Edge-grafting theorems on permanents of Laplacian matrices of graphs and their applications

Transcription

1 Electronic Journal of Linear Algebra Volume 6 Volume 6 (013) Article Edge-grafting theorems on permanents of Laplacian matrices of graphs and their applications Shuchao Li scmath@mailccnueducn Yan Li Xixi Zhang Follow this and additional works at: Recommended Citation Li, Shuchao; Li, Yan; and Zhang, Xixi (013), "Edge-grafting theorems on permanents of Laplacian matrices of graphs and their applications", Electronic Journal of Linear Algebra, Volume 6 DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository For more information, please contact scholcom@uwyoedu

2 Volume 6, pp 8-48, January EDGE-GRAFTING THEOREMS ON PERMANENTS OF THE LAPLACIAN MATRICES OF GRAPHS AND THEIR APPLICATIONS SHUCHAO LI, YAN LI, AND XIXI ZHANG Abstract The trees, respectiely unicyclic graphs, on n ertices with the smallest Laplacian permanent are studied In this paper, by edge-grafting transformations, the n-ertex trees of gien bipartition haing the second and third smallest Laplacian permanent are identified Similarly, the n-ertex bipartite unicyclic graphs of gien bipartition haing the first, second and third smallest Laplacian permanent are characterized Consequently, the n-ertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined Key words Laplacian matrix, Laplacian coefficient, Permanent, Tree, Unicyclic graph, Bipartition AMS subject classifications 05C50, 05C05 1 Introduction Let G = (V G,E G ) be a simple connected graph with ertex set V G = { 1,, n } and edge set E G The adjacency matrix A(G) = (a ij ) of G is an n n symmetric matrix with a ij = 1 if and only if i, j are adjacent and 0 otherwise Since G has no loops, the main diagonal of A(G) contains only 0 s Denote the degree of i by d G ( i ) (or d i ) for i = 1,,n, and let D(G) be the diagonalmatrix whose (i,i)-entry is d i,i = 1,,,n The matrix L(G) = D(G) A(G) is called the Laplacian matrix of G Of course, L(G) depends on the ordering of the ertices of G Howeer, a different ordering leads to a matrix which is permutation similar to L(G) The matrix Q(G) = D(G) + A(G) has been called the signless Laplacian matrix of G For surey papers on this matrix the reader is referred to [, 3, 4] If the ertex set of the connected graph G on n ertices can be partitioned into two subsets V 1 and V such that each edge joins a ertex of V 1 to a ertex of V, then G has a (p,q)-bipartition where V 1 = p and V = q Without loss of generality we may assume that p q A connected graph with n ertices and n edges is called a unicyclic graph For conenience, let Tn p,q (resp, Un p,q ) be the set ofall n-ertextrees (resp, bipartite unicyclic graphs) with a (p,q)-bipartition, and let U n be the set of all bipartite unicyclic graphs on n ertices Receied by the editors on March 6, 01 Accepted for publication on December 16, 01 Handling Editor: Raphael Loewy Faculty of Mathematics and Statistics, Central China Normal Uniersity, Wuhan , PR China (lscmath@mailccnueducn) 8

3 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 9 Throughout we denote by P n, S n and C n the path, star and cycle on n ertices, respectiely Let G,G u denote the graph obtained from G by deleting ertex V G, or edge u E G, respectiely (this notation is naturally extended if more than one ertex or edge is deleted) Similarly, G+u is obtained from G by adding edge u E G The distance between ertices u and in G is denoted by d G (u,) Let PV(G) denote the set of all pendant ertices of G The permanent of X = (x ij ) M n n, denoted by perx, is the quantity perx = n x tσ(t), σ S n t=1 where S n is the symmetric group of degree n; see [16] It was suggested in [15] to use the polynomial per(xi L(G)) to distinguish non-isomorphic trees For more progress on the quantity per( ), the reader is referred to [19] The first research paper on the permanent of the Laplacian matrix was [15], in which lower bounds for the permanent of L(G) were conjectured by Merris, Rebman and Watkins These lower bounds on perl(g) were proed by Brualdi and Goldwasser [1] and Merris [14] For more recent results on Laplacian (resp, signless Laplacian) permanent one is referred to [5, 11, 1] The Laplacian polynomial µ(g, λ) of G is the characteristic polynomial of its Laplacian matrix L(G), that is, n µ(g,λ) = det(λi n L(G)) = ( 1) k c k λ n k It is easy to see that c 0 (G) = 1,c 1 (G) = E G,c n (G) = 0,c n 1 (G) = nτ(g), where τ(g) denotes the number of spanning trees of G For two n-ertex graphs G 1 and G, we say that G 1 is dominated by G and write G 1 G, if c k (G 1 ) c k (G ) holds for all Laplacian coefficients c k,k = 0,1,,n If G 1 G and there exists j such that c j (G 1 ) < c j (G ), then we write G 1 G Note that the Laplacian coefficients hae combinatorial significance, hence the research on the Laplacian coefficients of graphs has receied great attention in recent years; see [6, 7, 8, 9, 10, 17, 18, 0] and the references therein Zhou and Gutman [1] showed that among all trees of order n, the kth coefficient c k is the largest when the tree is a path and is the smallest for a star, k = 0,1,,n In iew of Theorems 4 and 5 in [1] the counterparts of these results for the Laplacian permanent of trees are as follows k=0 Theorem 11 ([1]) Let T be a tree with n ertices Then (n 1) per L(T) (1+ ) n + + (1 ) n

