International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 10 No. II (August, 2016), pp. 215-230

MINIMUM COVERING PARTITION ENERGY OF A GRAPH

P. S. K. REDDY,1 K. N. PRAKASHA2 AND GAVIRANGAIAH K.3 1 Department of Mathematics, Siddaganga Institute of Technology, B. H. Road, Tumkur-572 103, India 2 Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru- 570 002, India 3 Department of Mathematics, Government First Grade Collge, Tumkur-562 102, India

Abstract The Partition energy of a graph was introduced by E. Sampathkumar et al. (2015). Motivated by this, we introduce the concept of minimum covering partition energy C C of a graph, Ep (G) and compute the minimum covering partition energy Ep (G) of few families of graphs. Also, we established the bounds for minimum covering partition energy.

1. Introduction The graph energy plays a vital role in chemistry to find the total π-electron energy of a molecule. The conjugated hydrocarbons can be represented by a graph called molecular

−−−−−−−−−−−−−−−−−−−−−−−−−−−− Key Words : Ninimum covering set, Minimum covering k-partition eigenvalues, Minimum cov- ering k-partition energy, k-complement, k(i)-complement. 2010 AMS Subject Classification : 05C50. c http: //www.ascent-journals.com

215 216 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K. graph. We can represent every carbon atom by a and every carbon-carbon bond by an edge and hydrogen atoms are ignored. Recently several matrices like adjacency matrix, Laplacian matrix, distance matrix, maximum degree matrix, minimum degree matrix, matrix of a subset S of V and color energy of a graph are studied in [1, 2, 4, 5, 10, 11 and 16].

2. Minimum Covering Partition Energy of a Graph

Let G be a simple graph of order n with vertex set V = v1, v2, v3, ..., 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 (see [4]). 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 partition matrix is given by  2 if vi and vj are adjacent where vi, vj ∈ Vr,   −1 if vi and vj are non-adjacent where vi, vj ∈ Vr,   1 if i = j and vi ∈ C, aij = 1 if vi and vj are adjacent between the sets   V and V for r 6= s, where v ∈ V and v ∈ V ,  r s i r j s  0 otherwise.

In this paper, we study minimum covering partition energy of a graph with respect to given partition of a graph. Further, we determine minimum covering partition energy of two types of complements of a partition graph called k-complement and k(i)-complement of a graph introduced by E. Sampathkumar in [13]. Definition 2.1 : The complement of a graph G is a graph G on the same vertices such that two distinct vertices of G are adjacent if and only if they are not adjacent in G.

Definition 2.2 [13] : Let G be a graph and Pk = {V1,V2, ..., Vk} be a partition of its vertex set V . Then the k-complement of G is obtained as follows: For all Vi and Vj in

Pk, i 6= j remove the edges between Vi and Vj and add the edges between the vertices of Vi and Vj which are not in G and is denoted by (G)k.

Definition 2.3 [13] : Let G be a graph and Pk = {V1,V2, ..., Vk} be a partition of its vertex set V . Then the k(i)-complement of G is obtained as follows: For each set

Vr in Pk, remove the edges of G joining the vertices within Vr and add the edges of G

(complement of G) joining the vertices of Vr, and is denoted by (G)k(i). MINIMUM COVERING PARTITION ENERGY OF A... 217

3. Some Basic Properties of Minimum Covering Partition Energy of a Graph

Let G = (V,E) be a graph with n vertices and Pk = {V1,V2,...,Vk} be a partition of V .

For 1 ≤ i ≤ k, let bi denote the total number of edges joining the vertices of Vi and ci be the total number of edges joining the vertices from Vi to Vj for i 6= j, 1 ≤ j ≤ k and di k k X X be the number of non-adjacent pairs of vertices within Vi. Let m1 = bi , m2 = ci i=1 i=1 k X C and m3 = di. Let Pk (G) be the minimum covering partition matrix. If the charac- i=1 C C n n−1 n−2 teristic polynomial of Pk (G) denoted by Φk (G, λ) is a0λ + a1λ + a2λ + ··· + an, C then the coefficient ai can be interpreted using the principal minors of Pk (G).

