Eigenvalue location for chain graphs

Linear Algebra and its Applications 505 (2016) 194–210

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Eigenvalue location for chain graphs Abdullah Alazemi a,∗ , Milica Anđelić a , Slobodan K. Simić b,c a b c

Department of Mathematics, Kuwait University, Safat 13060, Kuwait State University of Novi Pazar, Vuka Karadžića bb, 36 300 Novi Pazar, Serbia Mathematical Institute SANU, Kneza Mihaila 36, 11 000 Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 7 October 2015 Accepted 19 April 2016 Available online xxxx Submitted by R. Brualdi MSC: 05C50 Keywords: Chain graph Double nested graph Least positive eigenvalue Threshold graph Nested split graph

a b s t r a c t Chain graphs (also called double nested graphs) play an important role in the spectral graph theory since every connected bipartite graph of fixed order and size with maximal largest eigenvalue is a chain graph. In this paper, for a given chain graph G, we present an algorithmic procedure for obtaining a diagonal matrix congruent to A +xI, where A is the adjacency matrix of G and x any real number. Using this procedure we show that any chain graph has its least positive eigenvalue greater than 12 , and also prove that this bound is best possible. A similar procedure for threshold graphs (also called nested split graphs) is outlined. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Let G = (V (G), E(G)) be a simple graph, i.e. an undirected graph without loops or multiple edges. V (G) is its vertex set and ν = |V (G)| its order, while E(G) is its edge set and  = |E(G)| its size. A = A(G) is the (0, 1)-matrix with (ij)-th entry equal to 1 * Corresponding author. E-mail addresses: [email protected] (A. Alazemi), [email protected] (M. Anđelić), [email protected] (S.K. Simić). http://dx.doi.org/10.1016/j.laa.2016.04.030 0024-3795/© 2016 Elsevier Inc. All rights reserved.

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whenever vertices i and j are adjacent. det(xI − A) is the characteristic polynomial of G, and its zeros are the eigenvalues of G. Since A is real and symmetric, the eigenvalues of G are real, and then we assume that λ1 (G) ≥ · · · ≥ λν (G). In this paper we deal with chain graphs, which are {2K2 , C3 , C5 }-free graphs, and in spectral graph theory they feature as graphs whose largest eigenvalue within the connected bipartite graphs of fixed order and size is maximal (or least eigenvalue minimal – see [3,4] for more details). This important property of chain graphs was discovered independently in 2008 by Bhattacharya et al. in [5], and also by Bell et al. in [4] and named therein as double nested graphs (or DNGs for short). For more details on these graphs the reader is referred to [1], where not only chain graphs are considered, but also threshold graphs named therein as nested split graphs (or NSGs for short). Recall, that threshold graphs are {2K2 , P4 , C4 }-free graphs. In the foregoing text (Sections 2–4) we will prefer to address the above classes of graphs as DNGs and NSGs. Then, if not told otherwise, we will ignore their isolated vertices, or equivalently, we will assume that they are connected. Some authors (see, for example, [9]) define NSGs by using binary sequences (b1 , . . . , bν ), or equivalently, they toss a (binary) coin in order to generate a binary sequence, and hence a NSG (here bi is the outcome of i-th tossing). The generating procedure is as follows: • i = 1 (first tossing): We start with an isolated vertex. • i > 1 (i-th tossing): If bi = 0 we add a new isolated vertex to the graph obtained so far, while otherwise, if bi = 1 we add a new vertex which is a dominating vertex (that is adjacent to all vertices of the graph obtained so far). Note: If bν = 1 then the graph being generated is connected. Following [4], we now give the structure of a (connected) DNG, and also introduce a notation to be used further on. The vertex set of any DNG, say G, consists of two colour classes (or co-cliques). Both of them are partitioned into h non-empty cells U1 , . . . , Uh and V1 , . . . , Vh , respectively; all vertices in Us are joined by (cross) edges to all vertices h+1−s in k=1 Vk for s = 1, . . . , h. Denote by NG (w) the set of neighbours of a vertex w. Hence, if u ∈ Us+1 and u ∈ Us (v  ∈ Vt+1 and v  ∈ Vt ) then NG (u ) ⊂ NG (u ) (resp. NG (v  ) ⊂ NG (v  )), see Fig. 1.1. If ms = |Us | and ns = |Vs | for s = 1, . . . , h, then G is denoted by DN G(m1 , . . . , mh ; n1 , . . . , nh ). In [1] it was noted that any NSG (see also Section 4) can be obtained from a DNG by adding all edges to one co-clique or, in the other direction, by deleting edges from one clique. This observation makes it possible to define DNGs by binary sequences (as NSGs), but with somewhat modified generation procedure after tossing a coin. Namely we have: • i = 1 (first tossing): We start with an isolated vertex (say white).