4 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang The left equality holds if and only if T is a star, whereas the right equality holds if and only if T is a path Brualdi and Goldwasser [1] showed that T n,m is the unique tree among the n- ertex trees each of which contains an m-matching haing the minimum Laplacian permanent, where T n,m is the tree obtained from the star graph S n m+1 by attaching a pendant edge to each of certain m 1 non-central ertices of S n m+1 Ilić [8] showed that T n,m is also the unique n-ertex tree with gien matching number m which simultaneously minimizes all the Laplacian coefficients It is then natural to conjecture that among the class of graphs, if a particular graph has the smallest Laplacian permanent, then that particular graph also minimizes all of its Laplacian coefficients in that class of graphs, and ice ersa This mathematical phenomenon is further studied in [1, 7] We know from [7] that among the n-ertex trees of diameter d, the caterpillar T n,d, d/ has the minimum Laplacian coefficient c k, for eery k = 0,1,,n, whereas we know from [1] that among the n-ertex trees of diameter d, the broom T n,d, has the minimum Laplacian permanent Graphs T n,d, d/ and T n,d, are depicted in Fig 11 This implies that there is no monotone relationship between the Laplacian coefficients and the Laplacian permanent of graphs Yet we lack a better understanding of this relationship u u n d 1 u 1 u u 1 u n d d/ d T n,d, d/ d/ d d d+1 Fig 11 Graphs T n,d, d/ and T n,d, T n,d, An interesting fact is that among n-ertex trees with a gien bipartition, the extremal one that minimizes the Laplacian permanent [1] is exactly the one that simultaneously minimizes all Laplacian coefficients; see [13] Up to now, it is natural for us to find some interesting classes of graphs such that the extremal graphs which has smallest the Laplacian permanent and the Laplacian coefficients among these classes respectiely are the same Motiated by [1, 13], in this paper, we use a new and unified method to show some known results on the Laplacian permanent, as well we use the edge-grafting transformations to identify the n-ertex trees of gien bipartition haing the second and third smallest Laplacian permanent Similarly, we also characterize the n-ertex bipartite unicyclic graphs of gien bipartition haing the first, second and third smallest Laplacian permanent Consequently, we identify the n-ertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent

5 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 31 Three edge-grafting theorems on Laplacian permanent In this section, we introduce three edge grafting transformations and study their properties Definition 1 Let u be a pendant edge of an n-ertex bipartite graph G with d(u) = 1, n 3 Let w( ) be a ertex of G with d(w) d() Let G[ w;1] be the graph obtained from G by deleting the edge u and adding the edge uw In notation, G[ w;1] = G u +uw and we say G[ w;1] is obtained from G by Operation I Theorem Let G and G[ w;1] be the bipartite graphs defined as aboe Then perl(g) > perl(g[ w;1]) Proof Let d G () = r and d G (w) = t First we consider that w E G With an appropriate ordering of the ertices of G and G[ w;1] as u,,w,, we see that L(G) = r 0 x t x 0 y 1 y A and L(G[ w;1]) = r 1 0 x t+1 x 0 y 1 y A Let M 1 (resp, M ) be the matrix obtained from L(G) (resp, L(G[ w;1])) by eliminating the first row and the first column Let N 1 (resp, N ) be the matrix obtained from L(G) (resp, L(G[ w;1])) by eliminating the first rows and the first columns And let N be the matrix obtained from M by eliminating its second row and second column Then we hae perl(g) = perm 1 +pern 1, perl(g[ w;1]) = perm +pern Set ( 0 x S 1 = per y A ) ( 0 x1, S = per y 1 A ) Note that pern 1 = t pera+s 1, pern = (r 1) pera+s, perm = perm 1 +pern pern 1

6 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang Hence, (1) perl(g) perl(g[ w;1]) = ((t r) pera+pera+s 1 S ) By the choice of S 1 and S, we hae () pera+s 1 > S Note that t r, by (1) and () we get perl(g) perl(g[ w;1]) > 0 Now consider w E G By a similar argument as in the proof of the case w E G, we can also get perl(g) perl(g[ w;1]) > 0 We omit the procedure here Definition 3 Let w be an edge of a bipartite graph U with d(w) G is obtained from U and the star S k+ by identifying with a pendant ertex of S k+ whose center is u Let G[u w;] be the graph obtained from G by deleting all edges uz,z W and adding all edges wz,z W, where W = N G (u)\{} In notation, G[u w;] = G {uz : z W}+{wz : z W} andwesayg[u w;] isobtainedfromgbyoperation IIGraphsGandG[u w;] are depicted in Fig 1 U w G u u 1 u u k U w u u 1 u u k G[u w;] Fig 1 G G[u w;] by Operation II Theorem 4 Let G and G[u w;] be the bipartite graphs described as aboe Then perl(g) > perl(g[u w;]) Proof LetN G (u)\{} = {u 1,u,,u k }, wherek 1, u 1,u,,u k arependant ertices With an appropriate ordering of the ertices of G and G[u w;] as u 1,u,,u k,u,,w,, we hae L(G) = k d 1 x 1 1 d w x y 1 y A

7 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 33 and L(G[u w;]) = d 1 x d w +k x y 1 y A, where d and d w are degrees of and w in G with d w Let D 1 (resp, D ) be the matrix obtained from L(G) (resp, L(G[u w;])) by eliminating the first k + 1 rows and the first k + 1 columns; let M 1 (resp, M ) be the matrix obtained from D 1 (resp, D ) by eliminating the first row and the first column Let M be the matrix obtained from D 1 by eliminating the second row and the secondcolumn, and let M(i)(resp, N(i))(1 i k) be the matrixobtainedfrom L(G[u w;]) byeliminatingthe rowsandcolumnscorrespondingtou 1,u,,u k i and u (resp, u 1,u,,u k i,u and ) It is routine to check that {perm(i),1 i k} and {pern(i),1 i k}, respectiely, hae the recurrence relation perm(i) = perm(i 1)+perM, pern(i) = pern(i 1)+perA, i k with initial alue perm(1) = perd +perm and pern(1) = perm +pera Hence, (3) perm(k) = perd +k perm, pern(k) = perm +k pera By expanding the permanent of L(G) along the first (k +1) rows we obtain (4) perl(g) = (k +1) perd 1 +perm 1, and by expanding the permanent of L(G[u w;]) along the row corresponding to u, we get perl(g[u w;]) = perm(k)+pern(k) Note that perd = perd 1 +k perm and perm = perm 1 +k pera Hence, by (3) (5) perl(g[u w;]) = perd 1 +k perm +perm 1 +k pera For conenience, denote by D 1 the matrix obtained from D 1 by replacing d w with