The following proposition determines the first three coefficients of the characteristic C polynomial of Pk (G). C Proposition 3.1 : The first three coefficients of φk (G, λ) are given as follows:

(i) a0 = 1,

(ii) a1 = −|C| ,

(iii) a2 = |C|C2 − [4m1 + m2 + m3]. C Proof :(i) From the definition Φk(G, λ) = det[λI − Pk (G)], we get a0 = 1.

C (ii) The sum of determinants of all 1 × 1 principal submatrices of Pk (G) is equal C to the trace of Pk (G). 1 C ⇒ a1 = (−1) trace of [Pk (G)] = −|C|.

(iii)

2 X aii aij (−1) a2 = aji ajj 1≤i

2 218 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K.

C Proposition 3.2 : If λ1, λ2, . . . , λn are partition eigenvalues of Pk (G), then n X 2 λi = |C| + 2[4m1 + m2 + m3]. i=1 We know that n n n X 2 X X λi = aijaji i=1 i=1 j=1 n X 2 X 2 = 2 (aij) + (aii) i

= |C| + 2[4m1 + m2 + m3].

2

Theorem 3.3 : Let G be a graph with n vertices and Pk be a partition of G. Then

EC (G) ≤ pn(|C| + 2[4m + m + m ]) Pk 1 2 3 where m1, m2, m3 are as defined above for G. C Proof : Let λ1, λ2, . . . , λn be the eigenvalues of Pk (G). Now by Cauchy - Schwartz inequality we have

n !2 n ! n ! X X 2 X 2 aibi ≤ ai bi . i=1 i=1 i=1

Let ai = 1 , bi =| λi |. Then then

n !2 n ! n ! X X X 2 |λi| ≤ 1 |λi| i=1 i=1 i=1

[EC ]2 ≤ n(|C| + 2[4m + m + m ]) Pk 1 2 3

[EC ] ≤ pn(|C| + 2([4m + m + m ]) Pk 1 2 3 MINIMUM COVERING PARTITION ENERGY OF A... 219

.Which is upper bound. 2 C Theorem 3.4 : Let G be a partition graph with n vertices. If R= det Pk (G), then q C 2 E (G) ≥ (|C| + 2[4m + m + m ]) + n(n − 1)R n . Pk 1 2 3

Proof : By definition,

n !2 2 X EC (G) = | λ | Pk i i=1 n n X X = | λi | | λj | i=1 j=1 n ! X 2 X = | λi | + | λi || λj | . i=1 i6=j Using arithmetic mean and geometric mean inequality, we have

1   n(n−1) 1 X Y | λ || λ | ≥ | λ || λ | . n(n − 1) i j  i j  i6=j i6=j

Therefore,

1 n   n(n−1) X Y [EC (G)]2 ≥ | λ |2 +n(n − 1) | λ || λ | Pk i  i j  i=1 i6=j 1 n n ! n(n−1) X 2 Y 2(n−1) ≥ | λi | +n(n − 1) | λi | i=1 i=1 n X 2 2 = | λi | +n(n − 1)R n i=1 2 = (|C| + 2[4m1 + m2 + m3]) + n(n − 1)R n .

Thus, q C 2 E (G) ≥ (|C| + 2[4m + m + m ]) + n(n − 1)R n . Pk 1 2 3 2 Theorem 3.5 : If the minimum covering partition energy of a graph is a rational number, then it must be a positive even number. 220 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K.

Proof of this theorem is similar to the proof of Theorem 2.12 in [6].

4. Energy of Some Partition Graphs and Their Complements

Theorem 4.1 : The minimum covering 1-partition energy of a Kn is √ EC (K ) = (n − 2) + 4n2 + 4n − 7. P1 n Proof : Let Kn be the complete graph with vertex set {v1, v2, v3....Vn}. Consider all the vertices is in one partition. The minimum covering set = C = {v1, v2, v3....vn−1}. The minimum covering 1-partition matrix is

1 2 2 ... 2 2 2 1 2 ... 2 2   2 2 1 ... 2 2 P C (K ) =   . 1 n ......  ......    2 2 2 ... 1 2 2 2 2 ... 2 0

Characteristic equation is

(λ + 1)n−2(λ2 − (2n − 3)λ − (4n − 4)) = 0 √ √ (2n−3)+ 4n2+4n−7 (2n−3)− 4n2+4n−7 ! C −1 and the spectrum is Spec (Kn) = 2 2 . P1 n − 2 1 1 √ Therefore, EC (K ) = (n − 2) + 4n2 + 4n − 7. 2 P1 n Theorem 4.2 : The minimum covering 1-partition energy of star graph K1,n−1 is

C p 2 EP1 (K1,n−1) = (n − 2) + n + 14n − 15.