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Fig. 1.1. The structure of a connected DNG.

• i > 1 (i-th tossing): If bi = bi−1 and the last added vertex was white we add an isolated white vertex to the graph obtained so far, while otherwise, if the last added vertex was black we add a black vertex adjacent to all white vertices in the graph obtained so far. If bi = bi−1 and the last added vertex was white we add a black vertex adjacent to all white vertices in the graph obtained so far, while otherwise, if the last added vertex was black we add an isolated white vertex to the graph obtained so far. Note: By choosing after the first tossing that the added vertex is white we have also avoided to have isolated vertices coloured in black; if the last added vertex is black then we obtain a connected DNG. If 0 · · · 0 1 · · · 1 0 · · · 0 1 · · · 1 · · · 0 · · · 0 1 · · · 1 (ti , si = 0) t1

s1

t2

s2

th

sh

is the generating binary sequence of G, then G = DN G(t1 , . . . , th ; sh , . . . , s1 ). Note that vertices within each Ui and each Vj have constant degrees, which strictly decrease if i, or j, increases. These facts suggest a vertex ordering so that white vertices (from the Ui ’s) precede black vertices (from Vj ’s) and are ordered by non-increasing degrees within each colour class. If so then the adjacency matrix has two off-diagonal blocks, say H and H T (diagonal blocks are zero matrices), so that the matrix H which is a block in the first row and second column of A has a step-wise form (see Fig. 2.1). This canonical matrix can be also used for defining DNGs. Therefore a graph is a DNG whenever its adjacency matrix admits the above canonical matrix as an adjacency matrix. For any threshold graph G with adjacency matrix A and any real x an algorithmic procedure for finding a diagonal matrix congruent to A + xI is established (see [8]). The

A. Alazemi et al. / Linear Algebra and its Applications 505 (2016) 194–210

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Jn1 ×m1 ⎜ ⎜ Jn2 ×m1 ⎜ .. ⎜ ⎜ . ⎜ ⎝J

nh−1 ×m1

Jm1 ×n1 Jm2 ×n1 .. . Jmh−1 ×n1 Jmh ×n1

0 Jn1 ×m2 Jn2 ×m2

··· ··· . ..

Jn1 ×mh−1 Jn2 ×mh−1

Jm1 ×n2 Jm2 ×n2

··· ··· . ..

Jmh−1 ×n2

Jn1 ×mh

0

Jnh−1 ×m2

Jm1 ×nh−1 Jm2 ×nh−1

197

Jm1 ×nh

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Jnh ×m1 Fig. 2.1. An adjacency matrix of a connected DNG.

authors also presented several applications of this algorithm mainly for the eigenvalue location problem. In this paper, for any DNG, say G, with adjacency matrix A and any real x, we offer an algorithmic procedure for finding a diagonal matrix congruent to A + xI. We also show that this procedure can be reduced to diagonalization of the symmetric, tridiagonal matrix of order h. We apply this algorithm to prove that there is no DNG with eigenvalues in intervals (0, 12 ] and [− 12 , 0). This also proves that the least positive eigenvalue of any DNG is greater than 12 . In addition, we construct an infinite sequence of DNGs whose least positive eigenvalue approaches 12 from above as ν approaches infinity. It is worth mentioning that our diagonalization procedure for finding a diagonal matrix congruent to the adjacency matrix of DNG can be also applied to NSGs. For the latter class of graphs the diagonalization process can be also reduced to the diagonalization of the symmetric, tridiagonal matrix of order h, where h is the number of 0-subsequences (or 1-subsequences) in the binary sequence that generates a (connected) threshold graph. The rest of the paper is organized as follows: in Section 2 we present our diagonalization procedure; in Section 3 we provide some applications which concern the least positive (or largest negative) eigenvalue of DNGs; in Section 4 we draw some attention on NSGs. 2. A diagonalization of A(G) + xI if G is a DNG