8 Electronic Journal of Linear Algebra ISSN Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang d w 1 In iew of (4) and (5), we get perl(g) perl(g[u w;]) = k perd 1 k perm k pera = k(perd 1 +perm) k perm k pera = k(perd 1 pera) k[(d (d w 1)+1) pera pera] > 0 Definition 5 Let G be an n-ertex graph obtained from C k = 1 i j k 1 (k 3) and two stars S ni+1,s nj+1 by identifying i (resp, j ) with the center of S ni+1 (resp, S nj+1), where 4 < i < j, n = k +n i +n j ; see Fig Let G = G Then we say that G is obtained from G by Operation III n i C k Ck i j n j i j ni n j G G Fig G G by Operation III Theorem 6 Let G and G be the bipartite unicyclic graphs described as aboe Then perl(g) > perl(g ) Proof Ordering the ertices of G as 1,, 3, 4,, we hae 1 x L(G) = x 4 y 1 y 4 A Ordering the ertices of G as, 3, 4, 1,, we see that L(G ) = x 4 1 x 1 y 4 y 1 A Let D (resp, D ) be the matrix obtained from L(G) (resp, L(G )) by eliminating the rows and columns corresponding to and 3 ; and let M 1 (resp, M ) be obtained

9 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 35 from L(G) (resp, L(G )) by eliminating rows and columns corresponding to, 3, 4 (resp, 1,, 3 ) And for conenience, denote N 1 = ( 0 x1 A y 4 ), N = ( 0 x4 A y 1 ) Expanding the rows corresponding to and 3 of L(G) and L(G ), respectiely, yields perl(g) = pern 1 +perm +pera+5perd +perm 1 pern, perl(g ) = 3perD +perm 1 Together with perd = perd +perm 1 +pera pern 1 +pern, we obtain and hence perl(g ) = 3perD +4perM 1 +3perA 3perN 1 3perN perl(g) perl(g ) =perd +perm +pern 1 +pern perm 1 pera (6) =(perd perm 1 )+(perm pera)+pern 1 +pern By ordering the ertices of G as 1,, 3, 4,, i,, j,, k,, and by direct calculation, we hae pern 1 = pern = 1 And note that 3 0 x 1 (perd perm 1 )+(perm pera) = per 0 1 x 4 3 y 1 y 4 A Together with (6), we hae perl(g) perl(g ) 3+( 1)+( 1) = > 0 3 Applications 31 Laplacian permanents of trees among T p,q n We denote by D(p,q) a double star with n ertices, which is obtained from an edge w by attaching p 1 (resp,q 1)pendanterticesto (resp, w), wheren = p+q LetD (p 1,q 1)(resp, D (p 1,q 1))beann-ertextreeobtainedfromD(p 1,q 1)byattachingapendant path of length to w (resp, ) Graphs D(p,q), D (p 1,q 1), D (p 1,q 1) are depicted in Fig 31 Let T(n,k,a) be an n-ertex tree obtained by attaching a and n k a pendant ertices to the two end-ertices of P k, respectiely In particular, D(p,q) = T(n,,p 1) The following lemma is routine to check

10 Electronic Journal of Linear Algebra ISSN Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang p 1 w q 1 p w D(p,q) D (p 1,q 1) D (p 1,q 1) q p w q Fig 31 Trees D(p,q),D (p 1,q 1) and D (p 1,q 1) Lemma 31 Let p and q be positie integers, then perl(d (p 1,q 1)) = (p 3)(6q 5)+3, perl(d (p 1,q 1)) = (q 3)(6p 5)+3 From Lemma 31, a direct calculation yields (31) perl(d (p 1,q 1)) < perl(d (p 1,q 1)) for q > p > From [1] we knowthat D(p,q) minimizes the Laplacianpermanent oftrees among n In this subsection, we use a new and unified method to determine the tree in which has the first, second, and third smallest Laplacian permanent, respectiely T p,q Tn p,q Theorem 3 ([1]) Let T be a tree with a (p, q)-bipartition Then perl(t) (p 1)(q 1)+1 with equality if and only if T is a double-star D(p,q) Proof Choose a tree T with a(p, q)-bipartition such that its Laplacian permanent is as small as possible Let V 1, V be the bipartition of the ertices of T with V 1 = { 0, 1,, p 1 }, V = {u 0,u 1,,u q 1 } For conenience, let 0 (resp, u 0 ) be the ertex of maximal degree among V 1 (resp, V ) in T and let A = N T ( 0 ) PV(T) Hence, in order to complete the proof, it suffices to show the following claims s 0 u 1 1 u r 1 r 1 u 0 n r s Fig 3 Tree T(n,r,s) with some labelled ertices Claim 1 T = T(n,r,s) (see Fig 3) with r 1 and s 0 Proof of Claim 1 Otherwise, T must contain a pendant ertex w N T (u 0 ) N T ( 0 ) Without loss of generality, we may assume w V and its unique neighbor is

