Pyramidal Sum Labeling In Graphs

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1 ISSN : 48-96, Vol 8, Issue 4, ( Part -I) April 8, pp-9 RESEARCH ARTICLE Pyramidal Sum Labeling In Graphs OPEN ACCESS HVelet Getzimah, D S T Ramesh ( department Of Mathematics, Pope s College, Sayerpuram-68,Ms Uniersity,Tamil Nadu, India, ( department of Mathematic Margoschis College, Nazareth 68 6,Ms Uniersity, Tamil Nadu, India, Corresponding Auther: HVelet Getzimah ABSTRACT Let G = (V, E) be a graph ith p ertices and q edges A graph G is said to admit Pyramidal Sum labeling if its ertices can be labeled by nonnegatie integers,, p q such that the induced edge labels obtained by the Sum of the labels of the end ertices are the first q pyramidal numbers here p q is the q th pyramidal number A graph G hich admits a Pyramidal Sum labeling is called a Pyramidal Sum graph In this Paper e proe that the one ertex union of t copies of any Path, the graphs got by attaching the roots of different Stars to one ertex, Comb graph P n K and all Lobsters are Pyramidal Sum Graphs and sho that some graphs are Unpyramidal By a graph e mean a finite, undirected graph ithout multiple edges or loops For graph theoretic terminology, e refer to Harary [4] and Bondy and Murty [] For number theoretic terminology, e refer to M Apostal [] and Nien and Herbert S Zuckerman [] Key Words: Comb, Helm, Lobsters, Pyramidal Number, Unpyramidal Date Of Submission: Date Of Acceptance I INTRODUCTION All graphs in this Paper are finite, simple, undirected and connected Graph labeling is an assignment of labels to the ertices or edges or to both the ertices and edges subject to certain conditions In 99 Harary introduced the notion of Sum Graphs The concept of Triangular Sum labeling as introduced by S Hedge and P Sankaran Based on these ideas here a ne labeling called as Pyramidal Sum labeling is introduced In this Paper e gie some necessary conditions for a graph to admit Pyramidal Sum labeling and sho that some families of graphs are Pyramidal Sum graphs hile some others are termed as Unpyramidal as they do not satisfy the condition II PYRAMIDAL SUM LABELING Definition : A Triangular number is a number obtained by adding all positie integers less than or equal to a gien positie integer n If n th Triangular number is denoted by T n then T n = n(n+) Triangular numbers are found in the third diagonal of Pascal s Triangle starting at ro 3 They are, 3, 6,,, Definition : The sum of consecutie Triangular numbers is knon as Tetrahedral Numbers They are found in the fourth diagonal of Pascal s Triangle These numbers are, + 3, , They are, 4,,, 3 Definition 3: The Pyramidal numbers or Square Pyramidal numbers are the sums of consecutie pairs of Tetrahedral numbers The folloing are some Pyramidal Numbers: + 4, 4 +, +, + 3 They are,, 4,, Remark 4: The Pyramidal numbers are also calculated by the formula: P n = n(n+)(n+) 6 Definition : A Pyramidal Sum labeling of a graph G is a one-one function defined as f:v G,, p q that induces a bijection : E(G) {P,P,, P q } of the edges of G defined by f + (u) = f(u) +f() for all e = u E(G) here p i is the i th pyramidal number, i =,,3, The graph hich admits such a labeling is called as Pyramidal Sum Graph Notation: The Notation p i is used for each Pyramidal number here i =,,3 Theorem 6: The One Vertex Union of t isomorphic copies of any Path is a Pyramidal Sum graph Proof: Let P n be a Path ith n ertices Let there be t copies of P n Let the central ertex be denoted by, Let the ertices around the central ertex in the first layer be denoted by i,j, The ertices in the second layer by,j and so on Let e i,j be the corresponding edges, here j t, i n f + Define f : V(G),,,, p q as follos: f(, ) = f(,j ) = p j Where j t f(,j ) = p t+j p j Where j t ijeracom DOI: 99/ P a g e

