A Note on Soenergy of Cocktailparty and Crown Graphs

International Journal of Applied Science and Mathematics Volume 3, Issue 1, ISSN (Online): 2394-2894

A Note on soEnergy of Cocktailparty and Crown Graphs

S. P. Jeyakokila P. Sumathi

Abstract – Let G be finite non trivial connected graph. Definition 1.5 idegree, odegree, oidegree of a minimal dominating set were Let G be a graph and D be a dominating set, oEnergy of introduced and oEnergy of a graph with respect to the ε dominating set were calculated, an Algorithm to get the a graph with respect to D denoted by o D (G) is the soEnergy is being introduced and it is found out for some summation of all oid if od > id or otherwise zero . standard graphs in the earlier papers. In this paper soEnergy Definition 1.6 of complete graph, cocktail graph, complete bipartite graph Let G be a graph and D be a minimal dominating set , and crown graphs with respect to the given minimal th en energy curve is the curve obtained by joining the dominating sets were found out. oEnergies with respect to and for ≤ ≤ , taking Di−1 Di 1 i n Keywords – oEnergy, soEnergy. the number of vertices of D along the x axis and the i oEnergy with respect to the D along the y axis. I. INTRODUCTION i Definition 1.7 Let G = (V, E) be a finite non trivial connected graph. A Let G be a graph and D be a minimal dominating set, |V −D| set D is a dominating set of G if every vertex in V -D is soEnergy of a graph with respect to D is ε ∑o D (G) i+1 adjacent to some vertex in D. A dominating set D of G is i=0 called a minimal dominating set if no proper subset of D is where D = D ∪V , V is a singleton vertex with a dominating set. i+1 i i+1 i+1 Cocktail party graph is the graph consisting of two rows minimum oidegree of V-Di and D 0 is a minimal ≤ ≤ − of paired nodes in which all nodes but the paired on es are dominating set where 0 i |V D|, it is denoted by connected with a graph edge [5].A crown graph on 2n ε so D (G) . vertices is an undirected graph with two sets of vertices u i Definition 1.8 and v and with an edge from u to v whenever i ≠ j[5]. In i i j Let G be a graph and MDS(G) be the set of all minimal this paper soEnergy of complete graphs, complete bipartite dominating set of G, then Hardihood + of a graph G is graphs, cocktail party graphs and crown graphs are being max{ soε ( G ) } is denoted as HD +(G). calculated with respect to their minimal dominating sets . MDS( G ) Definition 1.9 II. PRE LIMINARIES Let G be a graph and MDS(G) be the set of all minimal - dominating set of G, then Hardihood of a graph G is min{ ε } is denoted as HD -(G). Definition 1.1 soMDS( G ) ( G ) Let G be a graph and S be a subset of V(G). Let v ∈V- Given below is the algorithm to find the soEnergy of S, the idegree or indegree of v with respect to S is the any given graph with respect to the given dominating set. number of neighbours of v in V-S and it is denoted by Algorithm 1.10 . id S (v) 1. Find a minimal dominating set D . Definition 1.2 2. Find the idegree, odegree and oidegree for the vertices Let G be a graph and S be a subset of V(G). Let v ∈V- in the graph induced by . S, the odegree or outde gree of v with respect to S is the 3. If oideree of the vertices are greater than zero, proceed to step 5. number of neighbours of v in S and is denoted as od( v ). S 4. If oidegree ≤ 0 then put oenergy = 0 and goto step 6. Definition 1.3 ∈ 5. Shift the vertex with minimum positive oidegree Let G be graph and S be a subset of V(G). Let v V-S, which appears first to the set D. the oidegree or outindegree of v with respect to S is 6. If no s uch positive oidegree exists, shift a vertex with − > odS() v id S () v if odS() v id S () v and it is denoted by oidegree zero to the set D otherwise shift a vertex oid S (v) . minimum iodegree to the set D. Definition 1.4 7. Find the idegree, odegree and iodegree and then oenergy for the vertices in the new with Let G be a graph and S be a subset of V(G). Let v ∈V- respect to new D set. S, the iodegree or inoutdegree of v with respect to S is 8. Rep eat steps 2 to 7 until |V -D|=0. id() v− od () v if id() v> od () v and it is denoted by S S S S 9. Find the sum of all the oEnergies at each step to get iod( v ). S the soEnergy.

