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Discrete Mathematics Contents lists available at ScienceDirect Discrete Mathematics journal homeage: wwwelseviercom/locate/disc Maximizing the number of sanning trees in K n -comlements of asteroidal grahs Stavros D Nikolooulos, Leonidas Palios, Charis Paadooulos Deartment of Comuter Science, University of Ioannina, GR-45110 Ioannina, Greece a r t i c l e i n f o a b s t r a c t Article history: Received 13 July 2005 Received in revised form 8 August 2008 Acceted 8 August 2008 Available online xxxx Keywords: Sanning trees Comlement-sanning-tree matrix Star-like grahs Maximization Interconnection networks In this aer we introduce the class of grahs whose comlements are asteroidal starlike grahs and derive closed formulas for the number of sanning trees of its members The roosed results extend revious results for the classes of the multi-star and multicomlete/star grahs Additionally, we rove maximization theorems that enable us to characterize the grahs whose comlements are asteroidal grahs and ossess a maximum number of sanning trees 2008 Elsevier BV All rights reserved 1 Introduction The number of sanning trees of a grah G is an imortant, well-studied quantity in grah theory, and aears in a number of alications Its most notable alication is in the field of network reliability: in a network modeled by a grah, intercommunication between all nodes of the network imlies that the grah must contain a sanning tree; thus, maximizing the number of sanning trees is a way of maximizing reliability [2,15,12,20] Other alication fields arise from enumerating certain chemical isomers [3], and counting the number of Eulerian circuits in a grah [10,11] Thus, both for theoretical and for ractical uroses, we are interested in deriving formulas for the number of sanning trees of a grah G, and also of the K n -comlement of G For any subgrah H of the comlete grah K n, the K n -comlement of H, denoted by K n H, is defined as the grah obtained from K n by removing the edges of H; note that if H has n vertices then K n H coincides with the comlement H of H Many different tyes of grahs K n H have been examined: for examle, there exist closed formulas for the cases where H is a airwise-disjoint set of edges [22], a chain of edges [13], a cycle [7], a star [19], a multi-star [18,23], a multi-comlete/star grah [4], a labeled molecular grah [3], and more recently if H is a circulant grah [8,12,24], a quasi-threshold grah [17], and so on see Berge [1] for an exosition of the main results A common aroach for determining the number of sanning trees of a grah G relies on a classic result known as the comlement-sanning-tree matrix theorem [21], which exresses the number of sanning trees of G as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of G, ie, adjacency matrix, adjacency lists, etc Calculating the determinant of the comlement-sanning-tree matrix seems to be a romising aroach for comuting the number of sanning trees of families of grahs of the form K n H, where H is a grah that exhibits symmetry see [1,4, 7,18,17,16,23,24] In this aer, we define two classes of grahs, namely, the comlete-lanet and the star-lanet grahs, which generalize well-known classes of grahs; we call these two classes of grahs asteroidal grahs It turns out that comuting the number of Corresonding author E-mail addresses: stavros@csuoigr SD Nikolooulos, alios@csuoigr L Palios, charis@csuoigr C Paadooulos 0012-365X/$ see front matter 2008 Elsevier BV All rights reserved doi:101016/jdisc200808008 Mathematics 2008, doi:101016/jdisc200808008

2 SD Nikolooulos et al / Discrete Mathematics sanning trees of these grahs is not difficult However, comuting the number of sanning trees of their K n -comlements is more interesting; we derive closed formulas for the number of sanning trees of i the K n -comlement of a comlete-lanet grah, and ii the K n -comlement of a star-lanet grah Our roofs are based on the comlement-sanning-tree matrix theorem and use standard techniques from linear algebra and matrix theory Our formulas generalize revious roosed formulas of classes of grahs such as comlete grahs, star grahs, wheel grahs, gem grahs, multi-star grahs, multicomlete/star grahs, etc Although the roblem of maximizing the number of sanning trees of a grah is difficult in general, it is ossible to achieve an efficient solution for some non-trivial classes of grahs [4,9,18] In this aer we also rove maximization results for the K n -comlements of asteroidal grahs In articular, we characterize the grahs whose comlements are asteroidal grahs and ossess a maximum number of sanning trees Our maximization results generalize and extend revious maximization results for the class of multi-star grahs [4] 2 Preliminaries We consider finite undirected grahs with no loos