Proof : Consider all the vertices is in one partition. The minimum covering set = C =

{v1}. The minimum covering 1-partition matrix is 1 2 2 ... 2 2  2 0 −1 ... −1 −1   2 −1 0 ... −1 −1 P C (K ) =   . 1 1,n−1 ......  ......    2 −1 −1 ... 0 −1 2 −1 −1 ... −1 0

Characteristic equation is

(λ − 1)n−2[λ2 + (n − 3)λ − (5n − 6)] = 0 MINIMUM COVERING PARTITION ENERGY OF A... 221

√ √ −(n−3)+ n2+14n−15 −(n−3)− n2+14n−15 ! C 1 spectrum is Spec (K1,n−1) = 2 2 . P1 n − 2 1 1 √ Therefore, EC (K ) = (n − 2) + n2 + 14n − 15. 2 P1 1,n−1 0 Definition 4.3 : The Crown graph Sn for an integer n ≥ 3 is the graph with vertex set 0 {u1, u2, ··· , un, v1, v2, ··· , vn} and edge set {uivi : 1 ≤ i, j ≤ n, i 6= j}. Sn is therefore equivalent to the complete Kn,n with horizontal edges removed. 0 Theorem 4.4 : The minimum covering 1-partition energy of Crown graph Sn is √ C 0 p 2 EP1 (Sn) = 37(n − 1) + 16n − 48n + 37

0 Proof : Consider all the vertices is in one partition. Let Sn be a crown graph of order

2n with vertex set {u1, u2, ··· , un, v1, v2, v3...vn} and minimum covering set = C =

{u1, u2, u3...un}.The minimum covering 1-partition matrix is

 1 −1 −1 ... −1 −1 2 ... 2 2  −1 1 −1 ... −1 2 −1 ... 2 2    −1 −1 1 ... −1 2 2 ... −1 −1    ......   ......    C 0 −1 −1 −1 ... 1 2 2 ... 2 −1 P (S ) =   . 1 n −1 2 2 ... 2 0 −1 ... −1 −1    2 −1 2 ... 2 −1 0 ... −1 −1    ......   ......     2 2 −1 ... 2 −1 −1 ... 0 −1 2 2 2 ... −1 −1 −1 ... −1 0

Characteristic equation is

[λ2 + (2n − 3)λ − (3n2 − 9n + 7)][λ2 − 3λ − 7]n−1 = 0

C 0 spectrum is SpecP (Sn) √ √1 √ √ ! 3+ 37 3− 37 (2n−3)+ 16n2−48n+37 (2n−3)− 16n2−48n+37 = 2 2 2 2 . Therefore, (n − 1) (n − 1) 1 1 √ √ EC (S0) = 37(n − 1) + 16n2 − 48n + 37. 2 P1 n Definition 4.5 : The double star graph Sn,m is the graph constructed from K1,n−1 and K1,m−1 by joining their centers v0 and u0. V (Sn,m) = V (K1,n−1) ∪ V (K1,m−1) and

E(Sn,m) = {v0u0; v0vi; u0uj : 1 ≤ i ≤ n − 1, 1 leqi ≤ n − 1} Therefore, double star graph is bipartite graph. 222 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K.

Theorem 4.6 : The minimum covering 1-partition energy of Double star graph Sn,n is √ C p 2 EP1 (Sn,n) = (2n − 4) + 4n + 4n − 4 + 2( 9n − 8).

