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Cycle Related Graphs on Square Difference Labeling Turkish Journal of Computer and Mathematics Education Vol.12 No. 10 (2021), 1482-1486 Research Article Cycle Related Graphs on Square Difference Labeling J. Rashmi Kumar 1 and K. Manimekalai2 1,2 Department of Mathematics, Bharath Institute of Higher Education and Research, Chennai, India. 1Corresponding author: [email protected] Article History Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published online: 28 April 2021 Abstract: In this study, we prove that the graphs cycle 퐶푛 with parallel chords, 2 − 푡푢푝푙푒 of 푍 − 푃푛, Durer-graph, Moser- spindle, Herchel graph are Square difference graph (SDG). Keywords: Square difference Labeling (푆퐷퐿), Z-Pn, Durer-graph, Moser-spindle, Herchel graph, 2-tuple graph. 1.Introduction All graphs in this paper are simple, undirected and finite. We refer J. A. Gallian for detailed study [1] and follow [2] for all terminology and notation. The Square difference labeling is introduced by Shiama [6]. A function of a graph G admits one to one and onto function 푓: 푣(퐺) → {0,1,2, … … 푝 − 1} such that the 1-1 function 푓∗: 퐸(퐺) → 푁 given by 푓∗(푢푣) = | [푓(푢)2 – 푓(푣)2|, for all 푢푣 휖 퐸(퐺), distinct are said to be Square Difference graph[푆퐷퐺] [1,4]. The concept of 2-tuple was introduced by P.L. Vihol [7]. P. Jagadeeswari investigated some various graphs for SDL. [4,5]. In this work, we prove Cycle related graphs on Square difference labeling. We Commenced some preliminaries which are helpful for our work. Definition 1.1[5]: The graph 푍 − 푃푛 is acquired from the two paths 푃푛′ and 푃푛′′. Let 푣푖 and 푢푖, 푖 = 1,2 … … 푛 − 1, are the vertices of ′′ 푡ℎ th path 푃푛′ and 푃푛 respectively. To determine 푍 − 푃푛 attach 푖 vertex of path 푃푛′ with (i+1) vertex of path 푃푛′′. for all 푖 = 1,2 … … 푛 − 1. Definition 1.2[5]: Let 퐺 = (푉, 퐸) be a simple graph and let 퐺′ = (푉′, 퐸′) be another copy of 퐺. Link each vertex 푉 of 퐺 to the equivalent vertex 푉′ of 퐺′ by an edge. The new graph thus gained is called 2-tuple of 퐺. We signify 2-tuple graph of 퐺 by the notation 푇2(퐺). Definition 1.3[5]: The Durer graph is the graph termed by the vertices and edges of the durer solid. It is a cubic graph of girth 3 and diameter 4. The Durer graph is Hamiltonian. It has exactly 6 Hamiltonian cycles, each pair of which may be mapped into each other by a symmetry of the graph. Definition 1.4[5]: The Moser graph which is also called Moser spindle is an undirected graph with 7 vertices and 11 