i Let A be the adjacency matrix of G = DN G(m1 , . . . , mh ; n1 , . . . , nh ), Ni = k=1 nk ,

i Mi = k=1 mk . We assume that the matrix A is represented in the form of its canonical matrix (see also Fig. 2.1), or shortly as  A=

O H HT O

 ,

where H = (Hij ) for 1 ≤ i, j ≤ h is the block matrix defined by

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 Hij =

Jmi ×nj

if i + j ≤ h + 1,

Omi ×nj

otherwise.

Jmi ×nj is the mi × nj all 1-matrix, and Omi ×nj is the mi × nj 0-matrix. We say that two (square) matrices X and Y are congruent if there exists a nonsingular matrix P such that Y = P T XP . In addition, it is well known that two symmetric congruent matrices have the same inertia (i.e., the same number of positive, negative and zero eigenvalues) – the Sylvester’s law of inertia. Namely, any such matrix can be reduced to a diagonal matrix by a sequence of elementary row and the same column operations. Here this fact will be applied in order to find diagonal matrix Dx which is congruent to the matrix A + xI, where A = A(G) is the adjacency matrix of a DNG G and x is a real number. The process of diagonalization will be divided into four stages. Stage 1 If m1 = 1 then the first block is processed. If m1 > 1 then the first m1 rows (columns) in H (resp. H T ) are pairwise equal. Then we perform the following row and column operations (Rm1 , Cm1 ) ← (Rm1 − Rm1 −1 , Cm1 − Cm1 −1 ), yielding the rows Rm1 , Rm1 −1 :  0 0

··· ···

0 x 0 −x

−x 0 · · · 2x 0 · · ·

0 0

··· ···

1 ··· 0 ···

 1 0

.

Next we perform 1 1 (Rm1 −1 , Cm1 −1 ) ← (Rm1 −1 + Rm1 , Cm1 −1 + Cm1 ) 2 2 and obtain



0 ··· 0 ···

0 0

1 2x

0

0 0 ··· 2x 0 · · ·

0 0

··· ···

1 ··· 0 ···

1 0

 .

Then we put focus on Rm1 −2 and Rm1 −1 , as well as on Cm1 −2 and Cm1 −1 (of the new matrix), and perform (Rm1 −1 , Cm1 −1 ) ← (Rm1 −1 − Rm1 −2 , Cm1 −1 − Cm1 −2 ) and

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2 2 (Rm1 −2 , Cm1 −2 ) ← (Rm1 −2 + Rm1 −1 , Cm1 −2 + Cm1 −1 ), 3 3 yielding Rm1 −2 and Rm1 −1 equal to 

0 ··· 0 ···

0 0

1 3x

0

0 0 ··· 3 2x 0 · · ·

0 0

··· ···

1 ··· 0 ···

 1 0

.

We continue until we annihilate the ones in the first m1 rows and columns except the first row and column. Then the first m1 diagonal entries become x m1 x (m1 − 1)x 3x 2x , , ,..., , . m1 m1 − 1 m1 − 2 2 1 We repeat the same procedure to all pairwise equal rows and columns in the range of Mi−1 + 1 to Mi−1 + mi , for i = 2, . . . , h and obtain the matrix congruent to A + xI with R1 , RM1 +1 , . . . , RMh−1 +1 of the following form x RMi−1 +1 = (0, . . . , 0, , 0, . . . , 0, 1, . . . , 1, 0, . . . , 0 )    mi          Mh −Mi−1 −1 Nh+1−i

Mi−1

(i = 1, . . . , h).

