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 vertex 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 complete graph 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 bipartite graph 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 complete bipartite graph 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 . 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