Proof : Consider all the vertices in the one partition. The minimum covering set

= C = {u0, v0}. The minimum covering 1-partition matrix is  1 2 2 ... 2 2 −1 −1 ... −1  2 0 −1 ... −1 −1 −1 −1 ... −1    2 −1 0 ... −1 −1 −1 −1 ... −1    ......   ......    C  2 −1 −1 ... 0 −1 −1 −1 ... −1 P (Sn,n) =   . 1  2 −1 −1 ... −1 1 2 2 ... 2    −1 −1 −1 ... −1 2 0 −1 ... −1    ......   ......    −1 −1 −1 ... −1 2 −1 0 ... −1 −1 −1 −1 ... −1 2 −1 −1 ... 0 Characteristic equation is

(λ − 1)2n−4[λ2 + (2n − 6)λ − (7n − 10)][λ2 − (9n − 8)] = 0

Hence, spectrum is √ √ √ √ (2n−6)+ 4n2+4n−4 (n−6)− 4n2+4n−4 ! C 1 9n − 8 − 9n − 8 Spec (Sn,n) = 2 2 . P1 2n − 4 1 1 1 1 √ √ Therefore, EC (S ) = (2n − 4) + 4n2 + 4n − 4 + 2( 9n − 8). 2 P1 n,n Theorem 4.7 : The minimum covering 1-partition energy of Cocktail party graph

Kn×2 is C p 2 EP1 (Kn×2) = (6n − 9) + 16n + 8n − 15. Proof : Consider all the vertices in one partition. The minimum covering set = C =

{u1, u2, ...un−1, v1, v2...vn−1}. The minimum covering 1-partition matrix is  1 −1 2 2 ... 2 2 2 2  −1 1 2 2 ... 2 2 2 2     2 2 1 −1 ... 2 2 2 2     2 2 −1 1 ... 2 2 2 2    C  ......  P1 (Kn×2) =  ......  .    2 2 2 2 ... 1 −1 2 2     2 2 2 2 ... −1 1 2 2     2 2 2 2 ... 2 2 0 −1 2 2 2 2 ... 2 2 −1 0 MINIMUM COVERING PARTITION ENERGY OF A... 223

Characteristic equation is

(λ − 1)1(λ − 2)n−1(λ + 4)n−2[λ2 − (4n − 9)λ − (20n − 24)] = 0

C Hence, spectrum is SpecP (Kn×2) = 1 √ √ (4n−9)+ 16n2+8n−15 (4n−9)− 16n2+8n−15 ! 1 2 −4 C 2 2 . Therefore, E (Kn×2) = 1 (n − 1) (n − 2) 1 1 P1 √ (6n − 9) + 16n2 + 8n − 15. 2 Theorem 4.8 : The minimum covering 1-partition energy of

Kn,n is C p 2 EP1 (Kn,n) = 3(n − 1) + 1 + 16n .

Proof : Consider all the vertices in the one partition. The minimum covering set

= C = {u1, u2..., un}. The minimum covering 1-partition matrix is

 1 −1 −1 −1 ... 2 2 2 2  −1 1 −1 −1 ... 2 2 2 2    −1 −1 1 −1 ... 2 2 2 2    −1 −1 −1 1 ... 2 2 2 2    C  ......  P1 (Kn,n) =  ......  .    2 2 2 2 ... 0 −1 −1 −1    2 2 2 2 ... −1 0 −1 −1    2 2 2 2 ... −1 −1 0 −1 2 2 2 2 ... −1 −1 −1 0

Characteristic equation is

(λ − 1)n−1(λ − 2)n−1[λ2 + (2n − 3)λ − (3n2 + 3n − 2)] = 0

C Hence, spectrum is SpecP (Kn,n) 1 √ √ 2 2 ! 1 2 −(2n−3)+ 16n +1 −(2n−3)− 16n +1 = 2 2 . (n − 1) (n − 1) 1 1 √ Therefore, EC (K ) = 3(n − 1) + 1 + 16n2. 2 P1 n,n Theorem 4.9 : The minimum covering 2-partition energy of star graph K1,n−1 in which the vertex of degree n − 1 is in one partition and vertices of degree 1 are in another √ partition is EC (K ) = (n − 2) + n2 + 2n − 3. P2 1,n−1 Proof : The 2-partition of star graph K1,n−1 in which the vertex of degree n − 1 is in one partition and vertices of degree 1 are in another partition The minimum covering 224 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K.