Nh −Nh+1−i

The remaining rows if any for some i contain all entries equal to zero except the diagonal ones. For each mi > 1 we obtain mi − 1 diagonal entries equal to 3x 2x mi x (mi − 1)x , ,..., , . mi − 1 mi − 2 2 1 Next by permuting the rows and columns we move rows R1 , RM1 +1 , . . . , RMh−1 +1 to be in the last h positions among the first Mh rows. At the end of Stage 1, the partially diagonalized matrix has the following form ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

D1 ..

. Dh x m1

..

.

BT

B x mh

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠

xINh

where Di = diag( and

mi x 2x ,..., ) mi − 1 1

(i = 1, . . . , h)

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⎛

1···1 ⎜1···1 ⎜ B=⎜ ⎝ 1···1

1···1 1···1

1···1

⎞ ⎟ ⎟ ⎟ ⎠

is h × Nh matrix with exactly Nh+1−i ones in the i-th row. Stage 2 We now put focus on the (h + Nh ) × (h + Nh ) matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

x m1

⎞ ..

.

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

B x mh

x ..

BT

. x

The first n1 columns of B are pairwise equal, as well as next n2 and so on up to nh . We perform similar row and column operations as in Stage 1 in order to remove all ones from pairwise equal columns of B except the first one in the sequence. For any i, the remaining ni − 1 columns if any have all entries equal to zero except the diagonal ones which are equal to ni x (ni − 1)x 3x 2x , ,..., , . ni − 1 ni − 2 2 1 Again, we may permute the rows and columns by pushing to the end rows and columns with ones. Stage 3 In this stage we consider only the following 2h × 2h matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

x m1

1 .. . 1

..

.

··· . ..

x mh

1

1 .. . 1 x n1

··· . ..

..

1

.

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

x nh

We start from Rh+1 and Rh+2 . By performing (Rh+1 , Ch+1 ) ← (Rh+1 − Rh+2 , Ch+1 − Ch+2 ) most of ones will be removed from Rh+1 and Ch+1 .

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We repeat the same argument to the remaining columns. At the end we obtain 2h × 2h matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

x m1

⎞

1 ..

..

. x mh

1 ( nx1 +

1

..

x n2 )

− nx2

.

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

− nx2 .. . .. .

..

.

..

.

x − nh−1 x ( nh−1 + nxh ) − nxh x − nxh nh

x − nh−1

1

(2.1)

Stage 4 If x = 0 then we have: Lemma 2.1. Let G = DN G(m1 , . . . , mh ; n1 , . . . , nh ) and A its adjacency matrix. Then A is congruent to a matrix diag(0, . . . , 0, 1, . . . , 1, −1, . . . , −1).          ν−2h

h

h



 Ih O . The remaining O −Ih ν − 2h diagonal entries have been processed in Stages 1 and 2 and for x = 0 all of them are equal to 0. This completes the proof. 2 Proof. If x = 0 then the matrix given in (2.1) is congruent to

Hence DN G(m1 , . . . , mh ; n1 , . . . , nh ) has h negative and h positive eigenvalues. Therefore mG (0) = ν − 2h. Moreover the multiplicity of each nonzero eigenvalue is equal to 1 (for more details see [1]). If x = 0 then, for i = 1, . . . , h, we perform (R2h−i+1 , C2h−i+1 ) ← (R2h−i+1 −

mi mi Ri , C2h−i+1 − Ci ) x x

and obtain the matrix ⎛ x ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

m1

..

. x mh

( nx + nx − 1 2 − nx 2

mh x

) ( nx 2

− nx 2 + nx − 3

..

.

mh−1 x

)

− nx

3

..

..

.

−nx

h−1

(nx

h−1

.

+ nx − h − nx h

m2 x

)

− nx h − mx1

x nh

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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Our final task is to diagonalize the tridiagonal matrix ⎛

a1

⎞

b1

⎜ ⎜b ⎜ 1 ⎜ Tx = ⎜ ⎜ ⎜ ⎝

..