or multile edges For a grah G, we denote by VG and EG the vertex set and edge set of G, resectively Let S be a subset of the vertex set of a grah G Then, the subgrah of G induced by S is the grah G[S] = S, E, where u, v E if and only if u, v S and u, v EG Moreover, we denote by G S the subgrah G[VG S] The neighborhood Nx of a vertex x is the set of all the vertices of G which are adjacent to x The closed neighborhood of vertex x is defined as N[x] = {x} Nx The degree of a vertex x in the grah G, denoted dx, is the number of edges incident on x; thus, dx = Nx If two vertices x and y are adjacent in G, we say that x sees y; otherwise we say that x misses y We extend this notion to vertex sets: V i VG sees misses V j VG if and only if every vertex x V i sees misses every vertex y V j By K n we denote the comlete grah on n vertices Moreover, for symmetry, we denote by S n+1 a tree on n + 1 vertices with one vertex having degree n and call it a star grah it is commonly denoted by S 1,n ; we call the vertex of S n+1 with degree n its center vertex The chordless ath res cycle on n vertices v 1 v 2 v n with edges v i v i+1 res v i v i+1 and v 1 v n, 1 i < n, is denoted by P n res C n Let G 1 and G 2 be two grahs Their union G = G 1 G 2 has VG = VG 1 VG 2 and EG = EG 1 EG 2 Their join is denoted G 1 + G 2 and consists of G 1 G 2 lus all edges joining VG 1 with VG 2 For any connected grah G, we write mg for the grah with m comonents, each isomorhic with G, m 2 Thus, the grah mk n res ms n, mp n, mc n consists of m disjoint coies of K n res S n, P n, C n Note that, S n+1 = K 1 + nk 1 Throughout the aer, we refer to comlete grahs, star grahs, ath grahs, and cycle grahs as cliques, stars, aths, and cycles, resectively 21 Asteroidal grahs A grah G on n vertices is called a comlete-lanet res star-lanet if its vertex set VG admits a vertex-disjoint artition into sets A and B such that: S1 A = {v 1, v 2,, v m } and G[A] = K m res A = {v 1, v 2,, v m, c} and G[A] = S m+1, m 1; S2 B = B 1 B 2 B m, and for each i = 1, 2,, m, B i 0 and B i induces α ij, β ij, γ ij, and δ ij disjoint coies of cliques, stars, aths, and cycles, resectively, on j vertices; that is, G i G[B i ] = j α ij K j β ij S j γ ij P j δ ij C j, 1 i m; S3 The vertex v i A sees all the vertices of G i and misses all the vertices in B VG i, 1 i m We collectively call the above defined grahs asteroidal grahs; Fig 1 shows the general form of a comlete-lanet grah and a star-lanet grah Let G be an asteroidal grah and let A and B be the artition sets of VG according to the above definition: we call the grah G[A] the sun-grah of G, the grah G[B] the lanet-grah, and the grahs G 1, G 2,, G m the lanet-subgrahs; by definition, G[B] = m G i A maximal connected subgrah of a lanet-subgrah G i is called lanetcomonent of G i Let us denote by l i the cardinality of B i Then, the number of vertices of G i is l i Moreover, the definition of G i imlies that for each i = 1, 2,, m, it holds that αij + β ij + γ ij + δ ij j = li 1 j Clearly, B = m l i; we denote this sum by l Therefore, a comlete-lanet grah has exactly m + l vertices, whereas a star-lanet grah has m + l + 1 vertices Hereafter, we call the vertices of the grahs G[A] and G[B], sun-vertices and lanet-vertices resectively Throughout the aer, we use the following convention: any isolated vertex of a lanet-subgrah is considered to be a K 1 and not an S 1, P 1, or C 1 ; hence, for all i, β i1 = γ i1 = δ i1 = 0 Similarly, β i2 = γ i2 = δ i2 = 0, since K 2 = S 2 = P 2 = C 2 Mathematics 2008, doi:101016/jdisc200808008

SD Nikolooulos et al / Discrete Mathematics 3 a A comlete-lanet grah b A star-lanet grah Fig 1 Asteroidal grahs Table 1 Subclasses of comlete-lanet grahs Parameters m, l, α, β, γ, δ Comlete-lanet grah G c Reference m = k + 1, l = 0 or K k+1 [1,19] m = 1, αk = 1, αi = 0 i k, β = γ = δ = 0 m = 1, βk = 1, βi = 0 i k, α = γ = δ = 0 K 2 + kk 1 [1] m = 1, γ k = 1, γ i = 0 i k, α = β = δ = 0 K 1 + P k gem, for k = 4 [1] m = 1, δk = 1, δi = 0 i k, α = β = γ = 0 Wheel grah W k+1 [1] m 1, α1 0, αi = 0 i 1, β = γ = δ = 0 Multi-star grah [18,23] m 1, aroriate α 0, β = γ = δ = 0 Multi-comlete/star grah [4] Table 2 Subclasses of star-lanet grahs Parameters m, l, α, β, γ, δ Star-lanet grah G s Reference m 1, l = 0 S 1,m [1,19] m 1, α1 0, αi = 0 i 1, β = γ = δ = 0 Trees with diameter d 4 [17] Finally, γ i3 = δ i3 = 0, since K 3 = C 3 and S 3 = P 3 Therefore, for a lanet-comonent which is a star S n, then n 3, whereas if it is a ath P n or a cycle C n then n 4 Let G be an asteroidal grah on m sun-vertices and l lanet-vertices We denote by αj the number of the maximal cliques K j of the lanet-grah G[B] Similarly, we denote by βj, γ j, and δj the number of the maximal stars S j, aths P j, and cycles C j, resectively, of G[B] Thus, we have: m m m m αj = α ij, βj = β ij, γ j = γ ij, and δj = δ ij We define the vector α = [α1, α2,, αl] on the lanet-grah G[B], and we call it the clique vector of the asteroidal grah G; in a similar manner, we define the vectors β, γ, and δ and we call them star vector, ath vector, and cycle vector, resectively Clearly, the vectors α, β, γ, and δ of an asteroidal grah G determine the number of the maximal cliques, stars, aths, and cycles in G[B] Hereafter, we write α 0 to denote that there exists at least one j, 1 j l, such that αj 0, ie, α = 0 is equivalent to α1 = α2 = = αl = 0 We use a similar notation for the star vector, ath vector, and cycle vector For examle, if α = 0, β 0, γ 0 and δ 0, then the lanet-grah G[B] contains only stars, aths, and cycles Many grahs can be derived as secial cases from the asteroidal grahs, deending on the sun-grah and the values of the clique, star, ath, and cycle vectors For examle, given a comlete-lanet grah with K m and vectors α, β, γ, δ, and setting m = 1, γ 5 = 1, γ j = 0 for all j 5, and α = β = δ = 0, we get the grah K 1 + P 5, and when setting m = 1, δk = 1, δj = 0 for all j k, and α = β = γ = 0, we get the wheel grah W k+1, ie, the grah obtained from a chordless cycle on k vertices by adding a vertex that sees every vertex of the cycle see Fig 2 A listing of such results is resented in Tables 1 and 2 Comuting the number of sanning trees of asteroidal grahs is not very interesting because it is fairly easy Consider a comlete-lanet or a star-lanet grah G; since the vertices V ij of each clique, star, ath, or cycle of a lanet-subgrah G i see vertex v i and miss all vertices in VG V ij {v i }, any sanning tree of G consists of a sanning tree of the sun-grah and Mathematics 2008, doi:101016/jdisc200808008

4 SD Nikolooulos et al / Discrete Mathematics Fig 2 Some simle asteroidal grahs a sanning tree of G[V ij {v i }] for each clique, star, ath, or cycle of each G i Let k 1,j, s 1,j, 1,j, and c 1,j denote the numbers of sanning trees of K 1 + K j = K j+1, K 1 + S j, K 1 + P j, and K 1 + C j = W j+1 ie, the wheel grah on j + 1 vertices, resectively; from K j+1 and W j+1 we have that k 1,j = j + 1 j 1 and c 1,j = Luc2j 2 [14], where Luc2j denotes the 2jth Lucas number, 1 while from combinatorial arguments we obtain s 1,j = j+2 2 j 1 and 1,j = Fib2j 1 5 1+ 5 2 2j where Fib2j denotes the 2jth Fibonacci number 2 Then, the numbers τg c and τg s of sanning trees of a comlete-lanet grah G c and a star-lanet grah G s on m sun-vertices are equal to τg c = m m 2 τg s and τg s = j k αj 1,j s βj 1,j γ j 1,j c δj 1,j ; note that a comlete grah on m vertices has m m 2 sanning trees whereas a star grah has a single sanning tree In contrast, comuting the number of sanning trees of K n -comlements of asteroidal grahs is not so easy In order to facilitate the derivation of closed formulas for this number, we define the following ordering of the vertices of the grah G: for each lanet-subgrah G i of G in order, we lace first the vertices that belong to the maximal cliques of G i starting from the vertices of the smallest clique; the vertices that belong to each maximal star of G i are laced next with the star s central vertex last; the vertices of the aths follow in the order they are met along the ath, and after them, the vertices of the cycles in the order they are met around the cycle; in the end, we have the vertices that belong to the sun-grah of G in arbitrary order 22 Comlement-sanning-tree matrix Let G be a grah on n vertices v 1, v 2,, v n The comlement-sanning-tree matrix of the grah G is an n n matrix A defined as follows: 1 dv i if i = j, n A i,j = 1 if i j and v i v j is an edge of G, n 0 otherwise, where dv i is the degree of the vertex v i in G It has been shown [21] also known as the comlement-sanning-tree matrix theorem that the number of sanning trees τg of G is given by τg = n n 2 deta For G = K n, we have that A = I n deta = 1, and Eq 2 imlies Cayley s tree formula [10] which states that τk n = n n 2 Let us aly Eq 2 on G = K n H where VH = < n; then, the comlement-sanning-tree matrix A of G has the following form emty entries in the matrix reresent 0s: [ ] In A = M, where the submatrix M is a matrix which corresonds to the vertices in H Note that the submatrix I n corresonds to the n remaining vertices which have degree n 1 in G, and, thus, they have degree 0 in G From the form of the matrix A, we see that deta = detm Thus, we focus on the comutation of the determinant of matrix M The degree matrix of a grah H on vertices is a matrix D defined as follows: D i,i = dv i and D i,j = 0 for i j, 1 i, j Given the adjacency matrix B of H and the degree matrix D of H, we have M = I + 1 B 1 D If we multily n n each column or row of matrix M by n, we get the matrix M such that: M = ni + B D; clearly, detm = n detm Concluding, we have the following result: 2 1 The Lucas numbers satisfy the recurrence Lucn = Lucn 1 + Lucn 2 with Luc1 = 1 and Luc2 = 3 2 The Fibonacci numbers satisfy the recurrence Fibn = Fibn 1 + Fibn 2 with Fib1 = 1 and Fib2 = 1 Mathematics 2008, doi:101016/jdisc200808008

