Integral Root Labeling of Graphs 1 V.L. Stella Arputha Mary & 2 N. Nanthini

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1 International Journal of Mathematics Trends and Technology (IJMTT) Volume Number - February 08 Integral Root Labeling of Graphs V.L. Stella Arputha Mary & N. Nanthini Department of Mathematics, St. Mary s College (Autonomous), Thoothukudi-800. M.phil Scholar, St. Mary s College (Autonomous), Thoothukudi-800. ABSTRACT Let G = V, E be a graph with p vertices and q edges. Let f: V,, q + is called an Integral Root labeling if it is possible to label all the vertices v V with distinct elements from,, q + such that it induces an edge labeling f + : E,, q defined as f + (uv) = (f u ) +(f v ) +f u f(v) is distinct for all uv E. (i.e.) The distinct vertex labeling induces a distinct edge labeling on the graph. The graph which admits Integral Root labeling is called an Integral Root Graph. In this paper, we introduce Integral Root labeling and investigate Integral Root labeling of Path, Comb, Ladder, Triangular Snake and Quadrilateral Snake. KEY WORDS Integral Root labeling, Integral Root graph, Path, Comb, Ladder, Triangular Snake, and Quadrilateral Snake. INTRODUCTION By a graph G= (V (G), E (G)) we mean a finite undirected graph without loops or parallel edges. For all detailed survey of graph labeling we refer to Gallian[]. For all other standard terminology and notations we follow Harary[]. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. In this paper we investigate the Integral Root labeling of Path, Comb, Ladder, Triangular Snake, and Quadrilateral Snake.. BASIC DEFINITIONS Definition:. A walk in which u, u, u n are distinct is called a path. A path on n vertices is denoted by P n. Definition:. A Closed Path is called a Cycle. A cycle on n vertices is denoted by C n. Definition:. The graph obtained by joining a single pendent edge to each vertex of a path is called a Comb. Definition:. The Cartesian product of two graphs G =(V,E ) and G =(V,E ) is a graph G=(V,E) with V=V V and two vertices u=(u u ) and v=(v v ) are adjacent in G G whenever (u =v and u is adjacent to v ) or (u =v and u is adjacent to v ).It is denoted by G G. Definition:. The Corona of two graphs G and G is the graph G=G G formed by taking one copy of G and (G ) copies of G where the i th vertex of G is adjacent to every vertex in the i th copy of G. Definition:. The product graph P P n is called a ladder and it is denoted by L n. Example: Ladder graph of L is given below ISSN: -7 Page 7

2 International Journal of Mathematics Trends and Technology (IJMTT) Volume Number - February 08 u u u u v v v v Definition:.7 A Triangular Snake T n is obtained from a path u, u, u n by joining u i and u i+ to a new vertex v i for i n.that is every edge of a path is replaced by a triangle C. Example: Triangular Snake T is given below v v v u u u u Definition:.8 A Quadrilateral Snake Q n is obtained from a path u, u, u n by joining u i and u i+ to two new vertices v i and w i respectively and then joining v i and w i. That is every edge of a path is replaced by a cycle C. Example: Quadrilateral Snake Q is given below v w v w v w u u u u. MAIN RESULTS Definition:. ISSN: -7 Page 8

3 International Journal of Mathematics Trends and Technology (IJMTT) Volume Number - February 08 Let G = V, E be a graph with p vertices and q edges. Let f: V,, q + is called an Integral Root labeling if it is possible to label all the vertices v V with distinct elements from,, q + such that it induces an edge labeling f + : E,, q defined as f + (uv) = (f u ) +(f v ) +f u f(v) is distinct for all uv E. (i.e.) The distinct vertex labeling induces a distinct edge labeling on the graph. The graph which admits Integral Root labeling is called an Integral Root Graph. Theorem:. Any path P n is an Integral Root graph. Let G be a path graph P n. Let u, u,. u n be the vertices of path P n and e, e,. e n be the edges of path P n The path P n consists of n vertices and n edges. Define f: V P n,,. q + byf u i = i ; i n. f + e = u i u i+ = i +i i+ + i+ Hence P n is an Integral Root graph. Example:. The Integral Root labeling of P is given below = i +i +i+i ++i = i +i+ = i are distinct. ; i n Figure: Theorem:. Any Comb P n ʘk is an Integral Root graphs. n Let G be a comb graph P n ʘk. Let u, u,. u n, v, v, v n be the vertices of comb. The comb has n edges. Define a function f: V P n ʘk,,. q + by f u i = i ; i n, f v i = i ; i n. f + e = u i u i+ = i; i n ; f + e = u i v i = i + ; i n are distinct. Hence P n ʘk is an Integral Root graph. Example:. The Integral Root labeling of P 7 ʘk is given below. ISSN: -7 Page 9

