Applied Mathematical Sciences, Vol. 7, 2013, no. 111, 5525 - 5536 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.38477 Minimum Covering Distance Energy of a Graph M. R. Rajesh Kanna, B. N. Dharmendra Department of Mathematics Maharani’s Science College for Women J. L. B. Road, Mysore - 570 005, India [email protected], [email protected] R. Pradeep Kumar Research and Development Centre Bharathiar University Coimbatore 641 046, India [email protected] Copyright c 2013 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Recently Prof.Chandrashekar Adiga et al.[1] have defined the minimum covering energy, EC (G) of a graph G which depends on its particular minimum cover C. Motivated by this paper, we introduced the concept of minimum covering distance energy ECd(G) of a graph G and computed minimum covering distance energies of a star graph, complete graph, crown graph, bipartite graph and cocktail graphs. Upper and lower bounds for ECd(G) are also established. Mathematics Subject Classification: 05C50, 05C69 Keywords: Minimum covering set, Minimum covering distance matrix, Minimum covering distance eigenvalues, Minimum covering distance energy of a graph 1 Introduction The concept of energy of a graph was introduced by I. Gutman [10] in the year 1978. Let G be a graph with n vertices {v1,v2, ..., vn} and m edges. Let A =(aij) be the adjacency matrix of the 5526 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar graph. The eigenvalues λ1,λ2, ··· ,λn of A, assumed in non increasing order, are the eigenvalues of the graph G.AsA is real symmetric, the eigenvalues of G are real with sum equal to zero. The energy E(G) of G is defined to be the sum of the absolute values of the eigenvalues of G. n i.e., E(G)= |λi|. i=1 For details on the mathematical aspects of the theory of graph energy see the reviews[11] ,papers [6, 7, 12] and the references cited there in. The basic properties including various upper and lower bounds for energy of a graph have been established in [14, 18], and it has found remarkable chemical applications in the molecular orbital theory of conjugated molecules [8, 13]. The distance matrix of G is the square matrix of order n whose (i, j)-entry is the distance (= length of the shortest path) between the vertices vi and vj . Let ρ1,ρ2, ..., ρn be the eigenvalues n of the distance matrix of G. The distance energy DE is defined by DE = DE(G):= |ρi| i=1 . Detailed studies on distance energy can be found in [5, 9, 15, 16, 21]. Further studies on covering energy,maximum degree energy, dominating energies can be found in [1, 2, 19, 20]and the references cited there in. 2 Definitions and examples: 2.1 THE MINIMUM COVERING ENERGY A GRAPH Let G be a simple graph of order n with vertex set V = {v1,v2, ..., vn} and edge set E. A subset C of V is called a covering set of