SD Nikolooulos et al / Discrete Mathematics 5 Corollary 21 Let G = K n H be a grah where VH =, and let M be the matrix of H as defined above Then, τg = n n 2 detm Throughout the aer, emty entries in matrices reresent 0s Moreover we denote by 1 the vector of size whose entries are all equal to 1 3 The number of sanning trees Before roving closed formulas for the number of sanning trees of grah K n G, where G is an asteroidal grah, let us consider the j j matrices M K j, M P j, MC j, and M S j, which corresond to a comlete grah K j, a star S j, a ath P j, and a cycle C j on j vertices, resectively; that is, n j 1 1 1 n 2 1 1 n j 1 1 n 2 1 M K j = 1 1 n, MS j = j 1 n, 2 1 1 1 1 n j 1 1 1 1 n j n 2 1 n 3 1 1 1 n 3 1 1 n 3 1 M P j = 1 n, and MC j = 3 1 1 n 3 1 1 n 2 1 1 n 3 Notice here that by the definition of the comlement-sanning-tree matrix, the sum of the entries in each row and each column, in each of the three matrices, is n 1 It is easy to derive a formula for the determinants of matrices M K j and M S j by subtracting the first row from all the other rows and by adding all the columns to the first column We obtain: λk j detm K j = n 1 n j 1 j 1 and λs j detm S = j n 2j 1 n j j 1 n 2 For the matrices M P j and M C j, we define a recurrence which is solved using standard techniques similar results can be found in [7]; for n 5, we have: where λp j detm P j = n 1 r 2 j n 3 + r j n 3 r j and = n 1 2 j 1 j 1 r t n 3 j t 1 t=0 λc j detm C j = 1 2 j n 3 + r j + n 3 r j + 2 j+1 r = n 1 n 5 It is not difficult to see that the quantities λk j, λs j, λp j, and λc j are all non-negative 31 Comlete-lanet grahs Let K n be the comlete grah on n vertices and G c be a comlete-lanet grah on m sun-vertices {v 1, v 2,, v m } and l lanet-vertices such that VG c VK n We use Corollary 21 in order to derive a closed formula for the number of sanning trees of the grah G = K n G c For each grah G i + v i, 1 i m, where G i is the ith lanet-subgrah of G c, we construct a matrix U i which, based on our ordering scheme see end of Section 21 has the following form: M K 1 αi 1 βi 1 γi 1 δi M S U i = M P M C, 1 T α i 1 T β i 1 T γ i 1 T δ i n dv i Mathematics 2008, doi:101016/jdisc200808008

6 SD Nikolooulos et al / Discrete Mathematics where the submatrices M K, M S, M P, and M C corresond to the cliques, stars, aths, and cycles of G i resectively, and α i = jα j ij, β i = jβ j ij, γ i = jγ j ij, δ i = jδ j ij The matrix M K contains α i1 coies of matrix M K 1, α i2 coies of matrix M K 2, and so on; these coies are laced on the diagonals of M K and thus M K is a block diagonal matrix More recisely, matrix M K has exactly α i rows and columns, each corresonding to a vertex of one of the α ij comlete grahs K j of the grah G i The case is similar for the matrices M S, M P, and M C From its form shown above and Eq 1, we conclude that the matrix U i is of size l i + 1 l i + 1 In order to comute the determinant of matrix U i, we add one more row and one more column at the to and left of the matrix U i ; the resulting l i + 2 l i + 2 matrix U i has its 1, 1-entry and 1, l i + 2-entry equal to 1 whereas all other ositions of the first row and column are equal to 0 More recisely, matrix U i has the following form: By exanding with resect to the entries of the first column of matrix U i, we have detu = i detu i We subtract the first row of U i from all the rows of U i, excet the last row Next, we multily all the columns of U i, excet for the last column, by 1/n 1 and add them to the first column Recall that the sum of the elements of every column excet the last one is equal to n 1 Finally, we subtract the first column from the last column of matrix U i Thus, substituting the value dv i = m+l i 1 and since detu = i detu i, we obtain: M K M S detu i = M P = q i λkj α ij λs j β ij λp j γ ij λc j δ ij, M C j 1 T α i 1 T β i 1 T γ i 1 T δ i q i where q i = n m + l i 1 l i n 1 Now we are ready to comute the number τg of sanning trees for the grah G = K n G c using the comlementsanning-tree matrix theorem and Corollary 21 Thus we construct an m+l m+l matrix U= M for a comlete-lanet grah G c, based on our vertex ordering scheme end of Section 21 Then, we have: where τg = n n m l 2 detu U 1,1 U 2,2 1 l2 U m,m 1 lm U = 1 T l 1 n dv 1 1 1 1 T l 2 1 n dv 2 1 1 l1 1 T l m 1 1 n dv m is an m + l m + l matrix and the submatrices U i,i, 1 i m, are obtained from U i by deleting its last row and its last column which corresond to vertex v i Note that U consists of two blocks: the first block corresonds to the vertices of the lanet-subgrahs of G while the second block corresonds to the vertices of the sun-grah of G it is easy to check the adjacencies It now suffices to comute the determinant of matrix U Following a rocedure similar to the one we alied for the matrix U i, we obtain: l detu = λkj αj λs j βj λp j γ j λc j δj detd c 4 where q 1 1 1 1 q 2 1 D c = 1 1 q m Mathematics 2008, doi:101016/jdisc200808008 3

