vol10_pp155-162. 115KB Jun 04 2011 12:05:50 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 10, pp. 155-162, June 2003
ELA
http://math.technion.ac.il/iic/ela
THE PATH POLYNOMIAL OF A COMPLETE GRAPH∗
C. M. DA FONSECA†
Abstract. Let Pk (x) denote the polynomial of the path on k vertices. A complete description
of the matrix that is the obtained by evaluating Pk (x) at the adjacency matrix of the complete
graph, along with computing the effect of evaluating Pk (x) with Laplacian matrices of a path and
of a circuit.
Key words. Graph, Adjacency matrix, Laplacian matrix, Characteristic polynomial.
AMS subject classifications. 05C38, 05C50.
1. Introduction and preliminaries. For a finite and undirected graph G without loops or multiple edges, with n vertices, let us define the polynomial of G, PG , as
the characteristic polynomial of its adjacency matrix, A(G), i.e.,
PG (x) = det (xIn − A(G)) .
When the graph is a path with n vertices, we simply call PG the path polynomial and
denote it by Pn . Define An as the adjacency matrix of a path on n vertices.
For several interesting classes of graphs, A(Gi ) is a polynomial in A(G), where Gi
is the ith distance graph of G ([5]). Actually, for distance-regular graphs, A(Gi ) is a
polynomial in A(G) of degree i, and this property characterizes these kind of graphs
([14]).
In [4], Beezer has asked when a polynomial of an adjacency matrix will be the
adjacency matrix of another graph. Beezer gave a solution in the case that the original
graph is a path.
Theorem 1.1 ([4]). Suppose that p(x) is a polynomial of degree less than n.
Then p(An ) is the adjacency matrix of graph if and only if p(x) = P2i+1 (x), for some
i, with 0 ≤ i ≤ ⌊ n2 ⌋ − 1.
In the same paper, Beezer gave an elegant formula for Pk (An ) with k = 1, . . . , n,
and Bapat and Lal, in [1], completely described the structure of Pk (An ), for all
integers k. This result was also reached by Fonseca and Petronilho ([10]) in a noninductive way.
Theorem 1.2 ([1],[4],[10]). For 0 ≤ k ≤ n − 1, n being a positive integer,
1 if i + j = k + 2r , with 1 ≤ r ≤ min {i, j, n − k}
(Pk (An ))ij =
0 otherwise.
In [12], Shi Ronghua obtained some generalizations of the ones achieved by Bapat
and Lal. Later, in [10], Fonseca and Petronilho determined the matrix Pk (Cn ), where
Cn is the adjacency matrix of a circuit on n vertices.
∗ Received by the editors on 15 March 2003. Accepted for publication on 02 May 2003. Handling
Editor: Ravindar B. Bapat.
† Departamento de Matem´
atica, Universidade de Coimbra, 3001-454 COIMBRA, PORTUGAL
(cmf@mat.uc.pt). Supported by CMUC - Centro de Matem´
atica da Universidade de Coimbra.
155
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 10, pp. 155-162, June 2003
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156
C. M. da Fonseca
Consider the permutation σ = (12 . . . n).
Theorem 1.3 ([10]). For any nonnegative integer k,
Pk (Cn ) =
n−1
n−1
j=0 r=0
δ2r,k+2+j−n˙ P σ j ,
where δ is the Kronecker function, σ is the permutation (12 . . . n), P σ j is the permutation matrix of σ j and n˙ runs over the multiples of n.
According to Bapat and Lal (cf. [1]), a graph G is called path-positive of order
m if Pk (G) ≥ 0, for k = 1, 2, . . . , m, and G is simply called path-positive if it is pathpositive of any order. In [3], Bapat and Lal have characterized all graphs that are
path-positive. The following corollary is immediate from the theorem above.
Corollary 1.4. The circuit Cn is path-positive.
We define the complete graph Kn , to be the graph with n vertices in which each
pair of vertices is adjacent. The adjacency matrix of a complete graph, which we
identify also by Kn , is the n × n matrix
(1.1)
0
1
1
..
.
Kn =
1
1
In this note, we evaluate Pk (Kn ).
1 1
0 1
1 0
.. ..
. .
1 1
1 1
··· 1 1
··· 1 1
··· 1 1
. . .
..
. .. ..
··· 0 1
··· 1 0
2. The polynomial Pk . Let us consider the tridiagonal matrix Ak whose entries
are given by
1 if |i − j| = 1
(Ak )ij =
0 otherwise.
The expansion of the determinant
det (xIk − Ak ) = Pk (x)
along the first row or column gives us the recurrence relation
(2.1)
Pk (x) = xPk−1 (x) − Pk−2 (x),
for any positive integer k, with the convention P−1 (x) = 0 and P0 (x) = 1.