G if every edge of G is incident to at least one vertex of C.Any covering set with minimum cardinality is called a minimum covering set. Let C be a minimum covering set of a graph G. The minimum covering matrix of G is the n × n matrix defined by AC (G):=(aij), ⎧ ⎨ 1ifvivj ∈ E a i j v ∈ C where ij = ⎩ 1if= and i 0 otherwise The characteristic polynomial ofAC (G) is denoted by fn(G, λ)= det(λI − AC (G)). The minimum covering eigenvalues of the graph G are the eigenvalues of AC (G). Since AC (G)is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order n λ1 λ2 ··· λn. The minimum covering energy of G is then defined as EC (G)= |λi|. i=1 2.2 THE MINIMUM COVERING DISTANCE ENERGY Let G be a simple graph of order n with vertex set V = {v1,v2, ..., vn} and edge set E. Let C be a minimum covering set of a graph G. The minimum covering distance matrix of G is the n × n matrix defined by ACd(G):=(dij), 1ifi = j and vi ∈ D where dij = d(vi,vj) otherwise Minimum covering distance energy of a graph 5527 The characteristic polynomial of ACd(G) is denoted by fn(G, ρ)=det(ρI − ACd(G)). The minimum covering eigenvalues of the graph G are the eigenvalues of ACd(G). Since ACd(G)is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order ρ1 ρ2 ··· ρn. The minimum covering distance energy of G is defined as ECd(G): = n |ρi| i=1 Note that the trace of ACd(G)=|C|. In this paper, we are interested in studying mathematical aspects of the minimum covering distance energy of a graph. The application of minimum covering distance energy in other branches of science have to be investigated. EXAMPLE 1: The possible minimum covering sets for the following graph G [Figure 1] are i) C1={v1,v2,v5} ii) C2={v2,v4,v5} iii) C3={v1,v3,v5} vs1 sv2 vs3 s s s v4 v5 v6 FIGURE - 1 ⎛ ⎞ 112112 ⎜ ⎟ ⎜ 111212⎟ ⎜ ⎟ A G ⎜ 210212⎟ i) Cd1 ( )=⎜ ⎟ ⎜ 122012⎟ ⎝ 111111⎠ 222210 Characteristic equation is ρ6 − 3ρ5 − 33ρ4 − 50ρ3 +5ρ2 +21ρ − 5=0. Minimum covering distance eigenvalues are ρ1 ≈−2.4142,ρ2 ≈−2.2203,ρ3 ≈−1.0000,ρ4 ≈ 0.2837,ρ5 ≈ 0.4142,ρ6 ≈ 7.9366. Minimum covering distance energy, ECd1 (G) ≈ 14.2691 ⎛ ⎞ 012112 ⎜ ⎟ ⎜ 111212⎟ ⎜ ⎟ A G ⎜ 210212⎟ ii) Cd2 ( )=⎜ ⎟ ⎜ 122112⎟ ⎝ 111111⎠ 222210 5528 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar Characteristic equation is ρ6 − 3ρ5 − 33ρ4 − 53ρ3 − 6ρ2 +13ρ − 1=0. Minimum covering distance eigenvalues are ρ1 ≈−2.3348,ρ2 ≈−2.2587,ρ3 ≈−0.8188,ρ4 ≈ 0.0825,ρ5 ≈ 0.3520,ρ6 ≈ 7.9778 Minimum covering distance energy, ECd2 (G) ≈ 13.8246. ∴ Minimum covering distance energy depends on the covering set. 3 MINIMUM COVERING DISTANCE ENERGY OF SOME STANDARD GRAPHS Definition 3.1. The Cocktail party graph is denoted by Kn×2,is a graph having the vertex set n V = {ui,vi} and the edge set E = {uiuj,vivj : i = j} {uivj,viuj :1≤ i<j≤ n}. i=1 Theorem 3.1. The minimum covering distance energy of Cocktail party graph Kn×2 is 4n. n Proof. Let Kn×2 be the Cocktail party graph with vertex set V = {ui,vi}. The minimum i=1 n−1 covering set is C = {ui,vi}. Then i=1 ⎛ ⎞ 1211... 