.

..

.

..

. bh−1

⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ bh−1 ⎠ ah

where ai =

x x mh+1−i + − , ni ni+1 x

bi = −

x ni+1

(i = 1, . . . , h − 1) and ah =

x m1 − . nh x

We proceed from the bottom to top. The goal is to eliminate all non-diagonal entries. After h − k steps the partially diagonalized matrix reads ⎛

a1

⎜ ⎜ b1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

b1 .. . .. .

⎞ ..

.

..

.

bk−1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

bk−1 αk 0 0 dk+1 ..

. dh

where αk = ak . If αk = 0 we perform (Rk−1 , Ck−1 ) ← (Rk−1 −

bk−1 bk−1 Rk , Ck−1 − Ck ) αk αk

b2

and obtain dk = αk and αk−1 = ak−1 − k−1 αk . Otherwise, if αk = 0 and ak−1 + bk−1 =  0, we perform 1 1 (Rk−1 , Ck−1 ) ← (Rk−1 + Rk , Ck−1 + Ck ) 2 2

followed by (Rk , Ck ) ← (Rk − b2

bk−1 bk−1 Rk−1 , Ck − Ck−1 ) ak−1 + bk−1 ak−1 + bk−1

and obtain dk = − ak−1k−1 +bk−1 and αk−1 = ak−1 + bk−1 .

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For αk = 0 and ak−1 + bk−1 = 0 we perform (Rk , Ck ) ← (Rk + Rk−1 , Ck + Ck−1 ) and obtain dk = bk−1 and αk−1 = −bk−1 . The method used in Stage 4 shows that any symmetric tridiagonal matrix can be diagonalized in a linear time. This is interesting in its own right and hence we state it as a proposition. Proposition 2.2. Any symmetric tridiagonal matrix can be diagonalized in linear time. The outcome of diagonalization procedure is the diagonal matrix Dx congruent to A + xI for a given real x. The matrix Dx has the following diagonal entries: 3x 2x ni x (ni − 1)x 3x 2x mi x (mi − 1)x , ,..., , , , ,..., , mi − 1 mi − 2 2 1 ni − 1 ni − 2 2 1 x x ,..., , d1 , . . . , dh . m1 mh

(i = 1, . . . , h)

It is worth mentioning that the algorithm for diagonalization of A +xI can be reduced to the computation of diagonal matrix congruent to the tridiagonal matrix Tx . Moreover each of n(= ν) steps in our algorithm takes a constant number of arithmetic operation and therefore its running time is O(n). Theorem 2.3. Let G = DN G(m1 , . . . , mh ; n1 , . . . , nh ) and its A adjacency matrix. Then, for any nonzero x, A + xI is congruent to a diagonal matrix Dx with diagonal entries equal to 3x 2x ni x (ni − 1)x 3x 2x mi x (mi − 1)x , ,..., , , , ,..., , mi − 1 mi − 2 2 1 ni − 1 ni − 2 2 1 x x ,..., , d1 , . . . , dh , m1 mh

(i = 1, . . . , h);

where d1 , . . . , dh are obtained by the diagonalization of Tx . 3. Applications Similarly as in [8] one can prove: Theorem 3.1. Let D be the diagonal matrix congruent to A −αI, where A is the adjacency matrix of a double nested graph G. Then (i) The number of eigenvalues of G greater than α is the number of positive entries in D.