set = C = {v0}. The minimum covering 1-partition matrix is

1 1 1 ... 1 1  1 0 −1 ... −1 −1   1 −1 0 ... −1 −1 P C (K ) =   . 2 1,n−1 ......  ......    1 −1 −1 ... 0 −1 1 −1 −1 ... −1 0

Hence, its characteristic equation is (λ−1)n−2[λ2 +(n−3)λ−(2n−3)] = 0 Hence, spec- √ √ −(n−3)+ n2+2n−3 −(n−3)− n2+2n−3 ! C 1 trum is Spec (K1,n−1) = 2 2 . There- P2 (n − 2) 1 1 √ fore, EC (K ) = (n − 2) + n2 + 2n − 3. 2 P2 1,n−1 Theorem 4.10 : The minimum covering 2-partition energy of 2(i)-complement of star graph K1,n−1 in which the vertex of degree n − 1 is in one partition and vertices of √ degree 1 are in another partition is (n − 2) + 4n2 − 8n + 5.

Proof : Consider 2(i)-complement of star graph K1,n−1, in which the vertex of degree n − 1 is in one partition and remaining vertices are in other partition.The minimum covering set is{v2, v3...vn}. Its minimum covering partition matrix is

0 1 1 ... 1 1 1 1 2 ... 2 2   1 2 1 ... 2 2 P C ((K ) ) =   . 2 1,n−1 2(i) ......  ......    1 2 2 ... 1 2 1 2 2 ... 2 1

Hence, its characteristic equation is (λ+1)n−2[λ2 −(2n−3)λ−(n−1)] = 0 Hence, spec- √ √ (2n−3)+ 4n2−8n+5 (2n−3)− 4n2−8n+5 ! C −1 trum is Spec ((K1,n−1) ) = 2 2 . P2 2(i) (n − 2) 1 1 √ Therefore, EC ((K ) ) = (n − 2) + 4n2 − 8n + 5.. 2 P2 1,n−1 2(i) 0 Theorem 4.11 : The minimum covering 2-partition energy of Crown graph Sn is

C 0 p 2 EP2 (Sn) = 3(n − 1) + 4n − 8n + 5.

0 Proof : Consider the crown graph Sn whose vertex set is partitioned into Un =

{u1, u2, . . . , un}, Vn = {v1, v2, . . . , vn}.

The minimum covering set = C = {u1, u2, . . . , un}.The minimum covering 2-partition MINIMUM COVERING PARTITION ENERGY OF A... 225 matrix is  1 −1 −1 ... −1 0 1 ... 1 1  −1 1 −1 ... −1 1 0 ... 1 1    −1 −1 1 ... −1 1 1 ... 0 1     ......   ......    C 0 −1 −1 −1 ... 1 1 1 ... 1 0  P (S ) =   . 2 n  0 1 1 ... 1 0 −1 ... −1 −1    1 0 1 ... 1 −1 0 ... −1 −1    ......   ......     1 1 0 ... 1 −1 −1 ... 0 −1 1 1 1 ... 0 −1 −1 ... −1 0

Characteristic equation is [λ2 − 3λ + 1][λ2 + (2n − 3)λ − (n − 1)] = 0 minimum covering 2-partition eigenvalues are C 0 specχ (Sn) = √ √ √ √ ! 3+ 5 3− 5 −(2n−3)+ (4n2−8n+5) −(2n−3)− (4n2−8n+5) 2 2 2 2 . n − 1 n − 1 1 1 √ EC (S0) = 3(n − 1) + 4n2 − 8n + 5. 2 P1 n Theorem 4.12 : The minimum covering 2-partition energy of 2(i)-complement of 0 Crown graph Sn is

C 0 p 2 p 2 EP2 ((Sn)2(i)) = (2n − 4) + n + 2n − 3 + 9n + 6n − 11.