11 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 37 w Using Operation I, let T 0 = T ww +w 0 By Theorem we hae perl(t 0 ) < perl(t), a contradiction to the choice of T Claim In the tree described as aboe, u 0 is adjacent to 0 Proof of Claim If not, then d(u 0, 0 ) 3 Note that 0 is the maximal degree ertex among V 1, hence d T ( 0 ) 1, which implies A Using Operation II, let T 1 = T { 0 z : z A}+{ 1 z : z A} then perl(t 1 ) < perl(t) by Theorem 4, which contradicts the choice of T By Claims 1 and, we get that T = D(p,q) By direct computing, we hae perl(d(p,q)) = (p 1)(q 1)+1 Theorem 33 Among T p,q n (i) If p =, then all the members in T,n n are ordered as follows: perl(d(,n )) = perl(t(n,3,0)) < perl(t(n,3,1)) < perl(t(n,3,) ( ( )) n 3 < < perl(t(n,3,i)) < < perl T n,3, (ii) If p >, (a) for T Tn p,q \ {D(p,q)}, we hae perl(t) (p 3)(6q 5) + 3 The equality holds if and only if T = D (p 1,q 1) (b) for T Tn p,q \{D(p,q),D (p 1,q 1)} with q > p, we hae perl(t) (q 3)(6p 5)+3 The equality holds if and only if T = D (p 1,q 1) Proof (i) If p =, then T,n n = By a simple calculation, we get { T(n,3,0),T(n,3,1),,T(n,3,i),,T ( n,3, n 3 ( perl(t(n,3,i)) = 8 i n 3 ) +(n 3) +6n 14 )} Consider the function f(x) = 8(x n 3 ) + (n 3) + 6n 14 in x with By the monotonicity of f(x), we hae 0 x n 3 ( n 3 (3) f(0) < f(1) < < f(i 1) < f(i) < f(i+1) < < f )

12 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang Note that T(n,3,0) = D(,n ), together with (3) we get that (i) holds (ii) Choose T T p,q n \{D(p,q)} such that its Laplacian permanent is as small as possible LetV 1, V be thebipartitionoftheerticesoft withv 1 = { 0, 1,, p 1 }, V = {u 0,u 1,,u q 1 } For conenience, let 0 (resp, u 0 ) be the ertex of maximal degree among V 1 (resp, V ) in T and let A = N T ( 0 ) PV(T) In order to complete the proof, it suffices to show the following claims Claim 1 u 0 0 E T Proof of Claim 1 If not, then d T (u 0, 0 ) 3 In this case, we are to show that T = T(n,r,s) (see Fig 3) Otherwise, T must contain a pendant ertex w N T (u 0 ) N T ( 0 ) Assume that the unique neighbor of w is w Using Operation I, let T = T ww +w 0 if w V and T = T ww +wu 0 otherwise Note that T Tn p,q \{D(p,q)}, by Theorem we hae perl(t ) < perl(t), a contradiction to the choice of T Hence, T = T(n,r,s) On the one hand, d T (u 0, 0 ) 3, hence r ; on the other hand, 0 is of the maximal degree ertex in V 1 of T, hence s 1 Therefore, A Using Operation II, let T 0 = T { 0 z : z A}+{ 1 z : z A} We also hae T 0 Tn p,q \{D(p,q)} In iew of Theorem 4, perl(t 0 ) < perl(t), which also contradicts the choice of T Claim In the tree T described as aboe, there exists a pendant ertex, say w, in V T such that d T (w,u 0 ) =,d T (w, 0 ) = 3 or d T (w, 0 ) =,d T (w,u 0 ) = 3 Furthermore, for all PV(T)\{w}, is adjacent to either u 0 or 0 Proof of Claim Note that T = D(p,q), hence there must exist a ertex (not necessarily a pendant ertex), say w, such that d T (w,u 0 ) =,d T (w, 0 ) = 3 or d T (w, 0 ) =,d T (w,u 0 ) = 3 With loss of generality, we assume that d T (w,u 0 ) =,d T (w, 0 ) = 3 If w is not a pendant ertex, then w is on a path which joins u 0 and a pendant ertex, say r Denote the unique neighbor of r by r Let T = T rr +ru 0 if r V 1 and T = T rr +r 0 otherwise It is routine to check that T Tn p,q \{D(p,q)} By Theorem, perl(t ) < perl(t), a contradiction to the choice of T Hence, w must be a pendant ertex In what follows, we show that for all PV(T) \ {w}, either u 0 E T or 0 E T In fact, if there exist a ertex, say y, in PV(T)\{w} such that yu 0,y 0 E T Denote the unique neighbor of y by y Let ˆT = T yy + yu 0 if y V 1 and ˆT = T yy +y 0 otherwise It is straightforward to check that ˆT T p,q n \{D(p,q)} By Theorem, perl(ˆt) < perl(t), a contradiction to the choice of T By Claims 1 and, we obtain T = D (p 1,q 1), or T = D (p 1,q 1) If p = q, then D (p 1,q 1) = D (p 1,q 1) Together with Lemma 31, (ii) holds

13 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 39 obiously in this case If p < q, then combining with Lemma 31 and Inequality (31), (ii) follows immediately Remark 34 We know from [13] that D(p,q) D (p 1,q 1) T for all T Tn p,q \{D(p,q),D (p 1,q 1)} In iew of Theorems 3 and 33, D(p,q) (resp, D (p 1,q 1)) is the tree with a (p,q) bipartition which has the smallest (resp, second smallest) Laplacian permanent Hence, our result support the conjecture that trees minimizing the Laplacian permanent usually simultaneously minimize the Laplacian coefficients, and ice ersa Furthermore, in iew of Theorems 3 and 33 it is natural to conjecture that D(p,q) D (p 1,q 1) D (p 1,q 1) T for all T Tn p,q \{D(p,q),D (p 1,q 1),D (p 1,q 1)} with q > p 3 Laplacian permanent of trees with diameter at least d Let Q n be the matrix obtained from L(P n+1 ) by eliminating row 1 and column 1 It is routine to check that perq 1 = 1 and perq = 3 In particular, define perq 0 = 1 We know [1] that (33) perq n = 1 (1+ ) n + 1 (1 ) n and (34) perl(p n ) = (1+ ) n + + (1 ) n Lemma 35 ([1]) Let n,j and k be positie integers with 1 k < j 1 (n+1) Then ( 1) k (perq j 1 perq n j perq k 1 perq n k ) > 0 Lemma 36 ([5]) Let u be the only non-pendant edge incidence with in a tree T and let A = N T ()\{u} Let T = T {z : z A}+{uz : z A}, then we hae perl(t ) < perl(t) In this subsection, we use a new method to proe the following known result Theorem 37 ([1]) Let d be a positie integer, and let T be a tree with n ertices haing diameter at least d Then ( ) perl(t) n d+ (1+ ( ) ) d 1 + n d (1 ) d 1 The equality holds if and only if T = T n,d, ; see Fig 11 Proof Choose an n-ertex tree T of diameter at least d such that its Laplacian permanent is as small as possible If T = P d+1, then our result holds by Theorem 11 Hence, in what follows we consider that T = Pd+1 If T contains just two pendant