10. Fix the x axis as the number of vertices of the set D i and y axis as the oEnergy of the sets at each and every step. Plot the graph.

III. soENERGY OF COMPLETE GRAPHS AND COCKTAIL PARTY GRAPHS

A complete graph with n vertices denoted by Kn is a graph with n vertices in which each vertex is connected to each of the others (with one edge between each pair of vertices). soEnergy of complete graph is calculated below. Theorem 3.1 Let K n be the graph and D be a minimal dominating set , n then ε = − − − − so D (kn ) ∑|V Di [|| Di | |V Di | ]1 n i= 2 Proof: Fig.1 .

Let D be a minimal dominating set of K . n The cocktail party graph of order n, also called the hyper Claim: octahedral graph. It is the graph complement of the Q9 For any K soEnergy is 0 if |D | < n . n i ladder rung graph L and the dual graph of the hypercube 2 n graph Q [5] i.e) |D |-(|V-D |-1)<1 if |D | < n n. i i i For a cocktail party graph the possible minimal 2 dominating sets are minimum independent dominating set for, and connected dominating set. Connected dominating set when |D | < n i   is a dominating set D such that is a connected graph. 2 Definition 3.2 n Cocktail party graph i s denoted by K nx2 , is a graph |V-Di|=|n- | || n having the vertex set = and the edge set 2 V∪( Ui , V i ) n i=1 |n- n +1| or |n- | || E = {u u , v v ;i ≠ j}∪{u v ,v u 1; ≤ i ≤ n}[9]   2 i j i j i j i j 2 Corollary 3.3 =|n- n +1| or n   Let G be a cocktail party graph K nx2 , 2 2 2n (i) sokε ()= | VDD − |[||| −− VD |] where D is n n D nx2 ∑ i i i i.e, |V-Di|-1 ≥ or -1 i= n + 1   2 2 connected dominating set. 2n n  n n |D i|-(|V-Di|-1)< - or ( - ( n -1)) <1 (ii) so ε (k ) = 4 + | V − D [|| D | − | V − D |] where         D nx 2 ∑ i i i 2 2 2 2 i=n+1   D is minimum independent dominating set. Therefore soEnergy is 0 if |D i| < n 2 Proof: n Let G be a cocktail party graph K nx2 and D be a minimal After this stage, |D i| < , dominating set. 2 i) Let D be the connected dominating set. oidegree of vertices are |D |-(|V-D |-1) i i Number of vertices in a cocktail party graph is 2n . By by definition, oEnergy at ith stage is (|V -Di|){|D i|-|V-Di|-1} n replacing n by 2n in theorem 3. 1,we get soKε ()= | VD − |[||| D −−− VD |1] 2n Dn∑ ii i ε = − −− n sokD() nx2 ∑ | VDD i |[||| i VD i |] . i= 2 i= n + 1 Hence the proof. (ii) Let D be minimum independent dominating set. The cardinality of minimum independent dominating set Energy Curve of K 8 is 2. Since the vertices are independent, they gain 2 values for each vertices. 2n Therefore ε =+ − −− sokD( nx2 )4∑ | VDD i |[||| i VD i |] i= n + 1 where D is minimum independent dominating set. Corollary 3.4 Let G be a c ocktail party graph K nx2 ,