2 ISSN : 48-96, Vol 8, Issue 4, ( Part -I) April 8, pp-9 f( 3,j ) = p t+j p t+j Where j t In general, f( i,j ) = p (i-)t+j p (i-)t+j + p (i-3)t+j ++(-) k- p (i-k)t+j, here i n, k i and for each fixed i, j aries from to t Pyramidal Sum Labeling Of P 4 The Graph P 4 is the union of copies of the Path P 4 Here n = 4, t =, j =,,3 Since there are copies of P 4, the number of edges in P 4 = 3x = and the edge labels defined by the sum of the labels of the end ertices are the first Pyramidal numbers Theorem: : Eery Comb Graph P n K is a Pyramidal Sum Graph Proof: Let G = P n K be a Comb Graph ith n ertices and (n ) edges Let u i, i = to n be the ertices of the Path P n and j, j= to n be the end ertices Let e i, i= to n- be the edges Define f : V(G),,,, p q as follos: Theorem 8: All Lobsters admit Pyramidal Sum Labeling Proof: Let G = (V, E) be a Lobster ith p ertices and q edges A Lobster is a Tree ith the property that the remoal of the end points leaes a Caterpillar Let the different Stars in a Lobster be denoted by K, K, K, K, K, K3,, K, Km Let the Path in a Lobster be denoted by P n here n is the number of ertices in the Path Let the ertices of the Path be denoted by,, 3,, n Let i,ji denote the pendant ertices of each star here i =,,, m and j i =,, 3 k i, Define f : V(G),,,, p q as follos: f( i ) = i3 6, i i 3 6, 6 i n Define f( i,ji ) = p n+ K Label of + K + + K I +j i root ertices, here i =,, 3 n Such a labeling makes all the edges get distinct Pyramidal numbers Example: f(u i ) = i3 6 i 3 6 for i and for 6 i n f( ) = p n f( j ) = P n+i f(u i+ ), for j n, i n The folloing is an example: Pyramidal Sum Labeling Of Lobster Theorem 9: The Graph hich is constructed by attaching the roots of different Stars to one ertex is a Pyramidal Sum graph Pyramidal Sum Labeling Of P K Proof: Let K, n, K, n, K,n3,, K, nt be t Stars that are attached by their central ertices to one ertex Label the ertex as Then label the central ertices of the Stars as p, p p t respectiely The end ertices of the first Star are labeled as p t+ p, p t+ p,, p t+ n p, The end ertices of the second Star are labeled as, p t+ n + p, p t+ n + p,, p t+ n +n p Proceeding like this, The end ertices of the last Star are labeled as, ijeracom DOI: 99/ P a g e

3 ISSN : 48-96, Vol 8, Issue 4, ( Part -I) April 8, pp-9 p t+n +n + +n t + p t, p t+n +n + +n t + p t, p t+n +n + +n t +n t p t No the edge labels got by the sum of the labels of the end ertices are the distinct Pyramidal numbers Example: Consider the folloing graph constructed by attaching the roots of 4 Stars to one ertex Pyramidal Sum Labeling Of Roots Of 4 Stars Attached To One Vertex V III UNPYRAMIDAL GRAPHS Definition 3: The Graphs hich do not admit Pyramidal Sum labeling are termed as Unpyramidal Graphs Remark 3: There does not exist consecutie integers hich are Pyramidal numbers Remark 33: The difference beteen to consecutie Pyramidal numbers is a perfect square and the difference beteen any to Pyramidal numbers is atleast 4 The folloing are some necessary conditions in any Pyramidal Sum Graph Result 34: In a Pyramidal Sum Graph the ertices ith labels and are adjacent Proof: To get the edge label p = it is a must for the ertices ith labels and to be adjacent There is no other possibility Hence in eery Pyramidal Sum graph the ertices ith labels and are adjacent Result 3: In a Pyramidal Sum Graph G, and cannot be the label of ertices of the same triangle contained in G Proof: Let x, y and z be the ertices of a triangle If the ertices x and y are labeled ith and and z is labeled ith some m,,,, p q here m, m Then such a ertex labeling ill gie rise to the edge labels, m+ and m In order to admit a Pyamidal Sum labeling m and m+ must be Pyramidal numbers But this is not possible as there does not exist consecutie integers hich are Pyramidal numbers by Remark 3 Hence the result is proed Result 36: In a Pyramidal Sum graph and 4 cannot be the labels of the ertices of the same Triangle contained in G Proof: Let x,y,z be the ertices of the triangle Let the ertices x and y be labeled ith and 4 and Z be labeled ith some m,,,, p q here m, m 4 Such a ertex labeling ill gie rise to the edge labels, m+, m+4 In order to admit a Pyramidal Sum labeling m+ and m+4 must be Pyramidal numbers For any m, if m + is a Pyramidal number, definitely m + 4 does not yield a Pyramidal number since it is not possible to get 3 as the difference beteen any to pyramidal