14 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang ertices, ie, T = P n = 1 i n Let T = T 1 + i 1 Obiously, T is of diameter at least d By Theorem, we hae perl(t ) < perl(t), a contradiction to the choice of T Hence, T contains at least 3 pendant ertices That is to say, the maximal ertex degree in T is of at least 3 Without loss of generality, we may assume that w is just of the maximal degree ertex Let P = 1 i l+1 be one of the longest paths contained in T, where l d In order to complete the proof, it suffices to show the following claims Claim 1 T = T n,l,i, where T n,l,i is obtained from P by inserting n l 1 pendant ertices at i, i {,3,, (l+)/ } Proof of Claim 1 First we show that all the pendant ertices excluding the endertices of P are adjacent to w Assume to the contrary that PV(T)\{ 1, l+1 } satisfying w E T Denote the unique neighbor of by Set T = T +w It is straightforward to check that T is of an n-ertex tree of diameter at least d By Theorem, we hae perl(t ) < perl(t), a contradiction to the choice of T Now we show that w is on the path P Assume that w is not on the path P, then T must be the tree obtained by joining the center of a star S and a ertex of P by a path of length at least 1 Denote the unique neighbor of w which is not a pendant ertex by w Set A = N T (w)\{w } Let T = T {wz : z A}+{w z : z A} It is easytoseethatt isatreeofdiameteratleastd ByLemma36, perl(t ) < perl(t), a contradiction Claim In the tree T n,l,i described as aboe, we hae l = d,i =, ie, T n,l,i = T n,d, Proof of Claim If not, then l d + 1 In the tree described aboe, let T = T n,l,i l+1 l + i l+1 It is easy to see that T is an n-ertex tree of diameter at least d By Theorem, we hae perl(t ) < perl(t n,l,i ), a contradiction to the choice of T n,l,i So we obtain T = T n,d,i Expanding the permanent of L(T n,d,i ) along the row corresponding to ertex i gies perl(t n,d,i ) = perl(p d+1 )+(n d 1)perQ i 1 perq d i+1 This gies perl(t n,d,j ) perl(t n,d, ) = (n d 1)(perQ j 1perQ d j+1 perq 1perQ d +1 ) > 0 for j = 3,4,, 1 (d+), and the last inequality follows by Lemma 35

15 Electronic Journal of Linear Algebra ISSN Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 41 In iew of (33) and (34), we hae ( ) (35) perl(t n,d, ) = n d+ (1+ ( ) ) d 1 + n d (1 ) d 1 By Claims 1 and, and Eq (35), Theorem 37 follows immediately 33 Lower bounds for the Laplacian permanent of graphs in Un p,q In this subsection, we are to determine sharp lower bounds for the Laplacian permanent of graphs in Un p,q Let C 4(1 s1 k 1, 1 s k, 1 s3 k 3, 1 s4 k 4 ) be the graph obtained from C 4 = by inserting s i pendant ertices at i and joining i to the center of a star S ki by an edge, i = 1,,3,4; see Fig 33 In particular, denote by B(p, q) = C 4 (1 p 0, 1 q 0, 1 0 0, 1 0 0) k 4 1 s 4 4 s 3 3 k 3 1 k s 1 s k 1 Fig 33 Graph C 4 (1 s 1k 1,1 s k,1 s 3k 3,1 s 4k 4 ) Theorem 38 For any G Un p,q, one has perl(g) 0(p 1)(q 1) + 4n The equality holds if and only if G = B(p,q) Proof ChooseG Un p,q such that its Laplacian permanent is as small as possible Ifn = 4, 5, U p,q 4 = {B(,)}and U p,q 5 = {B(, 3)}, our result holds obiously Hence in what follows we consider n 6 Let C r be the unique cycle contained in G Note that, by Theorem 6, G = Cn Hence, PV(G) Let V 1, V be the bipartition of V G such that V 1 = p and V = q with 0 (resp, u 0 ) being of the maximal degree ertex among V 1 (resp, V ) in G Claim 1 For all u PV(G), either uu 0 E G or u 0 E G Proof of Claim 1 If not, then there exists a pendant ertex, say u, such that u is not in N G (u 0 ) N G ( 0 ) Denote the unique neighbor of u by u Using Operation I, let G = G uu +uu 0 if u V 1 and G = G uu +u 0 otherwise It is routine to check that G Un p,q By Theorem, perl(g ) < perl(g), a contradiction to the choice of G Letd(u,C r ) = min{d(u,) : V Cr } Inparticular,ifuisonC r,thend(u,c r ) =