Proof : Consider the 2(i)-complement of crown graph whose vertex set is partitioned into Un = {u1, u2, . . . , un}, Vn = {v1, v2, . . . , vn}. The minimum covering set = C =

{u1, u2, . . . , un−1, v1, v2, . . . , vn−1}.The minimum covering 2-partition matrix is

1 2 2 ... 2 0 1 ... 1 1 2 1 2 ... 2 1 0 ... 1 1   2 2 1 ... 2 1 1 ... 0 1   ......  ......    C 2 2 2 ... 0 1 1 ... 1 0 P ((S0) ) =   . 2 n 2(i) 0 1 1 ... 1 1 2 ... 2 2   1 0 1 ... 1 2 1 ... 2 2   ......  ......    1 1 0 ... 1 2 2 ... 1 2 1 1 1 ... 0 2 2 ... 2 0 226 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K.

Characteristic polynomial is [λ]n−2[λ + 2]n−2[λ2 − (n − 1)λ − (n − 1)][λ2 − (3n − 5)λ − (9n − 9)] = 0 minimum covering 2-partition spectra is SpecD ((S0) ) = P2 n 2 √ √  2 2  −2 0 (n−1)+ (n +2n−3) (n−1)− (n +2n−3)  2 2   n − 2 n − 2 1 1   √ √  .  (3n−5)+ (9n2+6n−11) (3n−5)− (9n2+6n−11)   −2 0 2 2  n − 2 n − 2 1 1 √ √ EC ((S0) ) = (2n − 4) + n2 + 2n − 3 + 9n2 + 6n − 11. 2 P2 n 2(i)

Definition 4.13 : The cocktail party graph, denoted by Kn×2, is a graph having vertex Sn set V = i=1{ui, vi} and edge set E = {uiuj, vivj, uivj, viuj : 1 ≤ i < j ≤ n}. This graph is also called as complete n-partite graph. Theorem 4.14 : The minimum covering 2-partition energy of 2-complement of Cocktail party graph Kn×2 is

C p 2 EP2 ((Kn×2)(2)) = 2[(n − 2) + 4n + 4n − 7].

Proof : Consider the 2-complement of Cocktail party graph (Kn×2)(2) whose vertex set is partitioned into Un = {u1, u2, . . . , un}, Vn = {v1, v2, . . . , vn}. The minimum covering set = C = {u1, v1}.The minimum covering 2-partition matrix is

1 2 2 ... 2 1 0 ... 0 0 2 0 2 ... 2 0 1 ... 0 0   2 2 0 ... 2 0 1 ... 0 1   ......  ......    C 2 2 2 ... 0 0 0 ... 1 0 P ((Kn×2) ) =   . 2 (2) 1 0 0 ... 0 1 2 ... 2 2   0 1 0 ... 0 2 0 ... 2 2   ......  ......    0 0 1 ... 0 2 2 ... 0 2 0 0 0 ... 1 2 2 ... 2 0

Characteristic polynomial is [λ + 3]n−2[λ + 1]n−2[λ2 − (2n − 5)λ − (4n − 4)][λ2 − (2n − 1)λ − 2] = 0 minimum covering 2-partition spectra is SpecC ((K ) ) = P2 n×2 (2) MINIMUM COVERING PARTITION ENERGY OF A... 227

√ √  2 2  −3 −1 (2n−5)+ (4n −4n+9) (2n−5)− (4n −4n+9)  2 2   n − 2 n − 2 1 1   √ √  .  (2n−1)+ (4n2−4n+9) (2n−1)− (4n2−4n+9)   −3 −1 2 2  n − 2 n − 2 1 1 √ EC ((K ) ) = 4(n − 2) + 2 4n2 − 4n + 9 2 P2 n×2 (2) Theorem 4.15 : The minimum covering 2-partition energy of complete bipartite graph

Kn,n is

C p 2 EP2 (Kn,n) = 3(n − 1) + 4n + 1.