edges. Definition 1.5[5]: The Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. Main Results Theorem 2.1: The 2 − 푡푢푝푙푒 graph of 푍 − 푃푛 admits 푆퐷퐿. Proof: Let G be the graph with 4푛 vertices and (8푛 − 6) edges. Consider the vertex set 푉 = {푢푖, 푣푖 / 1 ⩽ 푖 ⩽ 푛} and the edge set 퐸 = {푢푖푣푖, 푢푖푢푖+1, 푣푖푣푖+1, 푢푖푢푖+8, 푣푖푣푖+8 / 1 ⩽ 푖 ⩽ 푛 − 1}. Determine the 1-1 and onto function as: 푓(푢푖) = 2(푖 − 1) 푓표푟 1 ⩽ 푖 ⩽ 푛 푓(푣푖) = 2푖 − 1 ∗ 푓 (푢푖푢푖+1) = 8푖 − 4 푓표푟 1 ⩽ 푖 ⩽ 푛 − 1 ∗ 푓 (푣푖푣푖+1) = 8푖 ∗ 푓 (푢푖푣푖) = 4푖 − 3 ∗ 푓 (푢푖푢푖+8) ≡ 0(푚표푑 8) ∗ 푓 (푣푖푣푖+8) ≡ 0(푚표푑 8) ∗ ∗ 퐻푒푟푒 푓 (푢푖푢푖+8) > 푓 (푣푖푣푖+8), therefore all the edge labeling are dissimilar. Hence the theorem is verified. 1482 Turkish Journal of Computer and Mathematics Education Vol.12 No. 10 (2021), 1482-1486 Research Article Figure 1. 2-tuple of Z-P3 Theorem 2.2: Every cycle Cn (n ≥ 6) with parallel chords is Square difference graph. Proof: Contemplate the graph 퐺 with 푉 = {푣푖 / 0 ⩽ 푖 ⩽ 푛 − 1} and 퐸 = {푣푖푣푖+1, 푣푖푣푛−푖/0 ⩽ 푖 ⩽ 푛 − 1} .The labeling 푓 for the vertices and the labeling 푓∗ for the edges are given respectively in the following two cases depending on n being even and 푛 being odd . Also, |푉(퐺)| = 푛 and (3푛−3) , 푛 푖푠 표푑푑 2 | 퐸(퐺)| = {(3푛−2) , 푛 푖푠 푒푣푒푛 2 Case (i) : n is even ∗ Define the bijective function g and the edge labeling 푔 as: 푔(푣푖) = 푖, 0 ⩽ 푖 ⩽ 푛 − 1 ∗ 푔 (푣푖푣푖+1) = 2푖 + 1 ∗ 푔 (푣푖푣푛−푖) ≡ 0(푚표푑 푛) ∗ 2 푔 (푣0푣푛−1) = (푛 − 1) ∗ 푔 (푣0푣1) = 1 Figure 2. Cycle C8 with parallel chords 1483 Turkish Journal of Computer and Mathematics Education Vol.12 No. 10 (2021), 1482-1486 Research Article Case(ii): n is odd 푔 (푣푖) = 푖, 0 ⩽ 푖 ⩽ 푛 − 1 ∗ 푔 (푣푖푣푖+1) = 2푖 + 1 ∗ 푔 (푣0푣1) = 1 ∗ 푔 (푣푖푣푛−1) ≡ 0(푚표푑 푛) ∗ 2 푔 (푣0푣푛−1) = (푛 − 1) For the above labeling pattern, the induced edge labeling function 푔∗: 퐸 (퐺) → 푁 defined by 푔∗(푢푣) = 2 2 * * | [푔(푢) – 푔(푣) |, for every uv ∈ E(G) are all diverse. such that g (ei) ≠ g (ej) for every 푒푖 ≠ 푒푗. Hence the graph G admits SDL. Figure 3. Cycle C5 with parallel chords Theorem 2.3: The Durer graph acknowledges 푆퐷퐿. Proof: Contemplate the graph 퐺 with 12 vertices and 18 edges . Let 푣푖 be the vertex set for 0 ⩽ 푗 ⩽ 11. Now define the function 푓 ∶ 푉 → {0,1, … ,11} as : 푓(푣푗) = 푗 for 0 ⩽ 푗 ⩽ 푛 − 1 and the induced function f * satisfies the condition of square difference labeling and it yields the edge labels as ∗ 푓 (푣0푣1) = 1 ∗ 푓 (푣0푣5) = 25 ∗ 푓 (푣푖푣푖+1) = 2푖 + 1 , 푖 = 1 푡표 5 ∗ 푓 (푣푖푣푖+7) ≡ 0(푚표푑 7) 푖 = 1 푡표 4 ∗ 푓 (푣푖푣푖+2) ≡ 0(푚표푑 4) 푖 = 6 푡표 9 ∗ 푓 (푣푖푣푖+4) ≡ 0(푚표푑 8) 푖 = 6, 7 Thus the entire 11 edges acquire the