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(ii) The number of eigenvalues of G less than α is the number of negative entries in D. (iii) The multiplicity of eigenvalue α is the number of zero entries in the diagonal of D. Corollary 3.2. The number of eigenvalues in an interval (α, β], counting multiplicities, is the number of positive entries in the diagonalization A − αI, minus the number of positive entries in the diagonalization of A − βI. We now prove the following result. Theorem 3.3. A double nested graph has no eigenvalues in [− 21 , 0) and (0, 12 ]. Proof. By Corollary 3.2 we have to show that the difference between the number of positive entries in diagonalization of A − 12 I and the number of positive entries in diagonalization of A is 0. In the diagonalization of A − 12 I all diagonal entries except, possibly d1 , . . . , dh , are negative. First dh = − 2n1h + 2m1 is strictly greater then 1, for any nh , m1 . Actually, for any di , i < h − 1, di > 12 . So, if dk > 12 , then

dk−1 = − >−

1 1 1 − + 2mh+2−k − 2nk−1 2nk dk



−1 2nk

2

1 1 1 − + 2mh+2−k − 2 2nk−1 2nk 4nk 2

>2−

1 3 = . 2 2

Hence d1 , . . . , dh are positive, which means that A − 12 I has exactly h eigenvalues greater than 12 . Since, by Lemma 2.1, the same holds for A with respect to 0 it follows that the adjacency matrix of any DNG does not have any eigenvalue in interval (0, 12 ], and in [− 12 , 0) as well since the spectrum of any DNG is symmetric with respect to the origin. 2 Corollary 3.4. Any double nested graph has the least positive eigenvalue greater than 12 . Next we show that the least positive eigenvalue of Gh = DN G(1, . . . , 1; 1, . . . , 1), say       h

h

1 θ(Gh ), for h ≥ 1 approaches from above as h tends to ∞. 2 In [1] it was shown that the positive part of spectra of any DNG contains the reciprocal values of the square roots of eigenvalues of the following tridiagonal matrix:

A. Alazemi et al. / Linear Algebra and its Applications 505 (2016) 194–210

⎛

a1 ⎜ −b ⎜ 1 ⎜ ⎜ ⎜ ˆ Th = ⎜ ⎜ ⎜ ⎜ ⎝

−a1 a2 + b1 −b2

205

⎞ −a2 a3 + b2 −b3

−a3 .. . .. .

..

.

..

.

−ah−1 ah + bh−1

−bh−1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

where ai =

1 , mi nh−i+1

for i = 1, . . . , h,

and bi =

1 mi = ai , mi+1 nh−i+1 mi+1

for i = 1, . . . , h − 1 .

For Gh , the tridiagonal matrix Tˆh is equal to ⎞

⎛

1 −1 ⎜ −1 2 ⎜ ⎜ −1 ⎜ ⎜ ˆ Th = ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ .. ⎟, . ⎟ ⎟ ⎟ .. . −1 ⎠ −1 2

−1 2 −1 . −1 . . .. .

and it is similar to ⎞

⎛

1 1 ⎜1 2 1 ⎜ ⎜ 1 2 ⎜ ⎜ Sh = ⎜ ⎜ 1 ⎜ ⎜ ⎝

1 .. .

..

.

..

..

.

.

1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ 1⎠ 2

It is obvious that λ1 (Qh−1 ) ≤ λ1 (Sh ) ≤ λ1 (Qh ) where Qh is the signless Laplacian matrix of the path Ph . Since λ1 (Qh ) = 2 + 2 cos( πh ), by letting h → +∞ we obtain limh→+∞ λ1 (Sh ) = 4 and therefore lim θ(Gh ) =

h→+∞

1 

limh→+∞

λ1 (Sh )

=

1 . 2

In [7] a direct formula for computing the eigenvalues of Sh can be found.