SD Nikolooulos et al / Discrete Mathematics 7 In order to comute the value detd c, we multily the first row of matrix D c by 1 and add it to the m 1 remaining rows Then, we multily column i by q 1 1, 2 i m, and add it to the first column The matrix D q i 1 c becomes uer triangular with its 1, 1-entry equal to q 1 + m q 1 1 i=2 and each i, i-entry, 2 i m, equal to q q i 1 i 1 Thus, m q 1 1 m 1 detd c = q 1 + q i 1 = 1 + q i 1 q i 1 q i 1 i=2 i=2 By using to denote = q i 1 and by substituting the value of detd c into Eq 4, we obtain the following theorem Theorem 31 Let G c be a comlete-lanet grah on m sun-vertices and l lanet-vertices Then, the number of sanning trees of the grah K n G c, where n m + l, is equal to τk n G c = n n m l 2 λkj αj λs j βj λp j γ j λc j δj m 1 1 + where = n m n n 1 l i j i and αj βj, γ j, δj, ress the number of maximal cliques stars, aths,cycles, res on j vertices in the lanet-grah of G c For the quantities, we obtain the following result The roof is a straightforward derivation of the restrictions on the corresonding arameters of a comlete-lanet grah and, thus, it is omitted Lemma 31 Let G c be a comlete-lanet grah on m sun-vertices and l lanet-vertices Then, for the quantities, i = 1, 2,, m, we have: 1 if m = 1, then 1 1 with equality holding iff n = m + l = l + 1; 2 if m 2 and l = 0, then for all i = 1, 2,, m, = n m 0, where equality holds iff n = m + l = m; 3 if m 2, l > 0, and l t = l, then t n m n 1 i = n m > 0; 4 if m 2, l > 0, and i = 1, 2,, m, l i l, then > 0 It is imortant to note that although a may be negative see Cases 1 and 3 of Lemma 31, the number of sanning trees is never negative 32 Star-lanet grahs Let G s be a star-lanet grah on m + 1 sun-vertices and l lanet-vertices such that VG s VK n We use the comlement-sanning-tree matrix theorem in order to derive a closed formula for the number of sanning trees of the grah G = K n G s In the revious section, for the grah G = K n G c we have formed the l i +1 l i +1 matrix U i, and we have comuted its determinant It is easy to see that the matrix U i matches the corresonding matrix for the grah K n G s rovided that we use dv i = l i + 1, 1 i m We comute the determinant of U i using the same technique as the one we alied for the case K n G c, and we obtain: detu i = λkj α ij λs j β ij λp j γ ij λc j δ ij, j where = n l i + 1 l i = n 1 n l n 1 n 1 i Thus, by Corollary 21 we construct the m + l + 1 m + l + 1 matrix U= M, according to our vertex ordering scheme for a star-lanet grah G s For the number τg of sanning trees of the grah G = K n G s, we have the following formula: where τg = n n m l 3 detu U 1,1 1 l1 U 2,2 1 l2 U m,m 1 lm U = 1 T l 1 n dv 1 1 1 T l 2 n dv 2 1 1 T l m n dv m 1 1 1 1 n dc Mathematics 2008, doi:101016/jdisc200808008 5 6

8 SD Nikolooulos et al / Discrete Mathematics is an m + l + 1 m + l + 1 matrix As in the case of the grah K n G c, each submatrix U i,i, 1 i m, is obtained from U i by deleting its last row and its last column Note that l i detu i,i = λk j α ij λs j β ij λp j γ ij λc j δ ij Thus, for the determinant of matrix U, we have 1 1 l 2 1 detu = λkj αj λs j βj λp j γ j λc j δj m 1 1 1 1 n m l = λkj αj λs j βj λp j γ j λc j δj detd s, where D s is an m + 1 m + 1 matrix In order to comute the determinant of the matrix D s, we work as in the case of the comlete-lanet grah: We multily the first row of matrix D s by 1 and add it to each of the next m 1 rows Then, we multily column i by 1, 2 i m, and add it to the first column Finally, in order to make the matrix D s uer triangular, we multily the first column by 1 and add it to column m 1 + 1 Thus, detd s = n m 1 m 1 m 1 = n m 1 We have the following theorem: i=2 i i Theorem 32 Let G s be a star-lanet grah on m + 1 sun-vertices and l lanet-vertices Then, the number of sanning trees of the grah K n G s, where n m + l + 1, is equal to τk n G s = n n m l 3 λkj αj λs j βj λp j γ j λc j δj m 1 n m where = n 1 n n 1 l i j i and αj βj, γ j, δj, ress the number of maximal cliques stars, aths, cycles, res on j vertices in the lanet-grah of G s In this case, for the quantities, i = 1, 2,, m, we have: 7 Lemma 32 Let G s be a comlete-lanet grah on m + 1 sun-vertices and l lanet-vertices Then, 1 if l = 0, then for all i = 1, 2,, m, = n 1 > 0; 2 if l > 0 and m = 1, then 1 > 0; 3 if l > 0, m 2, and l t = l, then t 1 > 0 and for all i t, n 1 i = n 1 > 0; 4 if l > 0, m 2, and i = 1, 2,, m, l i < l, then > m 2 Lemma 32 imlies that