2n (i)Hardihood +(K ) =+ − −− where nx2 4∑ |VDi |[||| D i VD i |] i= n + 1 D is minimum independent dominating set. 2n - = − −− (ii)Hardihood (K nx2 ) ∑ |VDi |[| D i || VD i |] where i= n + 1 D is connected dominating set. Proof: Let G be a cocktail party graph Knx2. Minimum independent dominating set and connected dominating set are the minimal dominating sets of c ocktail party graphs. ε so D (knx 2 ) of both the dom inating sets differ only by 4. 2n Fig. 3. + − −− 4∑ |VDi |[||| D i VD i |] i= n + 1 Energy curves of cockt ail graph with even number of 2n ≥ − −− n ∑ |VDi |[| D i || VD i |] i= n + 1 + Therefore by the definition, Hardihood (K nx2 ) = 2n =+ − −− 4∑ |VDi | [| D i | | VD i |] . This is the soEnergy i= n + 1 when D is minimal independent dominating set. 2n - = − −− Hardihood (K nx2 ) ∑ |VDi |[| D i || VD i |] This is i= n + 1 the soEnergy when D is connected dominating set. + 2n Hence Hardihood (K nx2 ) =+ − −− 4∑ |VDi |[||| D i VD i |] i= n + 1 where D is minimum indepe ndent dominating set and 2n - = − −− Hardihood (K nx2 ) ∑ |VDi |[| D i || VD i |] where D = + i n 1 is connected dominating set. Fig. 4. Energy curves of Cocktail party graphs are given below. Energy curve for K3x2 and K 4x2 with respect to minimum independent set and conne cted dominating sets are traced . Energy Curves of Cockta il graph with odd number of n

Fig. 5.

Fig. 2. IV. SO ENERGY OF COMPLETE BIPARTITE GRAPH

A complete bipar tite graph is a bipartite graph i.e., a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent) such that every pair of graph vertices in the two

sets are adjacent. There are three possible minimal soEnergies of Kn,m are dominating sets for a complete bipartite graph. nmm(+ 1) nmn ( + 1) n+ m − 2 , ,(∑ od− id ) ,they are Definition 4.1 Di 2 2 i= n − 2 A dominating set of a complete bipartite graph K that n,m n+ m − 2 mnn(+ 1) nmm ( + 1) contains the vertices in a partite having the cardinality m ∑ (od− id ) ≤ ≤ Di and cardinality n are called m-partite dominating set and i= n − 2 2 2 + n-partite dominating set respectively . + nm( m 1) Hence Hardihood (K n,m )= . This is the soEnergy Theorem 4.2 2 - Let K n,m be the graph, then of K n,m when D is n-partite dominating set and Hardihood + n+ m − 2 nm (m )1 ()K=∑ ( od − id ) is the soEnergy when D is minimal (i) so ε (K ) = where D is n-partite n, m D i D n,m 2 i= n − 2 dominating set. dominating set. nm (n + )1 Energy Curve of K 3,2 (ii) so ε (K ) = where D is D n,m 2 m-partite dominating set. n+m−2 (iii) ε = − where D is minim um so D (K n,m ) ∑ (od id )( Di ) i=n−2 independent dominating set. Proof: Let G be a complete bipartite graph Kn,m . (i) Let D be the n-partite dominating set of the cardinality n, then id D=0 and od D=n for every v i in V -D. Hence oid D=n for all v i in V-D. oε (K ) = nm , n(m − ),1 n(m − 2),... n,0 D n,m respectively Fig. 6. mn (n + )1 Therefore so ε (K ) = where n>m. D n,m 2 (ii) Let D be the m-partite dominating set of the cardinality Energy curve of K4x3 with 4 - m, then id D=0 and od D=m for every v i in V -D. partite domination set mn (n + )1 As in case (i) so ε (K ) = where D is m-partite 20 D n,m 2 dominating set. 10 (iii) Let D be minimum independent dominating set, 0 taking one vertex from each partite, then the cardinality of 0 2 4 6 8 the dominating set is 2. For any K n,m soEnergy is 0 if |D i| od Di when i= 1,2,...(n-2), Fig. 7. n+m−2 By definition, ε = − so D (Kn,m ) ∑(od id )( Di ) i=n−2 Hence the proof. + If n>m, then Hardihood (K n,m ) occurs with n -partite - dominating set and Hardihood (K n,m ) occurs for minimum independent dominating set. Theorem 4.3