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At this place we introduce some facts needed in the rest of this section. Recall first that the multiplicity mG (λ) of an eigenvalue λ of a graph G changes by at most 1 if any vertex of G is deleted (an immediate consequence of the Interlacing theorem – see, for example, [6, p. 18]). If the multiplicity of λ in G − v, the vertex deleted subgraph of G, remains the same as in G, i.e. if mG (λ) = mG−v (λ) (resp. if it goes down by 1, or goes up by 1) then the vertex v of G is called neutral (resp. downer, or Parter vertex with respect to λ) – for more details see, for example, [10]. To end up this section we now show which vertex in G = DN G(m1 , . . . , mi , . . . , mh ; n1 , . . . , nh ) is a downer or Parter vertex with respect to 0. The multiplicity of 0 corresponds to the nullity of G. Recall, in any bipartite graph a vertex v is never neutral with respect to 0 (by the Interlacing theorem) and mG (0) = ν − 2h (as pointed in Lemma 2.1). Let v ∈ Ui (or either Vj ) and let θ(G) denote the least positive eigenvalue of G. Then, provided v ∈ Ui , we have that G − v is equal to: (a) DN G(m1 , . . . , mi − 1, . . . , mh ; n1 , . . . , nh ), if mi > 1. In this case v is a downer with respect to 0 since mG−v (0) = ν − 1 − 2h = mG (0) − 1. Then by the interlacing θ(G − v) ≤ θ(G). (b) DN G(m2 , . . . , mh ; n1 , . . . , nh−1 ) ∪ nh K1 , if i = 1 and m1 = 1. Then v is a Parter vertex with respect to 0 since mG−v (0) = ν − nh − 1 − 2(h − 1) + nh = mG (0) + 1. Here, by the interlacing θ(G − v) ≥ θ(G). (c) DN G(m1 , . . . , mi−1 , mi+1 , . . . , mh ; n1 , . . . , nh−i , nh+1−i +nh+2−i , nh+3−i , . . . , nh ), if i > 1, mi = 1. Then again v is a Parter vertex with respect to 0 since mG−v (0) = ν − 1 − 2(h − 1) = mG (0) + 1. Also, in this case θ(G − v) ≥ θ(G). From the above items (a)–(c) we can say when θ(G) non-increases or non-decreases, provided a vertex is deleted from G. In addition, if any of parameters mi , or nj , is increased then θ(G) is non-decreasing, and if h is increased then θ(G) need not increase. If G is a DNG, then it has no Parter vertices with respect to λ = 0 (recall, all nonzero eigenvalues of DNGs are simple). Next, due to some extensive computations, we have: Conjecture 3.1. A vertex v of any DN G is a downer with respect to nonzero eigenvalues. 4. Threshold graphs In this section we apply the strategy from Section 2 in order to diagonalize A + xI, where A is the adjacency matrix of a threshold graph G and x a real number. For this aim, G is viewed as a NSG (nested split graph). It is a split graph since it admits a partition of its vertex set into two parts, say U and V , such that the vertices of U induce a co-clique, while those of V a clique; all other edges are cross edges and join a vertex in

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207

U with a vertex in V . In addition, both U and V are partitioned into h non-empty cells U1 , . . . , Uh and V1 , . . . , Vh , and cross edges are added according to the following rule: each vertex in Ui is adjacent to all vertices in V1 ∪ · · · ∪ Vi . If mi = |Ui | and ni = |Vi |,

k then G is denoted by N SG(m1 , . . . , mh ; n1 , . . . , nh ). We denote by Nk = i=1 ni and

k Mk = i=1 mi , k = 1, . . . , h. Suppose that a connected threshold graph of order ν is given by generating binary sequence 0 · · · 0 1 · · · 1 0 · · · 0 1 · · · 1 · · · 0 · · · 0 1 · · · 1 t1

s1

t2

s2

th

(ti , si = 0).

sh

Let mh+1−i = ti and nh+1−i = si , 1 ≤ i ≤ h. Then the corresponding threshold graph can be seen as N SG(m1 , . . . , mh ; n1 , . . . , nh ). Example 4.1. Let G be a threshold graph with a generating sequence (0, 0, 1, 1, 1, 0, 1, 1). Then G = N SG(1, 2; 2, 3). If we follow the approach for defining chain graphs, then the same sequence defines DN G(2, 1; 2, 3). If we use a canonical vertex ordering (so vertices in U are ordered first by degrees in non-decreasing fashion, and next vertices in V in the same fashion) then the adjacency matrix of N SG(m1 , . . . , mh ; n1 , . . . , nh ) reads  A=

H O T H JNh − INh

 ,

where H = (Hij ) for 1 ≤ i, j ≤ h is the block matrix defined by Hij =

 Jmi ×nj Omi ×nj

if i + j ≤ h + 1, otherwise.