16 Electronic Journal of Linear Algebra ISSN Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang 0 Claim d( 0,C r ) = d(u 0,C r ) = 0 Proof of Claim Here we only show that d( 0,C r ) = 0 by contradiction With the same method, we can also show d(u 0,C r ) = 0 Assume that d( 0,C r ) = t 1 Set A = PV(G) N G ( 0 ) By Claim 1, we hae A Let P t = 0 w 1 w w t 1 w t be the shortest path connecting 0 and the cycle C r, where w t is on C r Let u V Cr N G (w t ) Using Operation II, let { G {z0 : z A}+{zu : z A}, if t = 1; Ḡ = G {z 0 : z A}+{zw : z A}, if t It is routine to check that Ḡ U n p,q By Theorem 4, we hae perl(ḡ) < perl(g), a contradiction to the choice of G Claim 3 r 6 Proof of Claim 3 If not, then r 8 By the structure of G described as aboe, then there must exist four consecutie ertices, say u k1, k1,u k, k on the cycle C r such that u 0, 0 {u k1, k1,u k, k } Without loss of generality assume that k1, k V 1 and u k1,u k V Using Operation III, let G 0 = G u k1 k1 +u k1 k It is routine to check that G 0 Un p,q By Theorem 6 perl(g 0) < perl(g), a contradiction to the choice of G Hence, by Claims 1 3, we obtain r = 4, then G = B(p,q) r = 6, then G = G 1 or G, where G 1,G are depicted in Fig 34 p 3 u 0 0 q 3 q 3 0 p 3 u 0 G 1 G Fig 34 Graphs G 1 and G If G = G, using Operation III on G, we obtain graph C 4 (1 0,1 q 3 0,1 p 3 0,1 0 0) which is in Un p,q By Theorem 6, we hae perl(c 4(1 0,1 q 3 0,1 p 3 0,1 0 0)) < perl(g), a contradiction to the choice of G So G = G If G = G 1, by a simple calculation, we get (36) perl(g 1 ) = 100(p )(q )+40n 140, perl(b(p,q)) = 0(p 1)(q 1)+4n

17 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 43 Note that p+q = n with 3 p q n 3, hence (37) pq 3(n 3) Hence, perl(g 1 ) perl(b(p,q)) = 80pq 144n (n 3) 144n+40 (by (37)) = 96n 480 > 0 Therefore, perl(g 1 ) > perl(b(p,q)) Thus, we obtain G = B(p,q) Together with Eq (36), we complete the proof n 6 n 6 n 7 n 6 n 5 n 7 n 6 Ĝ 1 Ĝ Ĝ 3 Ĝ 4 Ĝ 5 Ĝ 6 Ĝ 7 Fig 35 Graphs Ĝ1,Ĝ,,Ĝ7 Next we are to identify the graph in Un p,q Laplacian permanent with the second (resp, third) smallest Theorem 39 Among U p,q n (i) If p =, the ordering of all the members in Un,n with n 4 is as follows: perl(b(,n )) < perl(c 4 (1 1 0,1 0 0,1 n 5 0,1 0 0)) < < perl(c 4 (1 i 0,1 0 0, 1 n 4 i 0,1 0 0)) < < perl(c 4 (1 n 4 0,1 0 0,1 n 4 0,1 0 0)) (ii) If p = 3, then perl(b(3,n 3)) < perl(ĝ1) < perl(ĝ) < perl(g) for all G Un 3,n 3 \{B(3,n 3),Ĝ1,Ĝ} with n 0, where Ĝ1,Ĝ are depicted in Fig 35 (iii) If p 4, (a) for all G Un p,q \{B(p,q)} with n 8, one has perl(g) 36(p )(q 1) + 4p + 8q 4 with equality if and only if G = C 4 (1 q 0, 1 p 3 0,1 0 0,1 1 0) (b) for all G U p,q n \{B(p,q),C 4 (1 q 0,1 p 3 0,1 0 0,1 1 0)} with q > p, n 9, one has perl(g) 36(q )(p 1)+4q+8p 4 with equality if and only if G = C 4 (1 q 3 0,1 p 0,1 1 0,1 0 0) Proof (i) If p =, then U,n n = { C 4 (1 i 0,1 0 0,1 n 4 i 0,1 0 0 ) : 0 i n 4 }

18 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang By a simple calculation, we get ( perl(c 4 (1 i 0,1 0 0,1 n 4 i 0,1 0 0 )) = 16 i n 4 ) +4(n 1) Consider the function f(x) = 16(x n 4 ) +4(n 1) in x with 0 x n 4 By the monotonicity of f(x), we hae ( ) n 4 (38) f(0) < f(1) < < f(i) < < f Note that C 4 (1 0 0,1 0 0,1 n 4 0,1 0 0) = B(,n ), hence (i) follows immediately from (38) (ii) Note that p = 3, hence the cycle C r contained in T Un 3,n 3 is of length at most 6, ie, r 6 If r = 4, the bipartite unicyclic graph, say U (resp, U ), in Un 3,n 3 with the second (resp, third) smallest Laplacian permanent should satisfy the following property: Apply Operation I (or II) to U (resp, U ) only once to get the graph B(3,n 3) (resp, B(3,n 3) or U) Hence, U,U {Ĝ1,Ĝ,Ĝ3,Ĝ4,Ĝ5,Ĝ6,Ĝ7,Ĝ8}, where Ĝ1,Ĝ,,Ĝ8 are depicted in Fig 35 If r = 6, then graph G 1 (see Fig 34) is the possible graph with the smallest Laplacian permanent By direct calculation, we hae perl(ĝ1) = 7n 76, perl(ĝ) = 76n 35, perl(ĝ3) = 168n 804, perl(ĝ4) = 11n 516, perl(ĝ5) = 96n 40, perl(ĝ6) = 10n 580, perl(ĝ7) = 16n 1140, perl(g 1 ) = 140n 640 Based on the aboe direct computing, (ii) follows immediately (iii) We first determine the graph, say G, in Un p,q with the second smallest Laplacian permanent for p 4 In iew of the proof of Theorem 38, it is easy to see that the cycle C r contained in G is of length at most 6 Furthermore, if r = 6, only G 1 as depicted in Fig 34 is possible to be the particular graph G If r = 4, in iew of Theorems and 4, we know that B(p,q) can be obtained from G by Operation I (or, II) once Hence, based on Operation I, G may be C 4 (1 q 0, 1 p 3 0,1 0 0,1 1 0) or C 4 (1 q 3 0,1 p 0,1 1 0,1 0 0); whereas based on Operation II, G may be in the set or A = {C 4 (1 q 3 (t+1),1 p t 0,1 0 0,1 0 0) : 1 t p } B = {C 4 (1 q t 0,1 p 3 (t+1),1 0 0,1 0 0) : 1 t q } Combining with Operation I we see that A (resp, B) contains just two members C 4 (1 q 3,1 p 3 0,1 0 0,1 0 0) and C 4 (1 q 3 (p 1),1 0 0,1 0 0,1 0 0) (resp, C 4 (1 q 3 0,1 p 3,

