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 8 2 if vi and vj are adjacent where vi; vj 2 Vr, > > −1 if vi and vj are non-adjacent where vi; vj 2 Vr, > < 1 if i = j and vi 2 C, aij = 1 if vi and vj are adjacent between the sets > > V and V for r 6= s, where v 2 V and v 2 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 = fV1;V2; :::; Vkg 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 = fV1;V2; :::; Vkg 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 = fV1;V2;:::;Vkg 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 = −|Cj , (iii) a2 = jCjC2 − [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)] = −|Cj. (iii) 2 X aii aij (−1) a2 = aji ajj 1≤i<j≤n X = aiiajj − ajiaij 1≤i<j≤n X X = aiiajj − ajiaij 1≤i<j≤n 1≤i<j≤n 2 2 2 = jCjC2 − [(2) m1 + (1) m2 + (−1) m3] = jCjC2 − [4m1 + m2 + m3]: 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 = jCj + 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<j i=1 X 2 = 2 (aij) + jCj i<j = jCj + 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(jCj + 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 =j λi j. Then then n !2 n ! n ! X X X 2 jλij ≤ 1 jλij i=1 i=1 i=1 [EC ]2 ≤ n(jCj + 2[4m + m + m ]) Pk 1 2 3 [EC ] ≤ pn(jCj + 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) ≥ (jCj + 2[4m + m + m ]) + n(n − 1)R n : Pk 1 2 3 Proof : By definition, n !2 2 X EC (G) = j λ j Pk i i=1 n n X X = j λi j j λj j i=1 j=1 n ! X 2 X = j λi j + j λi jj λj j : i=1 i6=j Using arithmetic mean and geometric mean inequality, we have 1 0 1 n(n−1) 1 X Y j λ jj λ j ≥ j λ jj λ j : n(n − 1) i j @ i j A i6=j i6=j Therefore, 1 n 0 1 n(n−1) X Y [EC (G)]2 ≥ j λ j2 +n(n − 1) j λ jj λ j Pk i @ i j A i=1 i6=j 1 n n ! n(n−1) X 2 Y 2(n−1) ≥ j λi j +n(n − 1) j λi j i=1 i=1 n X 2 2 = j λi j +n(n − 1)R n i=1 2 = (jCj + 2[4m1 + m2 + m3]) + n(n − 1)R n : Thus, q C 2 E (G) ≥ (jCj + 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 p EC (K ) = (n − 2) + 4n2 + 4n − 7: P1 n Proof : Let Kn be the complete graph with vertex set fv1; v2; v3::::Vng. Consider all the vertices is in one partition. The minimum covering set = C = fv1; v2; v3::::vn−1g. The minimum covering 1-partition matrix is 21 2 2 ::: 2 23 62 1 2 ::: 2 27 6 7 62 2 1 ::: 2 27 P C (K ) = 6 7 : 1 n 6. .. .7 6. .7 6 7 42 2 2 ::: 1 25 2 2 2 ::: 2 0 Characteristic equation is (λ + 1)n−2(λ2 − (2n − 3)λ − (4n − 4)) = 0 p p (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 p 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.