It is well known that
x
(2.2)
, x ∈ C, (k = 0, 1, . . .) ,
Pk (x) = Uk
2
where Uk (x) are the Chebyshev polynomials of the second kind.
Electronic Journal of Linear Algebra ISSN 1081-3810
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The Path Polynomial of a Complete Graph
157
From (2.2), it is straightforward to prove that
k−1
Pk (x) − Pk (y)
=
Pℓ (x) Pk−1−ℓ (y).
x−y
(2.3)
ℓ=0
Then, from (2.1) and (2.3), we may conclude the following lemma.
Lemma 2.1. For any positive integer k and square matrices A and B,
Pk (A) − Pk (B) =
k−1
Pℓ (A) (A − B) Pk−1−ℓ (B).
ℓ=0
As in Bapat and Lal [1], note that a connected graph is path-positive if it has
a spanning subgraph which is path-positive. Thus we have this immediate corollary
from Corollary 1.4 .
Corollary 2.2. The complete graph Kn is path-positive.
3. Evaluating Pk of a complete graph. If a matrix A = (aij ) satisfies the
relation
aij = a1σ1−i (j)
we say that A is a circulant matrix. Therefore, to define a circulant matrix A is
equivalent to presenting an n−tuple, say (a1 , . . . , an ). Then
A=
n−1
i=0
ai P σ i ,
and its eigenvalues are given by
(3.1)
λh =
n−1
ζ hℓ aℓ ,
ℓ=0
where ζ = exp i 2π
n . Given a polynomial p(x), the image of A is
p(A) = p
n−1
n−1
n−1
n−1
i
−1
−hj
hℓ
ai P σ
ζ
p
ζ aℓ P σ j .
=n
n−1
n−1
n−1
i
ai P σ
= n−1
ζ −hj Pk (λh ) P σ j ,
i=0
j=0 h=0
ℓ=0
Then,
Pk
i=0
where λh is defined as in (3.1).
j=0 h=0
Electronic Journal of Linear Algebra ISSN 1081-3810
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C. M. da Fonseca
The matrix Kn , defined in (1.1), is a circulant matrix and it can be written
Kn =
n−1
i=1
P σi .
By (3.1), Kn has the eigenvalues λ0 = n − 1 and λℓ = −1, for ℓ = 1, . . . , n − 1.
Therefore,
n−1
i
P σ
Pk (Kn ) = Pk
= n−1
i=1
n−1
n−1
j=0 h=0
=n
−1
n−1
j=0
ζ −hj Pk (λh ) P σ j
Pk (n − 1) + Pk (−1)
n−1
h=1
ζ
−hj
P σj
n−1
P σj .
= Pk (−1) P σ 0 + n−1 (Pk (n − 1) − Pk (−1))
j=0
Note that P σ 0 is the identity matrix.
We have thus proved the main result of this section:
Theorem 3.1. For any nonnegative integer k, the diagonal entries of Pk (Kn )
are the weighted average n1 Pk (n − 1) + n−1
n Pk (−1) and the off-diagonal entries are
1
1
P
(n
−
1)
−
P
(−1).
k
k
n
n
We can easily evaluate the different values of each term of the sum Pk (Kn ).
According to (2.2),
−1 if k ≡ 1 mod 3
0 if k ≡ 2 mod 3 .
Pk (−1) =
1 if k ≡ 0 mod 3
Another relation already known ([11, p.72]) for Pk (x) is
⌊k/2⌋
Pk (x) =
(−1)
ℓ
ℓ=0
k−ℓ
ℓ
xk−2ℓ ,
where ⌊z⌋ denotes the greatest integer less or equal to z. Therefore we have also
⌊k/2⌋
Pk (n − 1) − Pk (−1) =
(−1)
ℓ
ℓ=0
k−ℓ
ℓ
⌊k/2⌋ k−2ℓ
=n
ℓ=0 j=1
(−1)k−j+ℓ
(n − 1)k−2ℓ − (−1)k−2ℓ
(k − ℓ)!
nj−1 .
ℓ!j!(k − 2ℓ − j)!
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159
The Path Polynomial of a Complete Graph
4. Evaluating Pk of some Laplacian matrices. Let G be a graph. Denote
D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix.
Then
L(G) = D(G) − A(G)
is the Laplacian matrix of G.
In this section, expressions for Pk (L(An )) and Pk (L(Cn )), the path polynomials
of the Laplacian matrices of a path and a circuit, respectively, with n vertices, are
determined.
Let us consider the following recurrence relation:
P0 (x) = 1,
Pk (x) = (x + 2)Pk−1 (x) − Pk−2 (x),
and
Therefore
and
P1 (x) = x + 1,
for 2 ≤ k ≤ n − 1,
Pn (x) = (x + 1)Pn−1 (x) − Pn−2 (x).
x
x
+ 1 − Uk−1
+1 ,
Pk (x) = Uk
2
2
for 2 ≤ k ≤ n − 1,
x
Pn (x) = xUn−1
+1 .