Proof : Consider the complete bipartite graph Kn,n whose vertex set is partitioned into Un = {u1, u2, . . . , un}, Vn = {v1, v2, . . . , vn}. The minimum covering set = C =

{u1, u2, ..un}.The minimum covering 2-partition matrix is

 1 −1 −1 −1 ... 1 1 1 1  −1 1 −1 −1 ... 1 1 1 1    −1 −1 1 −1 ... 1 1 1 1    −1 −1 −1 1 ... 1 1 1 1    C  ......  P2 (Kn,n) =  ......  .    1 1 1 1 ... 0 −1 −1 −1    1 1 1 1 ... −1 0 −1 −1    1 1 1 1 ... −1 −1 0 −1 1 1 1 1 ... −1 −1 −1 0

Characteristic equation is

(λ − 1)n−1(λ − 2)n−1[λ2 + (2n − 3)λ − (3n − 2)] = 0

√ √ −(2n−3)+ 4n2+1 −(2n−3)− 4n2+1 ! C 1 2 Hence, spectrum is Spec (Kn,n) = 2 2 . P2 (n − 1) n − 1 1 1 √ Therefore, EC (K ) = 3(n − 1) + 4n2 + 1. 2 P2 n,n

Theorem 4.16 : The minimum covering 2-partition energy of Double star graph Sn,n is √ C p 2 EP2 (Sn,n) = (2n − 2) + 2 n + 1 + 4n + 12n + 4.

Proof : In double star graph, the centers {u0, v0} are taken in one partition and remaining vertices are taken in other partition. The minimum covering set = C = 228 P.S.K. REDDY, K. N. PRAKASHA & GAVIRANGAIAH K.

{u0, v0}. The minimum covering 2-partition matrix is

1 1 1 ... 1 2 0 0 ... 0  1 0 −1 ... −1 0 −1 −1 ... −1   1 −1 0 ... −1 0 −1 −1 ... −1   ......  ......    C 1 −1 −1 ... 0 0 −1 −1 ... −1 P (Sn,n) =   . 2 2 0 0 ... 0 1 1 1 ... 1    0 −1 −1 ... −1 1 0 −1 ... −1   ......  ......    0 −1 −1 ... −1 1 −1 0 ... −1 0 −1 −1 ... −1 1 −1 −1 ... 0

Characteristic equation is

(λ − 1)2n−4[λ2 − n][λ2 + (2n − 6)λ − (7n − 10)] = 0

Hence, spectrum is √ √ √ √ −(2n−6)+ 4n2+4n−4 −(2n−6)− 4n2+4n−4 ! C 1 n − n Spec (Sn,n) = 2 2 . P2 2n − 4 1 1 1 1 √ √ Therefore, EC (S ) = (2n − 4) + 2 n + 4n2 + 4n − 4. 2 P2 n,n Theorem 4.17 : The minimum covering 2-partition energy of 2(i)-complement of

Double star graph Sn,n is √ C p 2 EP2 ((Sn,n)2(i)) = (2n − 4) + 2 n + 16n − 28n + 12.

Proof : In 2(i)-complement of double star graph, the centers {u0, v0} are taken in one partition and remaining vertices are taken in other partition. . The minimum covering set = C = {u1, u2...un, v1, v2...vn}. The minimum covering 2-partition matrix is

 0 1 1 ... 1 −1 0 0 ... 0  1 1 2 ... 2 0 2 2 ... 2    1 2 1 ... 2 0 2 2 ... 2    ......   ......    C  1 2 2 ... 1 0 2 2 ... 2 P ((Sn,n) ) =   . 1 2(i) −1 0 0 ... 0 0 1 1 ... 1    0 2 2 ... 2 1 1 2 ... 2    ......   ......     0 2 2 ... 2 1 2 1 ... 2 0 2 2 ... 2 1 2 2 ... 1 MINIMUM COVERING PARTITION ENERGY OF A... 229

Characteristic equation is

(λ + 1)2n−4[λ2 − n][λ2 − (4n − 6)λ − (5n − 6)] = 0

Hence, spectrum is √ √ √ √ (4n−6)+ 16n2−28n+12 (4n−6)− 16n2−28n+12 ! C −1 n − n Spec ((Sn,n) ) = 2 2 . P2 2(i) 2n − 4 1 1 1 1 √ √ Therefore, EC ((S ) ) = (2n − 4) + 2 n + 16n2 − 28n + 12. 2 P2 n,n 2(i)

Acknowledgement

The authors are thankful to the anonymous referee for valuable suggestions and com- ments for the improvement of the paper. Also, the first author is grateful to Dr. M. N. Channabasappa, Director and Dr. Shivakumaraiah, Principal, Siddaganga Institute of Technology, Tumkur, for their constant support and encouragement.

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