19 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs ,1 0 0) and C 4 (1 0 0,1 p 3 (q 1),1 0 0,1 0 0)) Hence, summarizing the discussion as aboe we get that G must be in U = {C 4 (1 q 0,1 p 3 0,1 0 0,1 1 0),C 4 (1 q 3 0,1 p 0, 1 1 0,1 0 0),C 4 (1 q 3,1 p 3 0,1 0 0,1 0 0),C 4 (1 q 3 0,1 p 3,1 0 0,1 0 0), C 4 (1 0 0,1 p 3 (q 1), 1 0 0,1 0 0),C 4 (1 q 3 (p 1),1 0 0,1 0 0,1 0 0),G 1 } By direct calculation, we obtain perl(c 4 (1 q 0,1 p 3 0,1 0 0,1 1 0)) = 36pq 3n 3q +68, perl(c 4 (1 q 3 0,1 p 0,1 1 0,1 0 0)) = 36pq 3n 3p+68, perl(c 4 (1 q 3,1 p 3 0,1 0 0,1 0 0)) = 60pq 68n 40q +144, perl(c 4 (1 q 3 0,1 p 3,1 0 0,1 0 0)) = 60pq 68n 40p+144, perl(c 4 (1 q 3 (p 1),1 0 0,1 0 0,1 0 0)) = 48pq 7n+4p+84, perl(c 4 (1 0 0,1 p 3 (q 1),1 0 0,1 0 0)) = 48pq 7n+4q +84 This gies (39) perl(c 4 (1 q 0,1 p 3 0,1 0 0,1 1 0)) < perl(c 4 (1 q 3 0,1 p 0,1 1 0,1 0 0)) < perl(ĝ) for all Ĝ U \{C 4 (1 q 0,1 p 3 0,1 0 0,1 1 0),C 4 (1 q 3 0,1 p 0,1 1 0,1 0 0)} for q > p 4 This completes the proof of the first part of (iii) Now we show the second part of (iii) By a similar discussion as in the proof of the first part of (iii), we know that the graph, say G, in Un p,q haing the third smallest Laplacian permanent is either the graph with the second smallest Laplacian permanent in U, or apply Operation I (or II) once to G to obtain the graph C 4 (1 q 0,1 p 3 0,1 0 0,1 1 0), which has the second smallest Laplacian permanent in Un p,q Hence, together with (39), we obtain that G is in the set U = {C 4 (1 q 3 0, 1 p 0, 1 1 0,1 0 0), C 4 (1 q 3,1 p 3 0,1 0 0,1 0 0), C 4 (1 q 3 0,1 p 4, 1 0 0,1 1 0), C 4 (1 q 3, 1 p 4 0, 1 0 0, 1 1 0), C 4 (1 q 3 0, 1 p 3 0, 1 0 0, 1 0 ), C 4 (1 q 3 0, 1 p 3 0, 1 1 0, 1 1 0), C 4 (1 0 0, 1 p 4 (q 1),1 0 0,1 1 0), C 4 (1 q 3 (p ),1 0 0,1 0 0,1 1 0), C 4 (1 0 0,1 p 3 0,1 0 0,1 0 (q 1))} By direct calculation, we hae perl(c 4 (1 q 3,1 p 3 0,1 0 0,1 0 0)) = 60pq 68n 40q+144, perl(c 4 (1 q 3 0,1 p 4,1 0 0,1 1 0)) = 108pq 4q 04n+464, perl(c 4 (1 q 3,1 p 4 0,1 0 0,1 1 0)) = 108pq 168q 13n+400, perl(c 4 (1 q 3 0,1 p 3 0,1 0 0,1 0 )) = 9pq +8q 17n+336, perl(c 4 (1 q 3 0,1 p 3 0,1 1 0,1 1 0)) = 68pq 18n+60, perl(c 4 (1 0 0,1 p 4 (q 1),1 0 0,1 1 0)) = 80pq 40q 10n+60, perl(c 4 (1 q 3 (p ),1 0 0,1 0 0,1 1 0)) = 88pq 10q 100n+7, perl(c 4 (1 0 0,1 p 3 0,1 0 0,1 0 (q 1))) = 64pq +8p 96n+13