2
where Uk (x) are the Chebyshev polynomials of the second kind.
Then the zeroes of Pn (x) are
λj = 2 cos
jπ
− 2,
n
j = 0, . . . , n − 1.
The recurrence relation above can be written in the following matricial
P0 (x)
P0 (x)
−1 1
0
P1 (x)
P1 (x)
1 −2 1
.
.
.
.
.
.
..
.
.
.
(x)
+
P
x
=
n
.
.
.
.
1 −2 1
Pn−2 (x)
Pn−2 (x)
0
1 −1
Pn−1 (x)
Pn−1 (x)
Thus, for j = 0, . . . , n − 1, the vector
jπ
P0 (λj )
cos 2n
jπ
cos 3 2n
P1 (λj )
−1
..
..
= cos jπ
(4.1)
.
.
2n
cos(2n − 3) jπ
Pn−2 (λj )
2n
jπ
cos(2n − 1) 2n
Pn−1 (λj )
way:
0
0
.. .
.
0
1
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C. M. da Fonseca
is an eigenvector associated to the eigenvalue λj of −L(An ).
Therefore the matrix −L(An ) is diagonalizable and, for 0 ≤ k ≤ n, the (i, j)th
entry of Pk (L(An )) is given by
(Pk (L(An )))ij = (−1)k
n−1
ℓ=0
which is equal to
(−1)k cos
n
kπ
2
+
Pi−1 (λℓ ) Pk (λℓ ) Pj−1 (λℓ )
2
n
s=1 Ps−1 (λℓ )
n−1
(−1)k 2
ℓπ
ℓπ
ℓπ
− 1 cos(2j − 1) .
cos(2i − 1) Uk cos
n
2n
n
2n
ℓ=1
If we define
αpm
=
n−1
cos m
ℓ=1
ℓπ
ℓπ
cosp ,
n
n
then
αpm =
and
(4.2)
αpm
1 p−1
αm−1 + αp−1
m+1
2
p
1 p
= p
α0m+2ℓ−p ,
ℓ
2
ℓ=0
with
1
α0m = nδm,2n˙ − (1 + (−1)m ),
2
where n˙ represents a multiple of n.
Using the trigonometric transformation formula and the Taylor formula
k
(p)
Uk (−1)
ℓπ
ℓπ
−1 =
cosp ,
Uk cos
n
p!
n
p=0
we can state the following proposition.
Theorem 4.1. For 0 ≤ k ≤ n, n being a positive integer,
(−1)k cos
(Pk L(An ))ij =
n
kπ
where αpm is defined as in (4.2).
2
+
k
(p)
(−1)k 2 Uk (−1) p
αi−j + αpi+j−1 ,
n
p!
p=0
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The Path Polynomial of a Complete Graph
161
(p)
Note that Uk (−1) can be easily evaluated, since
⌊k/2⌋
Uk (x) =
(−1)ℓ
ℓ=0
k−ℓ
ℓ
(2x)k−2ℓ ,
and then
⌊k/2⌋
(p)
Uk (−1) =
(−1)k−ℓ−p 2k−2ℓ
ℓ=0
(k − ℓ)!
.
ℓ!(k − 2ℓ − p)!
Now, we can find the matrix Pk (L(Cn )) using the same techniques of the last
section. L(Cn ) is the circulant matrix
2 −1
−1
−1 2 −1 0
..
..
..
.
.
.
.
0 −1 2 −1
−1
−1 2
Hence
L(Cn ) = 2P σ 0 − P (σ) − P σ n−1 .
The eigenvalues of L(Cn ) are
2 − 2 cos
2ℓπ
,
n
for ℓ = 0, . . . , n − 1 and thus
Pk L(Cn ) = Pk 2P σ 0 − P (σ) − P σ n−1
n−1
n−1
2ℓjπ
2ℓπ
e−i n Uk 1 − cos
= n−1
P σj
n
j=0
ℓ=0
= (−1)k
p
n−1
k
j=0 p=0 ℓ=0
(p)
Uk (−1)
δj+2ℓ−p,n˙ P σ j .
p
ℓ!(p − ℓ)!2
REFERENCES
[1] R.B. Bapat and A.K. Lal. Path-positive Graphs. Linear Algebra and Its Applications, 149:125–
149, 1991.
[2] R.B. Bapat and V.S. Sunder. On hypergroups of matrices. Linear and Multilinear Algebra,
29:125–140, 1991.
