Integral Root Labeling of Pm ∪ G Graphs

1. International Research Research and Development (IJTSRD) International Open Access Journal Integral Root Labeling of Pm∪ ∪ ∪ ∪G Graphs V. L. Stella Arputha Mary1, N. Nanthini2 Department of Mathematics,2M.phil Scholar St. Mary’s College (Autonomous), Thoothukudi,Tamil Nadu,India International Journal of Trend in Scientific Scientific (IJTSRD) International Open Access Journal ISSN No: 2456 ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume com | Volume - 2 | Issue – 5 Integral Root Labeling o G Graphs V. L. Stella Arputha Mary 1Department of Mathematics, St. Mary’s College (Autonomous), Thoothukudi India ABSTRACT Let ? ? ??,?? be a graph with ? vertices and labeling if it is possible to label all the vertices induces an edge labeling ??:? → ?1,2,… ?????? ? ?????????????????????????? ? distinct edge labeling on the graph. The graph which Graph. In this paper, we investigate the Integral Root labeling of ??,??∪ ???ʘ??,??, ??∪ ???ʘ??,??, ?? Key words: P_m∪P_n, P_m∪(P_n ʘK_1), P_m ʘK_1), P_m∪(P_n ʘK_1)ʘK_1,2 INTRODUCTION The graph considered here will be finite, undirected and simple. The vertex set is denoted by edge set is denoted by????. For all detailed survey of gr terminology and notations we follow Haray [ concept of Integral Root Labeling of graphs in [8]. In this paper we investigate Integral Root la ??∪ ? graphs. The definitions and other informations which are useful for the present investigation are given below. BASIC DEFINITIONS Definition: 3.1 A walk in which ??,??,…?? are distinct is called a Definition: 3.2 The graph obtained by joining a single pendent edge to each vertex of a path is called a Definition: 3.3 The Cartesian product of two graphs ? vertices ?=(?1?2) and ?=(?1?2) are adjacent in is adjacent to ?1) .It is denoted by ?1×?2 Definition: 3.4 The Corona of two graphs ?1 and ?2 is the graph of ?2 where the ith vertex of ?1 is adjacent to every vertex in the is adjacent to every vertex in the ith copy of ?2. vertices and ? edges. Let ?:? → ?1,2,…? ? 1? is called an if it is possible to label all the vertices ? ∈ ? with distinct elements from ?1 ? …?? defined as ? is called an Integral Root ?1,2,…? ? 1? such that it ? is distinct for all ? is distinct for all ?? ∈ ?. (i.e.) The distinct vertex labeling induces a (i.e.) The distinct vertex labeling induces a admits Integral Root labeling is called an Integral Root distinct edge labeling on the graph. The graph which admits Integral Root labeling is called an stigate the Integral Root labeling of ??∪ ? graphs like??∪ ?∪ ???ʘ???ʘ??,? ??,??∪ ???ʘ???, ??∪ K_1), P_m∪L_n, P_m∪(P_n ʘK_1,2), P_m∪(P_n (P_n ʘK_1,3), P_m∪(T_n The graph considered here will be finite, undirected and simple. The vertex set is denoted by ???? and the The graph considered here will be finite, undirected and simple. The vertex set is denoted by . For all detailed survey of graph labeling we refer to Gallian [1]. For all standard we follow Haray [2]. V.L Stella Arputha Mary and N.Nanthini introduced the concept of Integral Root Labeling of graphs in [8]. In this paper we investigate Integral Root la graphs. The definitions and other informations which are useful for the present investigation are given graphs. The definitions and other informations which are useful for the present investigation are given aph labeling we refer to Gallian [1]. For all standard 2]. V.L Stella Arputha Mary and N.Nanthini introduced the concept of Integral Root Labeling of graphs in [8]. In this paper we investigate Integral Root labeling of are distinct is called a Path. A path on ? vertices is denoted by oted by ?? Comb. The graph obtained by joining a single pendent edge to each vertex of a path is called a ?1=(?1,?1) and ?2=(?2,?2) is a graph ?=(?,?) with ) are adjacent in ?1×?2 whenever (?1=?1and ?2 is adjacent to 2. ) with ?=?1×?2 and two is adjacent to ?2) or (?2=?2and ?1 is the graph ?=?1⨀?2 formed by taking one copy of rmed by taking one copy of ?1 and |(?1)| copies @ IJTSRD | Available Online @ www.ijtsrd.com @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Aug 2018 Page: 2141

