Kragujevac Journal of Mathematics Volume 37(2) (2013), Pages 369 373 CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS PRABHAKAR R HAMPIHOLI 1 AND BASAVRAJ S DURGI 2 Abstract The characteristic polynomial of a graph G with p vertices is defined as φ(g : λ) = det(λi A(G)), where A is the adjacency matrix of G and I is the unit matrix The roots of the characteristic equation φ(g : λ) = 0, denoted by λ 1, λ 2,, λ p are the eigenvalues of G The graphs with large number of edges are referred as graph representations of inorganic clusters, called as Cluster graphs In this paper we obtain the characteristic polynomial of class of cluster graphs 1 Introduction Let G be a simple undirected graph with p vertices and q edges Let V (G) = {v 1, v 2,, v p } be the vertex set of G The adjacency matrix of G is defined as A(G) = [a ij ], where a ij = 1 if v i is adjacent to v j and a ij = 0, otherwise The characteristic polynomial of G is φ(g : λ) = det(λi A(G)), where I is an unit matrix of order p The roots of an equation φ(g : λ) = 0 denoted by λ 1, λ 2,, λ p are the eigenvalues of G and their collection is the spectrum of G [1] Characteristic polynomial of some cluster graphs are obtained in [2 5] In this paper we obtain the characteristic polynomial of some more cluster graphs 2 Some edge deleted cluster graphs I Gutman and L Pavlović [2] introduced four classes of graphs obtained from complete graph by deleting edges For the sake of continuity we produce these here Definition 21 [2] Let v be a vertex of the complete graph K p, p 3 and let e i, i = 1, 2,, k, 1 k p 1, be its distinct edges, all being incident to v The graph Ka p (k) or Cl(p, k) is obtained by deleting e i, i = 1, 2,, k from K p In addition Ka p (0) = K p Key words and phrases Spectra, Complete graph, Cluster graphs 2010 Mathematics Subject Classification 05C50 Received: January 13, 2013 Revised: June 21, 2013 369
370 P R HAMPIHOLI AND B S DURGI Definition 22 [2] Let f i, i = 1, 2,, k, 1 k p/2 be independent edges of the complete graph K p, p 3 The graph Kb p is obtained by deleting f i, i = 1, 2,, k from K p The graph Kb p (0) = K p Definition 23 [2] Let V k be a k- element subset of the vertex set of the complete graph K p, 2 k p, p 3 The graph Kc p (k) is obtained by deleting from K p all the edges connecting pairs of vertices from V k In addition Kc p (0) = Kc p (1) = K p Definition 24 [2] Let 3 k p, p 3 The graph Kd p (k) obtained by deleting from K p, the edges belonging to a k- membered cycle Theorem 21 [2] For p 3 and 3 k p 1, it holds that φ(ka p (k) : λ) = (λ + 1) p 3 [λ 3 (p 3)λ 2 (2p k 3)λ + (k 1)(p 1 k)] Theorem 22 [2] For p 3 and 0 k p/2 it holds that φ(kb p (k) : λ) = λ k (λ + 1) p 2k 1 (λ + 2) k 1 [λ 2 (p 3)λ 2(p k 1)] Theorem 23 [2] For p 3 and 0 k p it holds that φ(kc p (k) : λ) = λ k 1 (λ + 1) p k 1 [λ 2 (p k 1)λ k(p k)] Theorem 24 [2] For p 3 and 3 k p it holds that φ(kd p (k) : λ) = (λ + 1) p k 1 [λ 2 (p 4)λ (3p 2k 3)] k 1 ( ( ) ) 2πi λ + 2 cos + 1 k We introduce here another class of graphs obtained from K p and we denote it by Ka p (p, m, k) Two subgraphs G 1 and G 2 of graph G are independent subgraphs if V (G 1 ) V (G 2 ) is an empty set, where V (G) is the vertex set of the graph G Definition 25 [4] Let (K m ) i, i = 1, 2,, k, 1 k p/m, 1 m p, be independent complete subgraphs with m vertices of the complete graph K p, p 3 The graph K ap (m, k) obtained from K p by deleting all the edges of k independent complete subgraphs (K m ) i, i = 1, 2,, k In addition K ap (m, 0) = K ap (0, k) = K ap (0, 0) = K p We note that for k = 1, K ap (p, m, k) = K ap (p, m) Theorem 25 [4] Let p, m, k be positive integers Let mk p Then for p 3, 0 k p/m, 1 m p, the characteristic polynomial of K ap (p, m, k) is φ(k ap (p, m, k) : λ) = λ mk k (λ+1) p mk 1 (λ+m) k 1 [λ 2 (p m 1)λ m(p+k mk 1)]
CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS 371 Definition 26 [5] Let K m1 and K m2 be independent complete subgraphs of K p The graph obtained by deleting all the edges of k 1 + k 2 such that m 1 k 1 + m 2 k 2 p independent complete subgraphs K m1 (k 1 copies) and K m2 (k 2 copies) is denoted by K ap (p, (m 1, k 1 ), (m 2, k 2 )) Theorem 26 [5] Let p, m i, k i, i = 1, 2 are positive integers with 2 m ik i p Then for p 3, the characteristic polynomial of K ap (p, (m 1, k 1 ), (m 2, k 2 )) is φ(k ap (p, (m 1, k 1 ), (m 2, k 2 )) : λ) = λ 2 m ik i 2 k i (λ + 1) p 2 m ik i 1 { [ (λ + m 1 p) + k ] ( 2 )} 2(m 2 2 m 1 m 2 ) (λ + 1) + (m 1 1) m i k i p λ + m 2 (λ + m 1 ) k 1 1 (λ + m 2 ) k 2 Now we proceed to prove the main result Definition 27 Let K mi, i = 1, 2,, l are the independent complete subgraphs of K p The graph obtained by deleting all the edges of l k i, such that l m ik i p independent complete subgraphs K m1 (k 1 copies), K m2 (k 2 copies),, K ml (k l copies) is denoted by K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) We note that for k 1 = k, m 1 = m, k i = 0 for i = 2, 3,, l it holds that K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) = K ap (p, m, k) Theorem 27 Let p, m i, k i, i = 1, 2,, l are positive integers with l m ik i p Then for p 3, the characteristic polynomial of K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) is φ(k ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) : λ) = λ l m ik i l k i (λ + 1) p l m ik i 1 {[ ] ( l l )} k i (m 2 i m i m 1 ) (λ + m 1 p) + (λ + 1) + (m 1 1) m i k i p λ + m i (λ + m 1 ) k 1 1 i=2 l (λ + m i ) k i i=2 Proof Without loss of generality we assume that the vertices of K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) are the vertices of K p and the edges are the edges of K p that are not there in K m1 (k 1 copies), K m2 (k 2 copies),,k ml (k l copies) The characteristic polynomial of K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) is
372 P R HAMPIHOLI AND B S DURGI λ 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 λ 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 λ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 λ 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 λ 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 λ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 λ 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 λ 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 λ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 λ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 λ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 λ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 λ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 λ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 λ By elementary calculations on the above determinant, we get the result Here we give some examples in Table 1 p, (m 1, k 1 ), (m 2, k 2 ),, (m l k l ) The characteristic equations of graphs K ap 9,(2,1),(3,1),(4,1) λ 6 (λ 3 26λ 48) 11,(2,2),(3,1),(4,1) λ 7 (λ 4 44λ 2 152λ 144) 12,(2,1),(3,1),(4,1) λ 6 (λ 6 56λ 4 260λ 3 477λ 2 392λ 120) 15,(3,2),(4,1),(5,1) λ 11 (λ 4 83λ 2 402λ 540) 14,(2,1),(3,1),(4,1),(5,1) λ 10 (λ 4 71λ 2 308λ 360) Table 1 The characteristic equations of the graphs denoted by K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) Acknowledgement: Authors are thankful to the referees for suggestions References [1] D M Cvetković, M Doob and H Sachs, Spectra of Graphs, Academic Press, New York, 1980 [2] I Gutman and L Pavlović, The energy of some graphs with large number of edges, Bull Acad Serbe Sci Arts (Cl Math Natur) 118 (1999), 35 [3] P R Hampiholi, H B Walikar and B S Durgi, Energy of complement of stars, J Indones Math Soc 19(1) (2013), 15 21
CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS 373 [4] H B Walikar and H S Ramane, Energy of some cluster graphs, Kragujevac J Sci 23 (2001), 51 62 [5] P R Hampiholi and B S Durgi, On the spectra and energy of some cluster graphs, Int J Math Sci Engg Appl 7(2) (2013), 219 228 1 Department of Master of Computer Applications, Gogte Institute of Technology, Belgaum - 590008, India E-mail address: prhampi@yahooin 2 Department of Mathematics, KLE College of Engineering and Technology, Belgaum - 590008, India E-mail address: durgibs@yahoocom