Super Edge Anti Magic Graceful Labeling of Some Special Graphs

SUPER EDGE ANTI MAGIC GRACEFUL LABELING OF SOME SPECIAL GRAPHS

N. Mary Bernard Assistant Professor, TDMNS College, T. Kallikulam.

Abstract- Let 푮(푽, 푬) be a graph with p vertices and q Throughout, it is assumed that G is connected, finite, edges. A Super Edge Anti Magic Graceful Labeling simple and undirected graph. (푺푬푨푴푮푳) of a graph 푮(풑 , 풒) is a one to one and onto function 풇: 푬 → { ퟏ, ퟐ, ퟑ, … , 풒 } with the property that II. MAIN RESULT for each 풖 ∈ 푽 , ∑풆∈푵(풖) 풇(풖풗) = 풅풊풔풕풊풏풄풕 . Such a A Super Edge Anti Magic Graceful Labeling labeling is Super Edge Anti Magic Graceful (푺푬푨푴푮) (푆퐸퐴푀퐺퐿) of a graph 퐺(푝 , 푞) is a one to one and if |풇(풖) + 풇(풗) − 풇(풖풗)| = 풅풊풔풕풊풏풄풕. A graph G is onto function 푓: 퐸 → { 1, 2, 3, … , 푞 } with the called Super Edge Anti Magic Graceful (푺푬푨푴푮) if G property that for each 푢 ∈ 푉 , ∑ 푓(푢푣) = admits Super Edge Anti Magic Graceful Labeling. In 푒∈푁(푢) this paper we determine Friendship graph, Dumbbell 푑푖푠푡푖푛푐푡 . Such a labeling is Super Edge Anti Magic graph, Middle graph and Prism graph which are Graceful (푆퐸퐴푀퐺) if |푓(푢) + 푓(푣) − 푓(푢푣)| = Super Edge Anti Magic Graceful. (푺푬푨푴푮). 푑푖푠푡푖푛푐푡. Index Terms- Super edge anti-magic, Graceful labeling, Friendship graph, Dumbbell graph, Middle Definition: graph, Prism graph. A friendship graph 퐹푛 is a planar undirected graph with 2푛 + 1 vertices and 3푛 AMS Subject Classification: 05C78 edges. The friendship graph 퐹푛 can be I. INTRODUCTION constructed by joining 푛 copies of the cycle graph C3 with a common vertex. A graph labelling [3] is a function which has domain as graph elements such as vertices and / Theorem : or edges with co-domain as a set of numbers. A friendship graph 퐹푛 is 푆퐸퐴푀퐺. Usually, the co-domain has been taken as integers. Proof: Many of graph labelings are defined and well- Let 퐺 = 퐹푛 studied by so many authors. Let p be the number of vertices, and q be the In 1963, Sedlacek introduced the concept number of edges. of magic labelling [8] in graphs. A graph 퐺(푝, 푞) is Let 푣1, 푣2, 푣3, . . . , 푣푛 be the vertices of the Fan magic if the edges of G can be labeled by the graph. numbers {1, 2 ,3 , … , 푞 } so that the sum of labels of Let 퐸(퐺) = {1,2,3, … ,3푛} 푎푛푑 푉(퐺) = all the edges incident with any vertex is the same. {1,2,3, … ,2푛 + 1} (푖푒) 푉(퐺) = ∑푒∈푁(푢) 푓(푢푣). In 1967, Rosa introduced graceful labelling Define a function 푓 ∶ 푉(퐺) ∪ 퐸(퐺) → [11]. An injection f from the vertices of {1,2, … . 푞푛 + 푛} as in the following way; 퐺 푡표 {0, 1, 2, … , 푞} is called graceful labeling of G if Fix, 푓(푒 ) = 1 . 푓(푒 ) = 푖 푓표푟 1 ≤ 푖 ≤ 푛 − 1. when we assign each edge 푢푣, the label |푓(푢) − 1 푖 푓(푣 ) = 푓(푒 ) + 푓(푒 ), 푖 = 1,2,3, … , 푛 − 푓(푣)|, the resulting edges are distinct. 푖 푖 푖+1 1 In 1990, N. Hartsfield and G. Ringel = 푖 + 푖 + 1 = 2푖 + 1 , 푖 introduced the concepts called Anti magic = 1,2,4,5,7,8,10,11 … , 푖 ≠ 3푡, 푡 labeling and Anti magic graphs [7]. A graph G is = 1,2,3, … Anti Magic if the q edges of G can be distinctly And 푓(푣 ) = 푞푛 + 푛 labeled in such a way that when taking the sum of 푛 It is easy to observe that the Fan graph 퐹 is the edges labels incident to each vertex they all will 푛 have distinct constants is called the anti-magic 푆퐸퐴푀퐺. labelling.

