑖 βˆ’ 7 for 𝑖 = 2, 3, … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡2 (𝑣𝑛) = πœ‡2(𝑣𝑛𝑣1) + πœ‡2(𝑣𝑛 π‘£π‘›βˆ’1) + πœ‡2(𝑣𝑛 𝑣2𝑛 ) + πœ‡2(𝑣𝑛 𝑣2𝑛+1) + πœ‡2(𝑣𝑛𝑣2π‘›βˆ’1) = (5𝑛) + (5𝑛 βˆ’ 1) + (4𝑛 βˆ’ 1) + (4𝑛 βˆ’ 2) + (4𝑛 βˆ’ 3) = 22𝑛 βˆ’ 7, π‘€π‘‘πœ‡2 (𝑣𝑛+𝑖 ) = πœ‡2(𝑣𝑛+𝑖 𝑣𝑖 ) + πœ‡2(𝑣𝑛+𝑖 𝑣𝑖+1) + πœ‡2(𝑣𝑛+𝑖 𝑣2𝑛+1) for 𝑖 = 1, 2, … 𝑛 βˆ’ 1, = (4𝑖 βˆ’ 1) + (4𝑖 + 1) + (4𝑖) = 12𝑖 for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡2 (𝑣2𝑛) = πœ‡2(𝑣2𝑛𝑣1) + πœ‡2(𝑣2𝑛𝑣𝑛 ) + πœ‡2(𝑣2𝑛𝑣2𝑛+1) = 1 + (4𝑛 βˆ’ 1) + (4𝑛) = 8𝑛. Finally: π‘€π‘‘πœ‡2 (𝑣2𝑛+1) = βˆ‘ πœ‡2(𝑣2𝑛+1𝑣𝑖 ) 𝑛 𝑖=1 + βˆ‘ πœ‡2(𝑣2𝑛+1𝑣𝑛+𝑖 ) 𝑛 𝑖=1 IHJPAS. 36(1)2023 287 = βˆ‘(4𝑖 βˆ’ 2) 𝑛 𝑖=1 + βˆ‘(4𝑖) 𝑛 𝑖=1 = 2𝑛(𝑛 + 1) βˆ’ 2𝑛 + 2𝑛(𝑛 + 1) = 4𝑛2 + 2𝑛. Which means that the vertex weights are all distinct and the following notes are observed: 1- π‘€π‘‘πœ‡2 (𝑣𝑖 ) < π‘€π‘‘πœ‡2 (𝑣𝑖+1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, 2- π‘€π‘‘πœ‡2 (𝑣𝑛+𝑖 ) < π‘€π‘‘πœ‡2 (𝑣𝑛+𝑖+1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, 3- the weight of vertex (𝑣2𝑛+1) is greater than the weight of any other vertex in the strong face wheel graph π‘Šπ‘› βˆ—. Thus, the strong face wheel graph π‘Šπ‘› βˆ— is vertex antimagic edge labeling, for every 𝑛 β‰₯ 3, 𝑛 β‰’ 0 (mod 3). However, when 𝑛 = 3, we can easily show that π‘€π‘‘πœ‡2 (𝑣5) = π‘€π‘‘πœ‡2 (𝑣6) and similarly when 𝑛 = 6, π‘‘β„Žπ‘’π‘› π‘€π‘‘πœ‡2 (𝑣10) = π‘€π‘‘πœ‡2 (𝑣12) and so on. Theorem 3 The strong face fan graph 𝐹𝑛 βˆ—, is