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Definition: 3.5 The product graph ??× ?? is called a Ladder and it is denoted by ?? Definition: 3.6 The union of two graphs ??? ???,??) and ??= (??,??) is a graph ? = ??∪ ?? with vertex set ? = ??∪ ?? and the edge set ? = ??∪ ??. Definition: 3.7 The graph ??ʘ??,? is obtained by attaching ??,? to each vertex of ??. Definition: 3.8 The graph ??ʘ??,? is obtained by attaching ??,? to each vertex of ??. Definition: 3.9 A graph that is not connected is disconnected. A graph ? is said to be disconnected if there exist two nodes in ? such that no path in ? has those nodes as endpoints. A graph with just one vertex is connected. An edgeless graph with two (or) more vertices is disconnected MAIN RESULTS Theorem: 4.1 ??∪ ?? is an Integral Root graph. Proof: Let ??= ??,??,….,?? be a path on ? vertices. Let ??= ??,??,….,?? be another one path on ? vertices. Let ? = ??∪ ??. Define a function ?:?(?) → {1,2,…,? + 1} by ?(??) = ?; 1 ≤ ? ≤ ?; ?(??) = ? + ?; 1 ≤ ? ≤ ?. Then we find the edge labels ??(??????) = ?; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + ?; 1 ≤ ? ≤ ? − 1. Then the edge labels are distinct. Hence ??∪ ?? is a Integral Root graph. Example: 4.2 An Integral Root labeling of ??∪ ?? is show below. Figure: 1 Theorem: 4.3 ??∪ (??ʘ??) is a Integral Root graph. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 2142

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Proof: Let ??ʘ?? be a Comb graph obtained from a path ??= ??,??,….,?? by joining a vertex ?? to ??, 1 ≤ ? ≤ ?. Let ??= ??,??,…,?? be a path. Let ? = ??∪ (??ʘ??). Define a function ?:?(?) → {1,2,….,? + 1} by ?(??) = ?; 1 ≤ ? ≤ ?; ?(??) = ? + 2? − 1; 1 ≤ ? ≤ ?; ?(??) = ? + 2?; 1 ≤ ? ≤ ?. Then we find the edge labels are ??(??????) = ?; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + 2?; 1 ≤ ? ≤ ? − 1; ??(????) = ? + 2? − 1; 1 ≤ ? ≤ ? − 1. Then the edge labels are distinct. Hence ??∪ (??ʘ??) is an Integral Root graph. Example: 4.4 An Integral Root labeling of ??∪ (??ʘ??) is given below. Figure: 2 Theorem: 4.5 ??∪ ?? is an Integral Root graph. Proof: Let ??= ??,??,….,?? be a path. Let {??,??,…,?? ,??,??,…..,??} be the vertices of ladder. efine a function ?:?(?) → {1,2,….,? + 1} by ?(??) = ?; 1 ≤ ? ≤ ?; ?(??) = ? + 3? − 2; 1 ≤ ? ≤ ?; ?(??) = ? + 3? − 1; 1 ≤ ? ≤ ?. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 2143

4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Then we find the edge labels ??(??????) = ?; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + 3? − 1; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + 3?; 1 ≤ ? ≤ ? − 1. ??(????) = ? + 3 − 2?; 1 ≤ ? ≤ ?. Then the edge labels are distinct. Hence ??∪ ?? is an Integral Root graph. Example: 4.6 An Integral Root labeling of ??∪ ?? is displayed below. Figure: 3 Theorem: 4.7 Proof: Let ??ʘ??,? be a graph obtained by attaching each vertex of a path ?? to the central vertex of ??,?. Let ??= ??,??,…,?? be a path. Let ?? and ?? be the vertices of ??,? which are attaching with the vertex ?? of ?? 1 ≤ ? ≤ ? . Let ??= ??,??,….,?? be a path. Let ? = ??∪ (??ʘ??,?). Define a function ?:?(?) → {1,2,….,? + 1} by ?(??) = ?; 1 ≤ ? ≤ ?; ?(??) = ? + 3? − 1; 1 ≤ ? ≤ ?; ?(??) = ? + 3? − 2; 1 ≤ ? ≤ ?; ?(??) = ? + 3?; 1 ≤ ? ≤ ?. ??∪ (??ʘ??,?) is a Integral Root graph. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 2144

5. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Then we find the edge labels are ??(??????) = ?; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + 3?; 1 ≤ ? ≤ ? − 1; ??(????) = ? + 3? − 2; 1 ≤ ? ≤ ?; ??(????) = ? + 3? − 1; 1 ≤ ? ≤ ?. Hence ??∪ (??ʘ??,?) is a Integral Root graph. Example: 4.8 An Integral Root labeling of ??∪ (??ʘ??,?) is given below. Figure: 4 Theorem: 4.9 Proof: Let ??ʘ??,? be a graph obtained by attaching each vertex of a path ?? to the central vertex of ??,?. Let ??= ??,??,…,?? be a path. Let ??, ??and ?? be the vertices of ??,? which are attaching with the vertex ?? of ?? 1 ≤ ? ≤ ? . Let ??= ??,??,….,?? be a path. Let ? = ??∪ (??ʘ??,?). Define a function ?:?(?) → {1,2,….,? + 1} by ?(??) = ?; 1 ≤ ? ≤ ?; ?(??) = ? + 4? − 3; 1 ≤ ? ≤ ?; ?(??) = ? + 4? − 2; 1 ≤ ? ≤ ?; ?(??) = ? + 4? − 1; 1 ≤ ? ≤ ?; ?(??) = ? + 4?; 1 ≤ ? ≤ ?. Then we find the edge labels are ??(??????) = ?; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + 4?; 1 ≤ ? ≤ ? − 1; ??(????) = ? + 4? − 3; 1 ≤ ? ≤ ?; ??(????) = ? + 4? − 2; 1 ≤ ? ≤ ?; ??(????) = ? + 4? − 1; 1 ≤ ? ≤ ?. Then the edge labels are distinct. Hence ??∪ (??ʘ??,?) is a Integral Root graph. ??∪ (??ʘ??,?) is a Integral Root graph. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 2145

6. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Example: 4.10 An Integral Root labeling of ??∪ (??ʘ??,?) is given below. Figure: 5 Theorem: 4.11 ??∪ (??ʘ??)ʘ??,? is a Integral root graph. Proof: Let ? = ??∪ (??ʘ??)ʘ??,?. Let ??= ??,??,….,?? be a path. Let ?? be a comb and ?? be the obtained by attaching ??,? at each pendant vertex of ??. Let its vertices be ??,??,??,?? 1 ≤ ? ≤ ?. Define a function ?:?(?) → {1,2,….,? + 1} by ?(??) = ?; 1 ≤ ? ≤ ?; ?(??) = ? + 5? − 3; 1 ≤ ? ≤ ?; ???. ?(??) = ? + 5? − 2; 1 ≤ ? ≤ ?; ?(??) = ? + 5?; 1 ≤ ? ≤ ?. Then we find edge labels are ??(??????) = ?; 1 ≤ ? ≤ ? − 1; ??(??????) = ? + 5? − 1; 1 ≤ ? ≤ ? − 1; ??(????) = ? + 5? − 4; 1 ≤ ? ≤ ?; ??(????) = ? + 5? − 3; 1 ≤ ? ≤ ?; ??(????) = ? + 5? − 2; 1 ≤ ? ≤ ?. Then the edge labels are distinct. Hence ??∪ (??ʘ??)ʘ??,? is an Integral Root graph. Example: 4.12 An Integral Root labeling of ??∪ (??ʘ??)ʘ??,? is given below. ?? = ? + 5? − 4; 1 ≤ ? ≤ ?; @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 2146

7. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Figure: 6 REFERENCE 1.J. A. Gallian, 2010, “A dynamic Survey of graph labeling,” The electronic Journal of Combinatories17#DS6. 2.F. Harary, 1988, “Graph Theory,” Narosa Publishing House Reading, New Delhi. 3.S. Sandhya, S. Somasundaram, S. Anusa, “Root Square Mean labeling of graphs,” International Journal of Contemporary Mathematical Science, Vol.9, 2014, no.667-676. 4.S. S. Sandhya, E. Ebin Raja Merly and S. D. Deepa, “Heronian Mean Labeling of Graphs”, communicated to International journal of Mathematical Form. 5.S. Sandhya, E. Ebin Raja Merly and S. D. Deepa, “ Some results On Heronian Mean Labeling of Graphs”, communicated to Journal of Discrete Mathematical Science of crypotography. 6.S. Sandhya, S. Somasundaram, S. Anusa, “Root Square Mean labeling of Some Disconnected graphs,” communicated to International Journal of Mathematical Combinatorics. 7.S. S. Sandhya, S.Somasundaram and A.S.Anusa, “Root Mean Labeling of Some New Disconnected Graphs”, communicated to International journal of Mathematical Tends and Technology, Volume 15 no.2 (2014) Pg no. 85-92. 8.V. L Stella Arputha Mary, and N. Nanthini, “Integral Root labeling of graph” International Journal of Mathematics Trends Technology(IJMTT), vol.54, no.6(2018), pp.437-442. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 2147