the s are in all cases ositive Moreover, for case 3, Eq 7 imlies that t nn 2 l+1 n 1 1 n 1 = n 12 nl n 1 = 4 Maximization results As mentioned in the introduction, a uniformly-most reliable network [5,15] is defined to maximize the number of sanning trees Thus, it is interesting to determine the tyes of grahs which have the maximum number of sanning trees for fixed numbers of vertices and edges In this section, we rovide maximization results for the number of sanning trees of K n G, where G is a comlete-lanet or a star-lanet grah In order to kee the number of vertices and edges fixed, we assume that: the clique K n is fixed ie, n is fixed, the number m of vertices of the sun-grah is fixed, and the clique vector α, the star vector β, the ath vector γ, and the cycle vector δ are fixed; thus, our results are over the family of comlete-lanet grahs res, star-lanet grahs obtained by all ossible combinations of connecting each clique, star, ath, and cycle of each lanet-subgrah to a sun-vertex Mathematics 2008, doi:101016/jdisc200808008

SD Nikolooulos et al / Discrete Mathematics 9 41 Comlete-lanet grahs Let G c be a comlete-lanet grah on m sun-vertices and l lanet-vertices, where l = l 1 +l 2 + +l m and l i, 1 i m, is the number of vertices of its lanet-subgrahs G 1, G 2,, G m For notational convenience, we write the number τk n G c of sanning trees of the grah K n G c given by Theorem 31 as the roduct τk n G c = XG c YG c, where XG c = n n m l 2 λkj αj λp j βj λc j γ j λs j δj, i m 1 YG c = 1 + ; recall that = n m n l n 1 i Eq 5 Since we are interested in maximizing the number of sanning trees when the arameters n and m, as well as the clique star, ath, and cycle vectors are fixed, it suffices to maximize the factor YG c We will concentrate in the case where m 2 and l > 0; if m = 1 or l = 0, we have no flexibility in changing the grah G c We note that m 1 m YG c = 1 + = + j = m + m i j i j i j + and if we substitute m by S in YG c, where S = m = n m m n n, m, l are fixed, we get YG c = 1 + S + S j j i n 1 l which has a fixed value since We comute the maximum by comuting the artial derivative of YG c with resect to any t, 1 t m 1, and setting it equal to 0: YG c = + 1 + S j + S j t which through standard algebraic maniulations is simlified to YG c 1 = 1 + S t t Then, we can show the following: Lemma 41 If m 2 and l > 0, then for any t {1, 2,, m} 1 1 + > 0 j i j i,t Proof If for all i = 1, 2,, m, it holds that l i < l, then > 0 Case 4 of Lemma 31 and the lemma clearly follows Suose now that l j = l for some j in {1, 2,, m}; then Case 3 of Lemma 31 alies and j n m and n 1 i = n m > 0 for all i j Since > 0 for all i j, the lemma again readily follows if t = j, whereas if t j we need only show that j 1 + 1 i > 0: 1 j 1 + = j 1 + m 2 n m + 1 = j 1 + m 2 + 1 n 2 j n m n 1 + 1 = 1 n 1 > 0 since j n m and n m + l > m 2 n 1 Mathematics 2008, doi:101016/jdisc200808008

10 SD Nikolooulos et al / Discrete Mathematics In light of Lemma 41, the artial derivative YG c / t equals 0 if and only if t = S = m This equality holds for each t = 1, 2,, m 1; thus, the quantity YG c reaches an extremum if and only if 1 = 2 = = m or equivalently if l 1 = l 2 = = l m Moreover, since 2 YG c = 2 i t 1 1 + 1 1 < 0, we verify that the above extremum of YG c is a maximum Our result is stated in the following theorem Theorem 41 Let G c be a comlete-lanet grah with fixed clique, star, ath, and cycle vectors, and m + l vertices, where l = m l i and l i is the number of vertices of its ith lanet-subgrah G i Then, the number of sanning trees of the grah K n G c is maximized when the l i s are all equal, if this is ossible It is worth noting that maximizing the number of sanning trees of K n G c is NP-comlete; it follows from the wellknown Partition roblem [6] In [4], a maximization theorem was rovided for the grah K n G, where G is a multi-star grah, which follows as a consequence of Theorem 41; since the authors in [4] consider that the lanet-comonents can only be single vertices, then if it is not ossible to have l 1 = l 2 = = l m, it is certainly feasible to ensure that any two of the l i s differ by at most 1 Since for given clique, star, ath, and cycle vectors, achieving that l 1 = l 2 = = l m, if ossible, requires us to make a large number of combinations in general, below we give another result which when alied reeatedly hels us attain a maximum in this number of sanning trees, although this may not necessarily be the global maximum Let v i and v j be two arbitrary vertices of the sun-grah G c [A] and let G i and G j on l i and l j vertices, resectively be their corresonding lanet-subgrahs From G c, we construct the comlete-lanet grah G c by moving lanet-comonents between the lanet-subgrahs G i and G j to obtain lanet-subgrahs G i and G j on l i and l j vertices, resectively; then, the