VAEL, for every 𝑛 β‰₯ 4, 𝑛 β‰’ 3 (mod 5). Proof: we define the vertex and edge sets of 𝐹𝑛 βˆ— graph as follows: 𝑉( 𝐹𝑛 βˆ—) = {𝑣 , 𝑣𝑖 : 𝑖 = 1, 2, … 2𝑛 βˆ’ 1}, 𝐸( 𝐹𝑛 βˆ—) = {𝑣𝑣𝑖 ∢ 𝑖 = 1, 2, … , 𝑛} βˆͺ {𝑣𝑖 𝑣𝑖+1,𝑣 𝑣𝑛+𝑖 , 𝑣𝑖 𝑣𝑛+𝑖 , 𝑣𝑖+1𝑣𝑛+𝑖 ∢ 𝑖 = 1, 2 … , 𝑛 βˆ’ 1}. For 𝑛 β‰₯ 4, 𝑛 β‰’ 3 (mod 5) we define the labeling of 𝐹𝑛 βˆ—, as: πœ‡3: 𝐸( 𝐹𝑛 βˆ—) β†’ {1, 2, … , 5𝑛 βˆ’ 4}. Such that: πœ‡3(𝑣𝑣𝑖 ) = 5𝑖 βˆ’ 4 for 𝑖 = 1, 2, … , 𝑛, and for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1, πœ‡3(𝑣𝑖 𝑣𝑖+1) = 5𝑖 βˆ’ 3, πœ‡3(𝑣𝑖 𝑣𝑛+𝑖 ) = 5𝑖 βˆ’ 2, πœ‡3(𝑣𝑖+1𝑣𝑛+𝑖 ) = 5𝑖, πœ‡3(𝑣𝑣𝑛+𝑖 ) = 5𝑖 βˆ’ 1. For the vertex-weights we get: π‘€π‘‘πœ‡3 (𝑣1) = πœ‡3(𝑣1𝑣) + πœ‡3(𝑣1𝑣2) + πœ‡3(𝑣1𝑣𝑛+1) = 1 + 2 + 3 = 6, π‘€π‘‘πœ‡3 (𝑣𝑖 ) = πœ‡3(𝑣𝑖 𝑣) + πœ‡3(𝑣𝑖 𝑣𝑖+1) + πœ‡3(𝑣𝑖 π‘£π‘–βˆ’1) + πœ‡3(𝑣𝑖 𝑣𝑛+π‘–βˆ’1) + πœ‡3(𝑣𝑖 𝑣𝑛+𝑖 ) for 𝑖 = 2, 3 … , 𝑛 βˆ’ 1, = (5𝑖 βˆ’ 4) + (5𝑖 βˆ’ 3) + (5𝑖 βˆ’ 8) + (5𝑖 βˆ’ 5) + (5𝑖 βˆ’ 2) = 25𝑖 βˆ’ 22 for 𝑖 = 2, 3, … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡3 (𝑣𝑛) = πœ‡3(𝑣𝑛𝑣) + πœ‡3(𝑣𝑛 π‘£π‘›βˆ’1) + πœ‡3(𝑣𝑛 𝑣2π‘›βˆ’1) = (5𝑛 βˆ’ 4) + (5𝑛 βˆ’ 8) + (5𝑛 βˆ’ 5) = 15𝑛 βˆ’ 17, π‘€π‘‘πœ‡3 (𝑣𝑛+𝑖 ) = πœ‡3(𝑣𝑛+𝑖 𝑣𝑖 ) + πœ‡3(𝑣𝑛+𝑖 𝑣𝑖+1) + πœ‡3(𝑣𝑛+𝑖 𝑣) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, = (5𝑖 βˆ’ 2) + (5𝑖) + (5𝑖 βˆ’ 1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, = 15𝑖 βˆ’ 3 for 𝑖 = 1,2, … , 𝑛 βˆ’ 1. Finally: π‘€π‘‘πœ‡3 (𝑣) = βˆ‘ πœ‡3(𝑣𝑣𝑖 ) 𝑛 𝑖=1 + βˆ‘ πœ‡3(𝑣𝑣𝑛+𝑖 ) π‘›βˆ’1 𝑖=1 = βˆ‘(5𝑖 βˆ’ 4) 𝑛 𝑖=1 + βˆ‘(5𝑖 βˆ’ 1) π‘›βˆ’1 𝑖=1 = 5 ( 𝑛2 + 𝑛 2 ) βˆ’ 4𝑛 + 5 ( 𝑛2 βˆ’ 𝑛 2 ) βˆ’ 𝑛 + 1 IHJPAS. 