4 International Journal of Mathematics Trends and Technology (IJMTT) Volume Number - February Figure: Theorem:. Any Ladder L n is an Integral Root graph. Let G be a Ladder graph L n. Let u, u,. u n, v, v, v n be the vertices of ladder. Define a function f: V L n,,. q + as f u i = i ; i n, f v i = i ; i n. f + u i u i+ = i ; i n, f + u i v i = i ; i n, f + v i v i+ = i ; i n are distinct. Hence L n is an Integral Root graph. Example:.7 The Integral Root labeling of L is given below Theorem:.8 Any Triangular Snake T n is an Integral Root graph. Let G be a triangular snake T n. Define a function f: V T n,,, q + by f u i = i ; i n, Figure: ISSN: -7 Page 0

5 International Journal of Mathematics Trends and Technology (IJMTT) Volume Number - February 08 f v i = i ; i n. f + u i u i+ = i ; i n, f + v i u i+ = i; i n, f + u i v i = i ; i n are distinct. Hence T n is an Integral Root graph. Example:.9 The Integral Root labeling of T is given below Figure Theorem:.0 Any Quadrilateral Snake Q n is an Integral Root graph. Let G be a Quadrilateral Snake Q n. Define a function f: V Q n,,. q + by f u i = i ; i n, f v i = i ; i n, f w i = i ; i n. Then the edge labels f + u i v i = i ; i n, f + u i u i+ = i ; i n, f + u i+ w i = i; i n, f + v i w i = i ; i n are distinct. Hence Q n is an Integral Root graph. Example:. The Integral Root labeling of Q is given below Figure: 7 ISSN: -7 Page

6 International Journal of Mathematics Trends and Technology (IJMTT) Volume Number - February 08 Theorem:. Any cycle C n is not a Integral Root graph n. Let G be a graph of cycle C n. Let u, u,. u n be the vertices of cycle C n and e, e,. e n be the edges of cycle C n. The cycle C n consists of n vertices and n edges. Define f: V C n,,. q + by f u i = i ; i n f + e = u i u i+ = i; i n are distinct., and f(u n u ) is not distinct. Hence C n is not a Root graph. Example:. Cycle C 7 is not Integral Root Labeling of graph is given below 7 Figure REFERENCES [] J.A.Gallian, 00, A dynamic Survey of graph labeling, The electronic Journal of Combinatories7#DS. [] F.Harary, 988, Graph Theory, Narosa Publishing House Reading, New Delhi. [] S.Sandhya, S. Somasundaram, S. Anusa, Root Square Mean labeling of graphs, International Journal of Contemporary Mathematical Science, Vol.9, 0, no.7-7. [] S. S. Sandhya, E. Ebin Raja Merly and S. D. Deepa, Heronian Mean Labeling of Graphs, communicated to International journal of Mathematical Form. [] V.L. Stella Arputha Mary S. Navaneetha Krishnan and A. Nagarajan Z p Magic labeling on some special graphs, International journal of Mathematics and Soft Computing Vol., No: (0) (-70). [] V.L. Stella Arputha Mary S. Navaneetha Krishnan and A. Nagarajan Z p Magic labeling for some more special graphs, International journal of Physical Sciences, Ultra Scientist Vol., No: (0) (9-). ISSN: -7 Page