+ mn( m + 1) (i) Hardihood (Kn,m ) = whe n D is n-partite 2 dominating set. n+m−2 - − (ii) Hardihood (Kn,m ) = ∑ (od id )D where D is i i=n−2 minimum independent dominating set. Proof: Let G be a complete bipartite graph K n,m ,n>m. Fig. 8.

V. SO ENERGY OF CROWN GRAPHS n( n 2 − 1) (i) Hardihood +( S 0 )= where D is maximal n 2 The crown graph can be viewed as a complete bipartite independent dominating set. graph from which the edges of a perfect matching have 2n been removed, Biggs (1993) [4]. For a crown graph the (ii) Hardihood -( S 0 )= ∑(od− id ) where D is minimal n Di possible minimal dominating sets with crown graphs are i= n minimum and maximal independent domina ting set with independent dominating set. cardinality of dominating set as 2 and n. The number of Proof: edges of crown graph is n(n-1). Let G be a S 0 for an integer n ≥2. Definition:5.1 n 0 2 − 0 n( n 1) Crown graph Sn for an integer n ≥2 is the graph with soEnergies of a S for an integer n ≥2 are and n 2 vertex set {u1,u2 ,... un,v ,1 v ,2 ..., vn} and the edge set 2n 2n ∑(od− id ) . It is obvious that $ ∑(od− id ) ≤ ≤ ≤ ≠ Di Di {ui,v j 1; i, j n;i j}.[8] i= n i= n A dominating set D of G is called an independent n( n 2 − 1) . dominating set if the vertices in D are independent. A 2 dominating set D of G is called a maximum or minimum 2 − + 0 n( n 1) independent dominating set if D is a n independent Hence Hardihood ( Sn )= which is the dominating set with max imum or minimum cardinality. 2 Theorem : 5.2 soEnergy when D is n-partite dominating set. 2n Let G be a S 0 for an integer n ≥2 And Hardihood -( S 0 )= ∑(od− id ) which is the n n Di i= n n( n 2 − 1) (i) soε ( S 0 ) = where D is maximal soEnergy w hen D is minimal independent . D n 2 Energy Curves of Crown graph independent do minating set of cardinality n. 2n (ii) soε () S0 =∑ ( od − id ) where D is minimum D n D i i= n independent dominating set. Proof: 0 Let G be a Crown graph Sn for an integer n ≥2. Let D be the minimal dominating set. (i) Let D be the maximal independent domi nating set with cardinality n. Then id=0 and od=n-1 for all the n vertices in the set V - D. Therefore oε (S 0 ) = n(n − )1 D1 n By theorem 14 n(n-1)+(n-1)(n-1)+(n-2)(n -1)+...+(n-1)$ Fig. 9. (n− 1)( nn )( + 1) nn (2 − 1) = = 2 2 n( n 2 − 1) Therefore soε ( S 0 ) = D n 2 (ii) Let D be the minimal independent domi nating set with cardinality 2. 0 For any Sn , soEnergy is 0 if |D i| od when i= 1,2,...(n-1), Di Di 2n By definition of soε () S0 =∑ ( ododid − id ). D n D i i= n Hence the proof. Theorem : 5.3 Let G be a S 0 for an integer n ≥2 Fig. 10 . n

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AUTHOR 'S PROFILE

S. P. Jeyakokila working in Lady Doak College, Madurai, India as Assistant Professor. Her area of interest is Graph theory. Email id: [email protected]

P.Sumathi , Head and Associate professor , Department of Mathematics, C. Kandasamy Naidu College, Chennai , India. Her area of interest is Graph theory. She is guiding many research scholars. Already two of them have pursued the degree. Email Id: [email protected]