discrete edge labels. Hence the theorem is verified. Figure 4. Durer graph 1484 Turkish Journal of Computer and Mathematics Education Vol.12 No. 10 (2021), 1482-1486 Research Article Theorem 2.4: The Moser - Spindle graph is Square Difference graph. Proof: Consider the Moser Spindle graph with 7 vertices and 11 edges. Let 푢푗 be the vertex set for 0 ⩽ 푗 ⩽ 6 . Then the vertex function 푓 and edge function 푓∗ is defined as 푓( 푢푗) = 푗 , 0 ⩽ 푗 ⩽ 푛 − 1 ∗ 푓 (푢푗푢푗+1) = 2푗 + 1 , 푗 = 0 푡표 3 ∗ 2 푓 (푢0푢푗) = 푗 , 푗 = 1,4,5,6 ∗ 푓 (푢푗푢푗+3) ≡ 0(푚표푑 7) , 푗 = 2,3 ∗ 푓 (푢1푢5) ≡ 0(푚표푑 8) ∗ 푓 (푢1푢4) ≡ 0(푚표푑 4) Hence the theorem. Figure 5. Moser – Spindle graph Theorem 2.5: The Herschel graph is 푆퐷퐺. Proof: Consider the Herschel graph with 11 vertices and 18 edges . The vertex set 푉 = {푣, 푢}. Then the vertex valued function 푓 and edge function 푓∗ is defined as 푓(푢푖) = 2푖 + 1 , 0 ⩽ 푖 ⩽ 4 푓(푣푗) = 2푗 , 0 ⩽ 푗 ⩽ 5 ∗ 푓 (푣1푢1) = 5 ∗ 푓 (푣0푢2) = 25 ∗ 푓 (푣1푢3) = 45 ∗ 푓 (푣4푢1) = 55 ∗ 푓 (푣3푢0) = 35 ∗ 푓 (푣2푢4) = 65 ∗ 푓 (푣5푢2) = 75 ∗ ∗ Hence 푓 (푒푖) 푓 (푒푗)푒푖, 푒푗퐸(퐺). i.e., all the edge labeling are diverse. Therefore, the theorem is verified. 1485 Turkish Journal of Computer and Mathematics Education Vol.12 No. 10 (2021), 1482-1486 Research Article Figure 6. Herschel graph REFERENCES: 1. J. A. Gallian, “A Dynamic survey of graph labeling”, The electronics journal of Combinatories, 17(2010), # DS6. 2. F. Harary, Graph Theory, Narosa publication House Reading, New Delhi, 1998. 3. P.Jagadeeswai, “Considering Square difference labeling for validating Theta graphs of dynamic machinaries” , International journal of Innovative Technology and Exploring Engineering, ISSN : 2278- 3075, vol-9, Issue-2S4 dec2019. 4. P.Jagadeeswai, K.Manimekalai, K.Bhuvaneswari , “Square difference labeling of cycle , path and tree related graphs”, Advancement in Engineering, science & Technology J.Mech.Cont.&Math.sci.,Special issue, No.-2, August (2019) pp 627-631. 5. Mydeenbibi & M.Malathi , “ Equality labeling on special graphs” , International journal of modern trends in Engineering & Research, vol5 , issue 04 (Apr2018) ISSN(online) 2349-9745 ISSN(print) 2393-8161. 6. J.Shiama , “ Square difference labeling for certain graph “ International journal of computer applications (0975-08887), 44(4) (2012). 7. P. L. Vihol et. al., “Difference Cordial Labeling of 2-tuple Graphs of Some Graphs”, International Journal of Mathematics and its Applications, Volume 4, Issue 2–A (2016), 111–119. 1486 .
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