36(1)2023 288 = 5𝑛2 βˆ’ 5𝑛 + 1. So, the weights of all vertices are distinct and the following notes are observed: 1- π‘€π‘‘πœ‡3 (𝑣𝑖 ) < π‘€π‘‘πœ‡3 (𝑣𝑖+1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 1, 2- π‘€π‘‘πœ‡3 (𝑣𝑛+𝑖 ) < π‘€π‘‘πœ‡3 (𝑣𝑛+𝑖+1) for 𝑖 = 1,2, … , 𝑛 βˆ’ 2. Thus, the strong face fan graph 𝐹𝑛 βˆ— is vertex antimagic edge labeling, for every 𝑛, 𝑛 β‰₯ 4, 𝑛 β‰’ 3 (mod 5). However, when 𝑛 = 8, we can easily show that π‘€π‘‘πœ‡3 (𝑣5) = π‘€π‘‘πœ‡3 (𝑣8) and similarly when 𝑛 = 13, π‘‘β„Žπ‘’π‘› π‘€π‘‘πœ‡3 (𝑣8) = π‘€π‘‘πœ‡3 (𝑣13) and so on. Theorem4 The strong face prism graph (𝐢𝑛 Γ— 𝑃2) βˆ— is VAEL, for every 𝑛 β‰₯ 3, 𝑛 β‰’ 0 (mod 5). Proof: we define the vertex and edge sets of (𝐢𝑛 Γ— 𝑃2) βˆ—graph as follows; 𝑉((𝐢𝑛 Γ— 𝑃2) βˆ—) = {𝑣𝑖 : 𝑖 = 1, 2, … ,3𝑛 + 1}, 𝐸( (𝐢𝑛 Γ— 𝑃2) βˆ—) = {𝑣𝑖 𝑣𝑖+1, 𝑣𝑛+𝑖 𝑣𝑛+𝑖+1 ∢ 𝑖 = 1, 2, … 𝑛 βˆ’ 1} βˆͺ {𝑣𝑛 𝑣1, 𝑣2𝑛 𝑣𝑛+1, 𝑣3𝑛𝑣1, 𝑣3𝑛 𝑣𝑛+1} βˆͺ {𝑣3𝑛+1𝑣𝑛+𝑖 , 𝑣𝑛+𝑖 𝑣𝑖 ∢ 𝑖 = 1, 2 … 𝑛} βˆͺ {𝑣2𝑛+𝑖 𝑣𝑖 , 𝑣2𝑛+𝑖 𝑣𝑛+𝑖 ∢ 𝑖 = 1, 2 … 𝑛} βˆͺ {𝑣2𝑛+𝑖 𝑣𝑖+1, 𝑣2𝑛+𝑖 𝑣𝑛+𝑖+1 ∢ 𝑖 = 1, 2 … 𝑛 βˆ’ 1}. For 𝑛 β‰₯ 3 , 𝑛 β‰’ 0 (mod 5) we define the labeling of (𝐢𝑛 Γ— 𝑃2) βˆ— graph as: πœ‡4: 𝐸((𝐢𝑛 Γ— 𝑃2) βˆ—) β†’ {1, 2, … , 8𝑛}. Such that: For 𝑖 = 1, 2, … , 𝑛, πœ‡4(𝑣3𝑛+1𝑣𝑛+𝑖 ) = 𝑖 πœ‡4(𝑣𝑛+𝑖 𝑣𝑖 ) = 4𝑛 + 3𝑖 βˆ’ 1, πœ‡4(𝑣2𝑛+𝑖 𝑣𝑖 ) = 4𝑛 + 3𝑖, πœ‡4(𝑣2𝑛+𝑖 𝑣𝑛+𝑖 ) = 2𝑛 + 2𝑖 βˆ’ 1. And for 𝑖 = 1, 2, … 𝑛 βˆ’ 1, πœ‡4(𝑣𝑖 𝑣𝑖+1) = 7𝑛 + 