1111 ⎜ ⎟ ⎜ 2111... 1111⎟ ⎜ ⎟ ⎜ 1112... 1111⎟ ⎜ ⎟ ⎜ 1121... 1111⎟ ⎜ ⎟ A K ⎜ . .. ⎟ Cd( n×2)=⎜ . ⎟ ⎜ ⎟ ⎜ 1111... 1211⎟ ⎜ ⎟ ⎜ 1111... 2111⎟ ⎝ ⎠ 1111... 1102 1111... 1120 Characteristic equation is ( ρ − 1)n−2( ρ + 2)( ρ +1)n−1[ ρ2 − (2n +1)ρ +2]=0 Minimum covering distance eigenvalues are ρ = 1[(n − 2)times] , ρ = −2 [one time], √ (2n +1)± 4n2 +4n − 7 ρ = −1[(n − 1) times], ρ = [one time each] 2 Minimum covering distance energy, ECd(Kn×2) √ (2n +1)+ 4n2 +4n − 7 = |1|(n − 2) + |−2|(1) + |−1|(n − 1) + + √ 2 (2n +1)− 4n2 +4n − 7 2 =4n. Theorem 3.2. For n ≥ 3, the minimum covering distance energy of Star graph K1,n−1 is equal to 4n − 7. Minimum covering distance energy of a graph 5529 Proof. Consider the Star graph K1,n−1 with vertex set V = {v0,v1,v2, ..., vn−1}. The Minimum covering set C = {v0}. Then ⎛ ⎞ 111... 1 ⎜ ⎟ ⎜ 102... 2 ⎟ ⎜ ... ⎟ ACd(K1,n−1)=⎜ 120 2 ⎟ ⎜ . ⎟ ⎝ . .. ⎠ ... 122 0 n×n Characteristic equation is (ρ +2)n−2[ ρ2 − (2n − 3)ρ +(n − 3)] = 0 The minimum covering distance eigenvalues are√ (2n − 3) ± 4n2 − 16n +21 ρ = −2[(n − 2) times], ρ = [one time each]. 2 E K Minimum covering distance energy√ is, Cd( 1,n−1) √ (2n − 3) + 4n2 − 16n +21 (2n − 3) − 4n2 − 16n +21 = |−2|(n − 2) + + 2 2 =4n − 7. 0 Definition 3.2. The Crown graph Sn for an integer n ≥ 2 is the graph with vertex set 0 {u1,u2, ..., un,v1,v2, ..., vn} and edge set {uivj :1≤ i, j ≤ n, i = j}. ∴ Sn coincides with the Complete bipartite graph Kn,n with horizontal edges removed. 0 Theorem 3.3.√ The minimum covering distance energy of the Crown graph Sn is equal to if n √ 5 √ =2 if n 101 + 2 17√ =3 (4n − 3) + (n − 1) 17 if n > 3 0 Proof. For the Crown graph Sn with vertex set V = {u1,u2, ..., un,v1,v2, ..., vn}. The minimum covering set is C = {u1,u2, ..., un} .Then ⎛ ⎞ 122... 2311... 1 ⎜ ⎟ ⎜ 212... 2131... 1 ⎟ ⎜ ⎟ ⎜ 221... 2113... 1 ⎟ ⎜ ⎟ ⎜ . .. .. ⎟ ⎜ . ⎟ ⎜ ⎟ 0 ⎜ 222... 1111... 3 ⎟ ACd(Sn)=⎜ ⎟ ⎜ 311... 1022... 2 ⎟ ⎜ ⎟ ⎜ 131... 1202... 2 ⎟ ⎜ ⎟ ⎜ 113... 1220... 2 ⎟ ⎜ . ⎟ ⎝ . .. .. ⎠ ... ... 111 3222 0 (2n×2n) 5530 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar 2 2 i) Characteristic equation for n = 2 is (ρ − ρ − 1) =0. √ 1 ± 5 Minimum covering distance eigenvalues for n = 2 are ρ = [two √ 2 0 times each]. ∴ ECd(Sn)= 5. ii) Characteristic equation for n ≥ 3is [ ρ2 − (4n − 3)ρ +(3n2 − 10n − 2)](ρ2 +3ρ − 2)n−1 =0 Minimum covering√ distance eigenvalues for n ≥ 3 are √ (4n − 3) ± 4n2 +16n +17 −3 ± 17 ρ = [one time each], ρ = [(n − 1) times 2 2 each] 0 Minimum covering distance energy for n ≥ 3, ECd(Sn) √ √ √ (4n − 3) + 4n2 +16n +17 (4n − 3) − 4n2 +16n +17 −3+ 17 = + + 2 √ 2 2 −3 − 17 (n − 1) + (n − 1) 2 √ √ 0 Case 1 :Ifn = 3 then ECd(Sn)= 101 + 2 17.