grahs G c and G c have the same clique, star, ath, and cycle vectors, and l + i l = l j i + l j Let us find the number of sanning trees of the grahs G c and G c First, the quantity YG c can be written in terms of and j as YG c = j k + j k=1 m k =1 k k=1 i,j k k,i,j k + + j k = j Φ 1 + + j Φ 2 where Φ 1 = m k=1 k 1 + m k 1 =1 k i,j k for the grah G c we obtain YG = c i Φ j 1 + + i Φ j 2 Since the grahs G c and G c k=1 and Φ 2 = m k=1 k ; note that Φ 1 and Φ 2 are indeendent of and j Similarly, have the same clique, star, ath, and cycle vectors, we have that XG c = XG c In order to comare the numbers of sanning trees of K n G c and K n G c, we examine their difference: τk n G c τk n G c = YG c YG c XG c = i j j Φ 1 + i + j j Φ 2 XGc 8 From Eq 5 and the fact that l + i l = l j i + l j, it is easy to see that + i = j + j and i j j = n Thus, Eq 8 becomes n 1 2 l i l j l il j 2 n τk n G c τk n G c = l i n 1 l l j il j Φ1 XG c 9 In a fashion similar to the one used in the roof of Lemma 41, we can show that Φ 1 > 0 Additionally, XG c 0 Thus, in Eq 9 we have to consider the value of l i l j l il j We rove the following lemma Lemma 42 If l i l j < l i l j, then l i l j l il j > 0 Proof Without loss of generality, let l i l j and l i l j ; then, l i l j < l i l j l i l j < l i l j This inequality and the fact that l i + l j = l i + l j imly that l i > l i and l i l i = l j l j Let l i l i = l j l j = r > 0, that is, l i = l i r and l j = l j + r Since l i l j, it has to be that l i r l j + r l i l j 2r Additionally, l i l j = l il j + rl i l j r 2, and, thus, l i l j l il j = rl i l j r 2 r 2 > 0 Mathematics 2008, doi:101016/jdisc200808008

Therefore, we have the following theorem ARTICLE IN PRESS SD Nikolooulos et al / Discrete Mathematics 11 Theorem 42 Let G c be a comlete-lanet grah, v i, v j two vertices of its sun-grah and l i, l j the numbers of vertices of the lanet-subgrahs G i and G j associated with v i and v j, resectively If the cliques, stars, aths, and cycles of G i and G j are rearranged so that in the resulting grah G c the numbers of vertices of the lanet-subgrahs associated with v i and v j are l i and l j with l i l j < l i l j, then the number of sanning trees of K n G c is larger than that of K n G c Theorem 42 imlies that we can ick a air of vertices of the sun-grah of a comlete-lanet grah G c and rearrange the lanet-comonents of their lanet-subgrahs so as to minimize the absolute value of the difference of their vertex numbers We reeat the rocess for another air of vertices of the sun-grah and so on so forth until no air of vertices yields a larger number of sanning trees 42 Star-lanet grahs We rove similar results for the number τk n G s of sanning trees of the grah K n G s for a star-lanet grah G s on m + 1 sun-vertices and l lanet-vertices where l = l 1 + l 2 + + l m and l i, 1 i m, is the number of vertices of its lanet-subgrah G i Then, from Theorem 32 we have that τk n G s = XG s YG s, where XG s = n n m l 3 λkj αj λs j βj λp j γ j λc j δj, m 1 YG s = n m, i and = n 1 n l n 1 i Eq 7 In order to maximize the number τk n G s of sanning trees under the assumtion that the arameters n and m, and the clique, star, ath, and cycle vectors are fixed, it suffices to maximize the factor YG s It is well known that the roduct of k ositive numbers a 1, a 2,, a k whose sum A is constant is maximized when they are all equal; this is shown by relacing a k by A k 1 a i in the roduct of the a i s and by working with the artial k 1 derivative of the resulting exression with resect to any of the a i s In the same way, we can show that the sum of a i k ositive numbers a 1, a 2,, a k whose sum is constant is minimized when they are all equal Thus, since the sum of the s is n mm nl n 1 which is fixed, because n, m, l are assumed to be fixed, both factors of YG s are maximized when all the s are equal Thus, if we take into account that any two, j are equal if and only if l i = l j see Eq 7 we have a result similar to Theorem 41 Again observe that achieving a maximum number of sanning trees by making all l i s equal is NP-comlete [6] Theorem 43 Let G s be a star-lanet grah with fixed clique, star, ath, and cycle vectors, and m + l + 1 vertices, where l = m l i and l i is the number of vertices of its ith lanet-subgrah G i Then, the number of sanning trees of the grah K n G s is maximized when the l i s are all equal, if this is ossible Next, we consider a star-lanet grah G s and two arbitrary vertices v i and v j of its sun-grah G s [A] whose corresonding lanet-subgrahs contain l i and l j vertices, resectively We construct from G s the star-lanet grah G s by moving comonents between these subgrahs so that the resulting lanet-subgrahs of v i and v j have l i and l j vertices, resectively Then, l + i l = l j i + l j and XG s = XG s The quantity YG s can be written in terms of and j as YG s = n m j where Ψ 1 = k j k=1 m k =1 k k=1 i,j k k,i,j k + j k = j Ψ 1 + j Ψ 2, n m m k 1 =1 = m k k=1 i,j k k ; again, Ψ 1, Ψ 2 are indeendent of and j A similar exression holds for YG s in terms of i and j Since l i + l j = l i + l j, Eq 7 imlies that + j = j + j and m k=1 k and Ψ 2 k=1 i j j = n 2 n 1 l i l l j il j Thus, 2 n τk n G s τk n G s = l i n 1 l l j il j Ψ1 XG s Lemma 43 establishes that for any i, j, Ψ 1 > 0 Lemma 43 If m 2 and l > 0, then Ψ 1 = n m m k =1 k i,j 1 k m k=1 k > 0 Mathematics 2008, doi:101016/jdisc200808008