20 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang Based on the aboe direct computing, the second part of (iii) follows immediately Remark 310 In iew of Theorems 38 and 39, we hope to show that, among the set of all n-ertex unicyclic graphs with a (p,q)-bipartition(q > p 4), B(p,q) C 4 (1 q 0,1 p 3 0,1 0 0,1 1 0) C 4 (1 q 3 0,1 p 0,1 1 0,1 0 0) G for all G Un p,q \ {B(p,q),C 4 (1 q 0,1 p 3 0,1 0 0,1 1 0),C 4 (1 q 3 0,1 p 0,1 1 0,1 0 0)} in the future research If this is true, it will support the relationship between the Laplacian coefficients and the Laplacian permanent of n-ertex bipartite unicyclic graphs with a (p, q)- bipartition To conclude this subsection, we determine the first, second, third smallest Laplacian permanent of graphs in U n, the set of all bipartite unicyclic graphs on n ertices Theorem 311 Among U n with n 4, (i) for all G U n, we hae perl(g) 4n 60 with equality if and only if G = B(,n ) (ii) for all G U n \{B(,n )} with n 6, we hae perl(g) 40n 140 with equality if and only if G = C 4 (1 1 0,1 0 0,1 n 5 0,1 0 0) (iii) for all G U n \ {B(,n ),C 4 (1 1 0,1 0 0,1 n 5 0,1 0 0)} with n 6, we hae perl(g) 44n 160 with equality if and only if G = B(3,n 3) Proof It is routine to see that U n = Un,n Un 3,n 3 U n, n n Note that for all G Un p,q, by Theorem 38 one has perl(g) perl(b(p,q)) = 0(p 1)(q 1)+4n, with the equality if and only if G = B(p,q) Consider the function f(x) = 0(x 1)(n x 1)+4n in x with x n It is routine to check that f (x) = 0(n x) > 0(n x (n x)) = 0 Hence, f(x) is an increasing function for x n That is to say, f() < f(3) < < f ( n ), which implies (i) immediately Based on Theorems 38 and 39, and the proof in (i) as aboe, in order to determine the graph in U n haing the second minimal Laplacian permanent, it suffices to compare the alues between perl(c 4 (1 1 0,1 0 0,1 n 5 0,1 0 0)) and perl(b(3,n 3)) By an elementary calculation, we hae (310) perl(c 4 (1 1 0,1 0 0,1 n 5 0,1 0 0)) = 40n 140, perl(b(3,n 3)) = 44n 160 It is routine to check that perl(c 4 (1 1 0,1 0 0,1 n 5 0,1 0 0)) < perl(b(3,n 3)) Hence, (ii) holds immediately Similarly, in order to determine the third minimal Laplacian permanent among U n, it suffices to compare the alues between perl(c 4 (1 0,1 0 0,1 n 6 0,1 0 0)) and

21 Volume 6, pp 8-48, January Edge-Grafting Theorems on Permanents of the Laplacian Matrices of Graphs 47 perl(b(3,n 3)) Note that if n = 6 (resp, 7), it is straightforward to check that C 4 (1 0,1 0 0,1 n 6 0,1 0 0) does not exist and B(3,n 3) is the graph with the third minimal Laplacian permanent among U n For n 8, by direct calculation, we hae (311) perl(c 4 (1 0,1 0 0,1 n 6 0,1 0 0)) = 56n 5 In iew of the second equation in (310) and (311), it is routine to check that perl(c 4 (1 0,1 0 0,1 n 6 0,1 0 0)) > perl(b(3,n 3)) = 44n 160 Hence, (iii) holds immediately Acknowledgment This work is financially supported by the National Natural Science Foundation of China (no , no ) and the Special Fund for Basic Scientific Research of Central Colleges (no CCNU11A0015) The authors would like to express their sincere gratitude to the referee for a ery careful reading of the paper and for all of his or her insightful comments and aluable suggestions, which led to a number of improements in this paper REFERENCES [1] RA Brualdi and JL Goldwasser Permanent of the Laplacian matrix of trees and bipartite graphs Discrete Math, 48:1 1, 1984 [] D Cetkoić and S Simić Towards a spectral theory of graphs based on the signless Laplacian, I Publ Inst Math (Beograd), 85(99):19 33, 009 [3] D Cetkoić and S Simić Towards a spectral theory of graphs based on the signless Laplacian, II Linear Algebra Appl, 43:57 7, 010 [4] D Cetkoić and S Simić Towards a spectral theory of graphs based on the signless Laplacian, III Appl Anal Discrete Math, 4: , 010 [5] XY Geng, X Hu, and SC Li Further results on permanental bounds for the Laplacian matrix of trees Linear Multilinear Algebra, 58: , 010 [6] SS He and SC Li Ordering of trees with fixed matching number by the Laplacian coefficients Linear Algebra Appl, 435: , 011 [7] A Ilić On the ordering of trees by the Laplacian coefficients Linear Algebra Appl, 431:03 1, 009 [8] A Ilić Trees with minimal Laplacian coefficients Comput Math Appl, 59: , 010 [9] A Ilić and M Ilić Laplacian coefficients of trees with gien number of leaes or ertices of degree two Linear Algebra Appl, 431:195 0, 009 [10] A Ilić, A Ilić, and D Steanoić On the Wiener index and Laplacian coefficients of graphs with gien diameter or radius MATCH Commun Math Comput Chem, 63:91 100, 010 [11] SC Li and L Zhang Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs Linear Multilinear Algebra, 59: , 011 [1] SC Li and L Zhang Permanental bounds for the signless Laplacian matrix of a unicyclic graph with diameter d Graphs Combin, 8: , 01 [13] WQ Lin and WG Yan Laplacian coefficients of trees with a gien bipartition Linear Algebra Appl, 435:15 16, 011 [14] R Merris The Laplacian permanental polynomial for trees Czechosloak Math J, 3(107): , 198

22 Volume 6, pp 8-48, January SC Li, Y Li, and X Zhang [15] R Merris, KR Rebman, and WWatkins Permanent polynomials of graphs Linear Algebra Appl, 38:73 88, 1981 [16] H Minc Permanents Addison-Wesley, Reading, MA, 1978 [17] B Mohar On the Laplacian coefficients of acyclic graphs Linear Algebra Appl, 4: , 007 [18] D Steanoić and A Ilić On the Laplacian coefficients of unicyclic graphs Linear Algebra Appl, 430:90 300, 009 [19] FZ Zhang An analytic approach to a permanent conjecture Linear Algebra Appl, 438: , 013 [0] XD Zhang, XP L, and YH Chen Ordering trees by the Laplacian coefficients Linear Algebra Appl, 431:414 44, 009 [1] B Zhou and I Gutman A connection between ordinary and Laplacian spectra of bipartite graphs Linear Mutilinear Algebra, 56: , 008