Here we follow the same stages as in Section 2 and give only significant steps in the corresponding stages. Stage 1 For i = 1, . . . , h we extract 3x 2x mi x (mi − 1)x , ,..., , mi − 1 mi − 2 2 1 if mi > 1 and focus on diagonalization of the submatrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

x m1

⎞ ..

.

B x mh

BT

JNh + xINh

⎟ ⎟ ⎟, ⎟ ⎠

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where the h × Nh matrix B is equal to ⎛

⎞

1···1

⎜ B=⎜ ⎝1···1 1···1 1···1 1···1 ···

⎟ ⎟, ⎠ 1···1

with exactly Ni ones in the i-th row. Stage 2 In this stage for any 1 ≤ i ≤ h, such that ni > 1 we extract 3(x − 1) 2(x − 1) ni (x − 1) (ni − 1)(x − 1) , ,..., , ni − 1 ni − 2 2 1 and continue with the diagonalization of ⎛

x m1

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

..

. x mh−1

1

···

1

··· .. .

⎞

1 .. . 1 1

x mh

..

1

x+n1 −1 n1

1 .. . 1

1 .. . 1

. ··· ··· 1 .. . .. . ···

1 1 ··· .. . .. . 1

1 1 1 1 x+nh −1 nh

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Stage 3 The previous matrix is congruent to 

 D Ih

Ih Sx

,

where D = diag( mx1 , . . . , mxh ) and ⎛

(x − 1)( n11 + ⎜ − x−1 ⎜ n2 ⎜ ⎜ Sx = ⎜ ⎜ ⎜ ⎜ ⎝

1 n2 )

− x−1 n2 (x − 1)( n12 + .. .

⎞ 1 n3 )

− x−1 n3 .. . − nx−1 h−1

..

1 (x − 1)( nh−1 + x−1 − nh

Stage 4 In the final stage for x = 0 we put focus only on 

 D O

O Tx

,

. 1 nh )

− x−1 nh

x+nh −1 nh

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

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where ⎛

(x − 1)( n11 + n12 ) − ⎜ − x−1 n2 ⎜

⎜ ⎜ Tx = ⎜ ⎜ ⎜ ⎝

m1 x

− x−1 n2 (x − 1)( n12 + n13 ) − ..

⎞ m2 x

− x−1 n3 ..

.

.

..

.

− x−1 nh

− x−1 nh x+nh −1 − mxh nh

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

is congruent to diag (d1 , . . . , dh ). Therefore we have Theorem 4.1. Let G = N SG(m1 , . . . , mh ; n1 , . . . , nh ) and A its adjacency matrix. Then, for any nonzero x, the matrix A + xI is congruent to a diagonal matrix Dx with diagonal entries equal to 3x 2x mi x (mi − 1)x , ,..., , ; mi − 1 mi − 2 2 1 3(x − 1) 2(x − 1) ni (x − 1) (ni − 1)(x − 1) , ,..., , ni − 1 ni − 2 2 1 x x ,..., ; m1 mh

(i = 1, . . . , h);

d1 , . . . , dh , where d1 , . . . , dh are obtained by the diagonalization of Tx . Using Theorem 4.1 some known facts proved in [2] and [9] about the spectra and inertia of N SG(m1 , . . . , mh ; n1 , . . . , nh ) can be replicated. For x = 1 we easily obtain  mG (−1) =

Nh − h, Nh − h + 1,

if if

mh > 1; mh = 1.

In addition, for x = 0 we obtain the exact inertia of any NSG. If mh = 1, then G has h positive eigenvalues, Mh − h equal to 0, Nh − h + 1 equal to −1, and h − 1 less than −1. If mh > 1, then G has h positive eigenvalues, Mh − h equal to 0, Nh − h equal to −1, and h less than −1. Acknowledgements The authors wish to thank an anonymous referee for his/her valuable suggestions which led to some substantial improvements. This work was supported and funded by Kuwait University, Research grant No. SM03/15.

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