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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 Di along the x axis and the 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 <V -D>. 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 <V -D> 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. Copyright © 201 6 IJASM, All right reserved 48 International Journal of Applied Science and Mathematics Volume 3, Issue 1, ISSN (Online): 2394-2894 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 <D> 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 , Copyright © 201 6 IJASM, All right reserved 49 International Journal of Applied Science and Mathematics Volume 3, Issue 1, ISSN (Online): 2394-2894 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 |] .
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