numbers Hence the result is proed Result 3: If a Graph G contains a cycle of length 3 then G is an Unpyramidal Graph Proof : If G admits a Pyramidal Sum labeling, suppose to ertices of C 3 are labeled ith labels and respectiely then there is a Triangle haing to of the ertices labeled ith and hich contradicts Result 3 Thus G does not admit a Pyramidal Sum labeling If C 3 does not receie the labels and, as the sum of any to Pyramidal numbers is not a Pyramidal number it is not possible Result 38: The Sum of any to Pyramidal numbers is not a Pyramidal number Proof: Suppose the result is not true Then let p x and p y be to Pyramidal numbers such that their sum p z is also a Pyramidal number Therefore e hae, p x +p y = p z () By Remark 33 p -p = p 3 -p = 3 p n - p n = n Hence the Pyramidal number p x can be ritten as p x - p x = x Similarly for p y and p z e hae p y - p y = y, p z - p z = z Adding all e get, p x + p y + p z p x + p y + p z = x +y +z () From () e hae, p x +p y = p z (3) Applying () and (3) in () e get p z -p z = x +y +z Therefore, p z - p z = x +y +z hich is not a perfect square for any positie integers x,y,z here x #, y #, z # and x, y, z But the difference beteen any to Pyramidal numbers is a perfect square Hence this is a contradiction Therefore the result follos Remark 39: All Cycles are Unpyramidal Graphs because of Result 38 But Tte Cycle C alone can ijeracom DOI: 99/ P a g e

4 ISSN : 48-96, Vol 8, Issue 4, ( Part -I) April 8, pp-9 be embedded as an Induced Subgraph of a Pyramidal Sum graph as follos: Embedding Of C Remark 3: Result 3 implies that the Complete Graphs (n 3), Wheel Graphs, Triangular Crocodiles, Dutchindmill Graphs, Floer Graphs, Gear Graphs, Helm Graphs, Fan Graphs, Parachute Graphs are Unpyramidal Graphs 38 6 Theorem 3: The Graph <W n :W m > obtained by joining the apex ertices of Wheels W n and W m to a ne ertex x is an Unpyramidal graph (A) One of the ertices from,, n is labeled ith 4 Without loss of generality assume that f( i ) = 4 for some i {,,,n } Hence e get a Triangle haing to of its ertices labeled ith and 4 hich is a contradiction to Result 36 (B) If f( )=, To get the next edge label p 3 = 4 We hae the folloing Subcases Subcase : Assume that f( i ) = 3 for some i {,,,n } In this Case e get a Triangle hose ertex labels are,3 and x Then x + 3 and x + ill be the edge labels of to edges ith difference hich is not a perfect square Subcase : Assume that and or 3 and or 6 and 8 are the labels of to adjacent ertices from one of the to Wheels so there exists a Triangle hose ertex labels are either,, or,, and,3, or,3, and,6,8 or,6,8 In these cases e get an edge label hich is not a Pyramidal number Subcase 3: Assume that f( i ) = 9 for Some i {,,3,,m } In this case e get a Triangle hose ertex labels are,9 and x Then x+9 and x+ are the labels of to edges hich are not Pyramidal numbers for any x Proof: Let G = <W n :W m > Let us denote the apex ertex of W n by and the ertices adjacent to of the Wheel W n by,,, n Similarly denote the apex ertex of other Wheel W m By and the ertices adjacent to W of the Wheel W m by,,, m Let u be the ne ertex adjacent to the apex ertices of both the Wheels u n W n 3 m The Graph <W n :W m > Wm 3 Let f:v G,,,, p q be one of the possible Pyramidal Sum Labeling As and are the labels of any to adjacent ertices of the graph G, e hae the folloing cases: Case : If and are the labels of adjacent ertices in W n or W m, then there is a Triangle haing to of the ertices labeled ith and hich is a contradiction to Result 3 Case : If f(u) = then one of the ertices, is labeled ith Without loss of generality assume that f( ) = To get the edge label p = e discuss the folloing possibilities: Case 3: If f(u) =, then one of the ertices and must be labeled ith Without loss of generality assume that f( ) = To hae an edge label e hae the folloing cases: Subcase 3: If f( i ) = for some i {,,,n } then there is a Triangle haing ertex labels as,,x Therefore e hae to edge labels as x + and x hich are not Pyramidal numbers for any x Subcase 3: Assume that f( ) = 4 To obtain the edge label p 3 = 4 We discuss the