𝑖, πœ‡4(𝑣𝑛+𝑖 𝑣𝑛+𝑖+1) = 𝑛 + 𝑖, πœ‡4(𝑣2𝑛+𝑖 𝑣𝑖+1) = 4𝑛 + 3𝑖 + 1, πœ‡4(𝑣2𝑛+𝑖 𝑣𝑛+𝑖+1) = 2𝑛 + 2𝑖. Finally, πœ‡4(𝑣2𝑛 𝑣𝑛+1) = 2𝑛, πœ‡4(𝑣𝑛 𝑣1) = 8𝑛, πœ‡4(𝑣3𝑛 𝑣1) = 4𝑛 + 1, πœ‡4(𝑣3𝑛 𝑣𝑛+1) = 4𝑛. For the vertex-weights we get: π‘€π‘‘πœ‡4 (𝑣1) = πœ‡4(𝑣1 𝑣𝑛 ) + πœ‡4(𝑣1𝑣2) + πœ‡4(𝑣1𝑣𝑛+1) + πœ‡4(𝑣1𝑣3𝑛 ) + πœ‡4(𝑣1𝑣2𝑛+1) = (8𝑛) + (7𝑛 + 1) + (4𝑛 + 2) + (4𝑛 + 1) + (4𝑛 + 3) = 27𝑛 + 7, π‘€π‘‘πœ‡4 (𝑣𝑖 ) = πœ‡4(𝑣𝑖 𝑣𝑖+1) + πœ‡4(𝑣𝑖 π‘£π‘–βˆ’1) + πœ‡4(𝑣𝑖 𝑣2𝑛+𝑖 ) + πœ‡4(𝑣𝑖 𝑣2𝑛+π‘–βˆ’1) + πœ‡4(𝑣𝑖 𝑣𝑛+𝑖 ) for 𝑖 = 2, 3, … , 𝑛 βˆ’ 1, = (7𝑛 + 𝑖) + (7𝑛 + 𝑖 βˆ’ 1) + (4𝑛 + 3𝑖) + (4𝑛 + 3𝑖 βˆ’ 2) + (4𝑛 + 3𝑖 βˆ’ 1) = 26𝑛 + 11𝑖 βˆ’ 4 for 𝑖 = 2, 3, … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡4 (𝑣𝑛) = πœ‡4(𝑣𝑛𝑣1) + πœ‡4(𝑣𝑛 π‘£π‘›βˆ’1) + πœ‡4(𝑣𝑛 𝑣3𝑛 ) + πœ‡4(𝑣𝑛 𝑣2𝑛) + πœ‡4(𝑣𝑛𝑣3π‘›βˆ’1) = (8𝑛) + (8𝑛 βˆ’ 1) + (7𝑛) + (7𝑛 βˆ’ 1) + (7𝑛 βˆ’ 2) = 37𝑛 βˆ’ 4, π‘€π‘‘πœ‡4 (𝑣𝑛+1) = πœ‡4(𝑣𝑛+1𝑣1) + πœ‡4(𝑣𝑛+1𝑣3𝑛) + πœ‡4(𝑣𝑛+1𝑣2𝑛+1) + πœ‡4(𝑣𝑛+1𝑣𝑛+2) + πœ‡4(𝑣𝑛+1𝑣2𝑛 ) + πœ‡4(𝑣𝑛+1𝑣3𝑛+1) = (4𝑛 + 2) + (4𝑛) + (2𝑛 + 1) + (𝑛 + 1) + (2𝑛) + 1 = 13𝑛 + 5, IHJPAS. 36(1)2023 289 π‘€π‘‘πœ‡4 (𝑣𝑛+𝑖 ) = πœ‡4(𝑣𝑛+𝑖 𝑣𝑖 ) + πœ‡4(𝑣𝑛+𝑖 𝑣𝑛+𝑖+1) + πœ‡4(𝑣𝑛+𝑖 𝑣𝑛+π‘–βˆ’1) + πœ‡4(𝑣𝑛+𝑖 𝑣2𝑛+π‘–βˆ’1) +πœ‡4(𝑣𝑛+𝑖 𝑣2𝑛+𝑖 ) + πœ‡4(𝑣𝑛+𝑖 𝑣3𝑛+1) for 𝑖 = 2, 3, … , 𝑛 βˆ’ 1 = (4𝑛 + 3𝑖 βˆ’ 1) + (𝑛 + 𝑖) + (𝑛 + 𝑖 βˆ’ 1) + (2𝑛 + 2𝑖 βˆ’ 2) + (2𝑛 + 2𝑖 βˆ’ 1) + 𝑖 for 𝑖 = 2, 3, … , 𝑛 βˆ’ 1, = 10𝑛 + 10𝑖 βˆ’ 5 for 𝑖 = 2, 3, … 𝑛 βˆ’ 1, π‘€π‘‘πœ‡4 (𝑣2𝑛) = πœ‡4(𝑣2𝑛𝑣𝑛+1) + πœ‡4(𝑣2𝑛𝑣2π‘›βˆ’1) + πœ‡4(𝑣2𝑛𝑣3𝑛+1) + πœ‡4(𝑣2𝑛𝑣3𝑛 ) +πœ‡4(𝑣2𝑛𝑣𝑛 ) + πœ‡4(𝑣2𝑛 𝑣3π‘›βˆ’1) = (2𝑛) + (2𝑛 βˆ’ 1) + (𝑛) + (4𝑛 βˆ’ 1) + (7𝑛 βˆ’ 1) + (4𝑛 βˆ’ 2) = 20𝑛 βˆ’ 5, π‘€π‘‘πœ‡4 (𝑣2𝑛+𝑖 ) = πœ‡4(𝑣2𝑛+𝑖 𝑣𝑖 ) + πœ‡4(𝑣2𝑛+𝑖 𝑣𝑖+1) + πœ‡4(𝑣2𝑛+𝑖 𝑣𝑛+𝑖 ) + πœ‡4(𝑣2𝑛+𝑖 𝑣𝑛+𝑖+1) for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1, = (4𝑛 + 3𝑖) + (4𝑛 + 3𝑖 + 1) + (2𝑛 + 2𝑖 βˆ’ 1) + (2𝑛 + 2𝑖) for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1, = 12𝑛 + 10𝑖 for 𝑖 = 1, 2, … , 𝑛 βˆ’ 1, π‘€π‘‘πœ‡4 (𝑣3𝑛) = πœ‡4(𝑣3𝑛𝑣1) + πœ‡4(𝑣3𝑛𝑣𝑛 ) + πœ‡4(𝑣3𝑛𝑣2𝑛 ) + πœ‡4(𝑣3𝑛 𝑣𝑛+1) = (4𝑛 + 1) + (7𝑛) + (4𝑛 βˆ’ 1) + (4𝑛) = 19𝑛. Finally: π‘€π‘‘πœ‡4 (𝑣3𝑛+1) = βˆ‘ πœ‡3(𝑣3𝑛+1 𝑣𝑛+𝑖 ) 𝑛 𝑖=1 = βˆ‘(𝑖) 𝑛 𝑖=1 = 𝑛(𝑛 + 1) 2 . So, the weights of all vertices are distinct and the following notes are observed: 1- π‘€π‘‘πœ‡4 (𝑣𝑖 ) < π‘€π‘‘πœ‡4 (𝑣𝑖+1) for 𝑖 = 1,2, … , , 𝑛 βˆ’ 1. 2- The weight of vertex (𝑣𝑛 ) is greater than the weight of any other vertex in the (𝐢𝑛 Γ— 𝑃2) βˆ— graph. Thus, the strong face prism graph (𝐢𝑛 Γ— 𝑃2) βˆ— is vertex antimagic edge labeling, for every 𝑛, 𝑛 β‰₯ 3, 𝑛 β‰’ 0 (mod 5). However when 𝑛 = 5, we can easily show that π‘€π‘‘πœ‡4 (𝑣6) = π‘€π‘‘πœ‡4 (𝑣11) and similarly when 𝑛 = 10, then π‘€π‘‘πœ‡4 (𝑣9) = π‘€π‘‘πœ‡4 (𝑣11) and so on. Theorem 5 The strong face friendship graph (𝑇𝑛) βˆ— is VAEL for every 𝑛 β‰₯ 3. Proof: we define the vertex and edge sets of (𝑇𝑛) βˆ— graph as follows; 𝑉( (𝑇𝑛) βˆ— ) = {𝑣, 𝑣𝑖 : 𝑖 = 1, 2, … 2𝑛} βˆͺ { 𝑒𝑖 : 𝑖 = 1, 2, … 𝑛}, 𝐸((𝑇𝑛) βˆ— ) = {𝑣𝑣𝑖 ∢ 𝑖 = 1, 2, … 2𝑛} βˆͺ {𝑣𝑒𝑖 ∢ 𝑖 = 1, 2, … 𝑛} βˆͺ {𝑣𝑖 𝑣𝑖+1 ∢ 𝑖 = 1, 3, … 2𝑛 βˆ’ 1} βˆͺ {𝑒𝑖 𝑣2π‘–βˆ’1, 𝑒𝑖 𝑣2𝑖 ∢ 𝑖 = 1, 2, … , 𝑛}. For every 𝑛 β‰₯ 3, we define the labeling of (𝑇𝑛) βˆ— graph as: πœ‡5((𝑇𝑛) βˆ—) β†’ {1, 2, … , 6𝑛}. Such that: 1- πœ‡5(𝑣𝑣𝑖 ) = { 3𝑖 βˆ’ 2 for 𝑖 = 1, 3, … ,2𝑛 βˆ’ 1, 3𝑖 βˆ’ 3 for 𝑖 = 2, 4, … ,2𝑛, 2- πœ‡5(𝑣𝑖 𝑣𝑖+1) = 3𝑖 βˆ’ 1 for 𝑖 = 1, 3, … ,2𝑛 βˆ’ 1, 3- πœ‡5(𝑣𝑒𝑖 ) = 6𝑖 βˆ’ 2 for 𝑖 = 1, 2, … , 𝑛, 4- πœ‡5(𝑒𝑖 𝑣2π‘–βˆ’1) = 6𝑖 βˆ’ 1 for 𝑖 = 1, 2, … , 𝑛, 5- πœ‡5(𝑒𝑖 𝑣2𝑖 ) = 6𝑖 for 𝑖 = 1, 2, … , 𝑛. For the vertex-weights we get: IHJPAS. 36(1)2023 290 π‘€π‘‘πœ‡5 (𝑣𝑖 ) = { πœ‡5(𝑣𝑖 𝑣) + πœ‡5(𝑣𝑖 𝑣𝑖+1) + πœ‡5 (𝑣𝑖 𝑒𝑖+1 2 ) for 𝑖 = 1, 3, … ,2𝑛 βˆ’ 1, πœ‡5(𝑣𝑖 𝑣) + πœ‡5(𝑣𝑖 π‘£π‘–βˆ’1) + πœ‡5 (𝑣𝑖 𝑒 𝑖 2 ) for 𝑖 = 2, 4, … ,2𝑛, = { (3𝑖 βˆ’ 2) + (3𝑖 βˆ’ 1) + (3𝑖 + 2) for 𝑖 = 1, 3, … ,2𝑛 βˆ’ 1, (3𝑖 βˆ’ 3) + (3𝑖 βˆ’ 4) + 3𝑖 for 𝑖 = 2, 4, … ,2𝑛, = { 9𝑖 βˆ’ 1 for i = 1, 3, … ,2n βˆ’ 1, 9𝑖 βˆ’ 7 for i = 2, 4, … ,2n, π‘€π‘‘πœ‡5 (𝑒𝑖 ) = πœ‡5(𝑒𝑖 𝑣2π‘–βˆ’1) + πœ‡5(𝑒𝑖 𝑣2𝑖 ) + πœ‡5(𝑒𝑖 𝑣) for 𝑖 = 1, 2, … , 𝑛, = (6𝑖 βˆ’ 1) + (6𝑖) + (6𝑖 βˆ’ 2) = 18𝑖 βˆ’ 3 for 𝑖 = 1,2, … , 𝑛. Finally: π‘€π‘‘πœ‡5 (𝑣) = βˆ‘ πœ‡5(𝑣𝑣𝑖 ) 2𝑛 𝑖=1 + βˆ‘ πœ‡5(𝑣𝑒𝑖 ) 𝑛 𝑖=1 = (3𝑛2 βˆ’ 2𝑛) + (3𝑛2) + (3𝑛2 + 𝑛) = 9𝑛2 βˆ’ 𝑛. Which implies that: π‘€π‘‘πœ‡5 (𝑣1) < π‘€π‘‘πœ‡5 (𝑣2) < π‘€π‘‘πœ‡5 (𝑒1) < π‘€π‘‘πœ‡5 (𝑣3) < π‘€π‘‘πœ‡5 (𝑣4) < π‘€π‘‘πœ‡5 (𝑒2) < β‹― < π‘€π‘‘πœ‡5 (𝑣2π‘›βˆ’1) < π‘€π‘‘πœ‡5 (𝑣2𝑛) < π‘€π‘‘πœ‡5 (𝑒𝑛) < π‘€π‘‘πœ‡5 (𝑣). 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