12 SD Nikolooulos et al / Discrete Mathematics Proof If for all k = 1, 2,, m, l k l then k > m see Case 4 of Lemma 32, and thus Ψ 1 > n m m 2 m m 2 > n m 1 m m 2 > 0 m since n m + 1 + l > m + 1 Suose now that there exists t such that l t = l Then, t 1 k = n 1 Case 3 of Lemma 32 If t is either i or j then Ψ 1 = n m m 2 n 1 m 2 n 1 > n m 1 n 1 m 2 > 0 since n 1 > m 2 0 If t differs from both i and j then m 3 Ψ 1 = n m n 1 + 1 n 1 m 3 t = n 1 m 3 t because n > m m 3 n 1 n m m 3 n 1 m 3 > 1 n m > n m 1 n 1 and n 1 t 1 n 1 Additionally, XG s 0 Thus, from Lemmas 42 and 43, we have that: n 1 > 0 and for all k t, t 1 Theorem 44 Let G s be a star-lanet grah, v i, v j two vertices of its sun-grah and l i, l j the numbers of vertices of the lanetsubgrahs G i and G j associated with v i and v j, resectively If the cliques, stars, aths, and cycles of G i and G j are rearranged so that in the resulting grah G s the numbers of vertices of the lanet-subgrahs associated with v i and v j are l i and l j with l i l j < l i l j, then the number of sanning trees of K n G s is larger than that of K n G s Due to Theorem 44, a local minimum in the number of sanning trees can be obtained as in the case of comlete-lanet grahs > 0 Acknowledgments The authors thank the anonymous referees whose suggestions heled imrove the resentation of the aer References [1] C Berge, Grahs and Hyergrahs, North-Holland, 1973 [2] FT Boesch, On unreliability olynomials and grah connectivity in reliable network synthesis, J Grah Theory 10 1986 339 352 [3] TJN Brown, RB Mallion, P Pollak, A Roth, Some methods for counting the sanning trees in labelled molecular grahs, examined in relation to certain fullerenes, Discrete Al Math 67 1996 51 66 [4] K-L Chung, W-M Yan, On the number of sanning trees of a multi-comlete/star related grah, Inform Process Lett 76 2000 113 119 [5] CJ Colbourn, The Combinatorics of Network Reliability, Oxford University Press, New York, 1980 [6] MR Garey, DS Johnson, Comuters and Intractability: A Guide to the Theory of NP-Comleteness, WH Freeman, San Francisco, 1979 [7] B Gilbert, W Myrvold, Maximizing sanning trees in almost comlete grahs, Networks 30 1997 23 30 [8] MJ Golin, X Yong, Y Zhang, Chebyshev olynomials and sanning tree formulas for circulant and related grahs, Discrete Math 298 2005 334 364 [9] AK Kelmans, On grahs with the maximum number of sanning trees, Random Structures Algorithms 1 2 1996 177 192 [10] F Harary, Grah Theory, Addison-Wesley, Reading, MA, 1969 [11] Z Huaxiao, Z Fuji, H Qiongxiang, On the number of sanning trees and Eulerian tours in iterated line digrahs, Discrete Al Math 73 1997 59 67 [12] Z Lonc, K Parol, J Wojcieckowski, On the number of sanning trees in directed circulant grahs, Networks 36 2001 129 133 [13] W Moon, Enumerating labeled trees, in: F Harary Ed, Grah Theory and Theoretical Physics, 1967, 261 271 [14] BR Myers, Number of sanning trees in a wheel, IEEE Trans Circuit Theory 18 1971 280 282 [15] W Myrvold, KH Cheung, LB Page, JE Perry, Uniformly-most reliable networks do not always exist, Networks 21 1991 417 419 [16] SD Nikolooulos, C Nomikos, P Rondogiannis, A limit characterization for the number of sanning trees of grahs, Inform Process Lett 90 2004 307 313 [17] SD Nikolooulos, C Paadooulos, The number of sanning trees in K n -comlements of quasi-threshold grahs, Grahs Combin 20 2004 383 397 [18] SD Nikolooulos, P Rondogiannis, On the number of sanning trees of multi-star related grahs, Inform Process Lett 65 1998 183 188 [19] PV O Neil, The number of trees in a certain network, Notices Amer Math Soc 10 1963 569 [20] L Petingi, F Boesch, C Suffel, On the characterization of grahs with maximum number of sanning trees, Discrete Math 179 1998 155 166 [21] HNV Temerley, On the mutual cancellation of cluster integrals in Mayer s fugacity series, Proc Phys Soc 83 1964 3 16 [22] L Weinberg, Number of trees in a grah, Proc IRE 46 1958 1954 1955 [23] W-M Yan, W Myrvold, K-L Chung, A formula for the number of sanning trees of a multi-star related grah, Inform Process Lett 68 1998 295 298 [24] Y Zhang, X Yong, MJ Golin, The number of sanning trees in circulant grahs, Discrete Math 223 2000 337 350 Mathematics 2008, doi:101016/jdisc200808008