folloing possibilities: (A) If f( i ) = 4 for some i {,,,n }, then there is a Triangle haing ertex labels,4,x Therefore e hae to edge labels as x and x + 4 haing difference 4 hich is not possible (B) If f( i ) = for some i {,,,m }, then there is a Triangle ith ertex labels,4,x The edge labels are 4, 4 + x, + x hose difference is 6 As discussed aboe this is not possible (C) Assume that and, 3 and, and 9, 6 and 8 are the labels of to adjacent ertices from one of the to Wheels So there is a Triangle hose ertices are either,, and 4,, (Or),3, and 4,3, (Or),,9 and 4,,9 (Or),6,8 and 4,6,8 In all the aboe possibilities there exists an edge label hich is not a Pyramidal number Hence the Graph G = <W n : W m >is an Unpyramidal Graph Theorem 3: The Helm Graph H n is an Unpyramidal Graph ijeracom DOI: 99/ P a g e

5 ISSN : 48-96, Vol 8, Issue 4, ( Part -I) April 8, pp-9 Proof: The Helm Graph H n is obtained from a Case 3: Suppose one of the ertices from Wheel W n = C n +K by attaching a Pendant edge at u,u,,u n is labeled ith Without loss of each ertex of C n Let c be the apex ertex of the generality assume that f(u ) = Since and are Wheel and,,, n be the ertices adjacent to c alays adjacent in a Pyramidal Sum labeling e in the clockise direction Let the Pendant ertices hae f( ) = No, the edge label p = can be adjacent to,,, n be denoted by u,u,,u n obtained by using the ertex labels, or,4 As u respectiely is a pendant ertex the combination, is not possible Therefore one of the ertices from, n,c must be labeled ith 4 Hence there is a Triangle ith ertex labels as and 4 hich is a contradiction to Remark 36 Hence in each of the possibilities discussed aboe, H n does not admit Pyramidal Sum labeling Hence H n is an Unpyramidal Graph The Helm H 8 If H n admits a Pyramidal Sum labeling then e hae the folloing cases: Case : Suppose f(c ) = Since the labels and are alays adjacent in a Pyramidal Sum graph e assign the label to exactly one of the ertices,,, n In this case there is a Triangle ith ertex labels as and hich is a contradiction to Result 3 Case : Suppose one of the ertices from,,, n is labeled ith Without loss of generality assume that f( ) = Here e consider to subcases: Subcase : If one of the ertices from c,, n is labeled ith, then in each case there is a Triangle ith ertex labels as and hich is a contradiction to Result 3 Subcase : If f(u ) = then e discuss the possibility of the next edge label The label p = can be obtained by using the ertex labels, or,4 As u is a pendant ertex haing label the combination,4 is not possible Therefore one of the ertices, n,c must be labeled ith Thus there is a Triangle ith to ertices labeled as and Let the third ertex be labeled ith x such that x, x To admit a Pyramidal Sum labeling x and x+ must be distinct Pyramidal numbers But here the difference is hich is not a perfect square Hence it is not possible for the edges to receie the labels as Pyramidal numbers IV CONCLUSION As all Connected acyclic graphs satisfy the condition of Pyramidal Sum Labeling, Pyramidal numbers can be applied in problems connected ith Trees In the aboe Labeling, the Pyramidal numbers are brought into existence As the difference beteen any to Pyramidal nmbers is a perfect square and the difference is sufficiently large, Pyramidal numbers can be used as frequencies in Distance labelings such as L(3,,), L(4,3,,) and in Radio labelings to make the frequencies of the Transmitters sufficiently large for better transmission REFERENCES: [] M Apostal, Introduction To Analytic Number Theory, Narosa Publishing House, Second Edition, 99 [] JA Bondy And USR Murty, Graph Theory With Applications, Macmillan Press,London(96) [3] JA Gallian, A Dynamic Surey Of Graph Labeling, The Electronic Journal Of Combinatorics (4) [4] F Harary, Graph Theory Addison Wesley, Reading Mars, (968) [] I Nien And Herbert S Zuckerman, An Introduction To The Theory Of Numbers, Wiley Eastern Limited, Third Edition,99 HVelet Getzimah" Pyramidal Sum Labeling In Graphs"InternationalJournal of Engineering Research and Applications (IJERA), ol 8, no 4, 8, pp -9 ijeracom DOI: 99/ P a g e