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Definition: Let 푒1, 푒2, 푒3, . . . , 푒푛 be the edges of the dumbbell A dumbbell graph, denoted by퐷푎,푏.푐 is a graph , such that bicyclic graph consisting of two vertex- disjoint Label the edges are as follows: cycles 퐶푎, 퐶푏and a path 푃푐+3 (c ⩾ −1) joining them f (푒푖) = 푖 푓표푟 1 ≤ 푖 ≤ 푛 , and having only its end-vertices in common with the the vertices are labelled by 푓(푣푖) = two cycles ∑푒∈푁(푢) 푓(푢푣) 1 ≤ 푖 ≤ 푛. Theorem: A dumbbell graph admits 푆퐸퐴푀퐺. Also |푓(푢) + 푓(푣) − 푓(푢푣)| is distinct. Proof: It is clear that the dumbbell graph is 푆퐸퐴푀퐺. Let 푣1, 푣2, 푣3, . . . , 푣푛 be the vertices of the dumbbell graph,

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Definition: (i.e.) f (푒1) = 1 , f (푒2) = 2 , 푓(푒3) = 3 , …, f The middle graph of a graph is the graph whose (푒푛) = 푞 . vertex set is where two vertices are adjacent if and The vertices are label as, 푞 only if they are either adjacent edges of G or one is 푓(푣 ) = 푖 푓표푟 1 ≤ 푖 ≤ and 푖 2 a vertex and the other is an edge incident with it. f (푣푞 + 푖) = 푓 (푒푞 + 1) + 푓 (푒푞 + 2) + 푓(푒푖) ,1 ≤ Theorem: A Middle graph is Super Edge Anti 2 2 2 푞 Magic Graceful. 푖 ≤ 2 Proof: hence all the vertex labels are distinct. Let 퐺 (푝, 푞) be a Middle graph. Also |푓(푢) + 푓(푣) − 푓(푢푣)| is distinct. Let 푣1, 푣2, 푣3, . . . , 푣푛 be the vertices of the Middle, It is clear that G admits 푆퐸퐴푀퐺퐿. graph Thus, the middle graph is 푆퐸퐴푀퐺. Let 푒1, 푒2, 푒3, . . . , 푒푛 be the edges of the Middle graph The edges are label as {1,2,3, … , 푞} , where q is the number of edges.

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Definition: The edges are label as, 푓(푒푖) = 푖 , 푖 = The prism graph is the line graph of the complete 1,2,3, … , 푛 bipartite graph . The prism graph is isomorphic and the vertex labels are 푓(푣푖) = with the cubical graph ∑푒∈푁(푢) 푓(푢푣) , 푖 = 1,2,3, … , 푛 Theorem 3.5: A Prism graph is Super Edge Anti Also |푓(푢) + 푓(푣) − 푓(푢푣)| is distinct. Magic Graceful. It is clear that G admits 푆퐸퐴푀퐺퐿. Hence the prism graph is 푆퐸퐴푀퐺. Proof: Let G be a prism graph.

The edges of the graph are {푒1, 푒2, 푒3, . . . , 푒푛} and the vertices of a graph are {푣1, 푣2, 푣3, . . . , 푣푛}.

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