[3] R.B. Bapat and A.K. Lal. Path positivity and infinite Coxeter groups. Linear Algebra and Its
Applications, 196:19–35, 1994.
[4] Robert A. Beezer. On the polynomial of a path. Linear Algebra and Its Applications, 63:221–
225, 1984.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 10, pp. 155-162, June 2003
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162
C. M. da Fonseca
[5] N.L. Biggs. Algebraic Graph Theory. Cambridge University Press, Cambridge, 1974.
[6] T.S. Chihara. An introduction to orthogonal polynomials. Gordon and Breach, New York,
1978.
[7] D.M. Cvetkovi´c, M. Doob and H. Sachs. Spectra of Graphs, Theory and Applications. Academic
Press, New York, 1979.
[8] P.J. Davis. Circulant Matrices. John Wiley & Sons, New York, 1979.
[9] A. Erd´elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi. Higher Transcendental Functions.
Vol. II. Robert E. Krieger Publishing Co., Melbourne, FL, 1981.
[10] C.M. da Fonseca and J. Petronilho. Path polynomials of a circuit: a constructive approach.
Linear and Multilinear Algebra, 44:313–325, 1998.
[11] L´
aszl´
o Lov´
asz. Combinatorial Problems and Exercises. North-Holland, Amsterdam, 1979.
[12] Shi Ronghua. Path polynomials of a graph. Linear Algebra and Its Applications, 31:181–187,
1996.
[13] Alan C. Wilde. Differential equations involving circulant matrices. Rocky Mountain Journal
of Mathematics, 13:1–13, 1983.
[14] P.M. Weichsel. On distance-regularity in graphs. Journal of Combinatorial Theory Series B,
32:156–161, 1982.
A publication of the International Linear Algebra Society
Volume 10, pp. 155-162, June 2003
ELA
http://math.technion.ac.il/iic/ela
THE PATH POLYNOMIAL OF A COMPLETE GRAPH∗
C. M. DA FONSECA†
Abstract. Let Pk (x) denote the polynomial of the path on k vertices. A complete description
of the matrix that is the obtained by evaluating Pk (x) at the adjacency matrix of the complete
graph, along with computing the effect of evaluating Pk (x) with Laplacian matrices of a path and
of a circuit.
Key words. Graph, Adjacency matrix, Laplacian matrix, Characteristic polynomial.
AMS subject classifications. 05C38, 05C50.
1. Introduction and preliminaries. For a finite and undirected graph G without loops or multiple edges, with n vertices, let us define the polynomial of G, PG , as
the characteristic polynomial of its adjacency matrix, A(G), i.e.,
PG (x) = det (xIn − A(G)) .
When the graph is a path with n vertices, we simply call PG the path polynomial and
denote it by Pn . Define An as the adjacency matrix of a path on n vertices.
For several interesting classes of graphs, A(Gi ) is a polynomial in A(G), where Gi
is the ith distance graph of G ([5]). Actually, for distance-regular graphs, A(Gi ) is a
polynomial in A(G) of degree i, and this property characterizes these kind of graphs
([14]).
In [4], Beezer has asked when a polynomial of an adjacency matrix will be the
adjacency matrix of another graph. Beezer gave a solution in the case that the original
graph is a path.
Theorem 1.1 ([4]). Suppose that p(x) is a polynomial of degree less than n.
Then p(An ) is the adjacency matrix of graph if and only if p(x) = P2i+1 (x), for some
i, with 0 ≤ i ≤ ⌊ n2 ⌋ − 1.
In the same paper, Beezer gave an elegant formula for Pk (An ) with k = 1, . . . , n,
and Bapat and Lal, in [1], completely described the structure of Pk (An ), for all
integers k. This result was also reached by Fonseca and Petronilho ([10]) in a noninductive way.
Theorem 1.2 ([1],[4],[10]). For 0 ≤ k ≤ n − 1, n being a positive integer,
1 if i + j = k + 2r , with 1 ≤ r ≤ min {i, j, n − k}
(Pk (An ))ij =
0 otherwise.
In [12], Shi Ronghua obtained some generalizations of the ones achieved by Bapat
and Lal. Later, in [10], Fonseca and Petronilho determined the matrix Pk (Cn ), where
Cn is the adjacency matrix of a circuit on n vertices.
∗ Received by the editors on 15 March 2003. Accepted for publication on 02 May 2003. Handling
Editor: Ravindar B. Bapat.
† Departamento de Matem´
atica, Universidade de Coimbra, 3001-454 COIMBRA, PORTUGAL
(cmf@mat.uc.pt). Supported by CMUC - Centro de Matem´
atica da Universidade de Coimbra.
155
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156
C. M. da Fonseca
Consider the permutation σ = (12 . . . n).
Theorem 1.3 ([10]). For any nonnegative integer k,
Pk (Cn ) =
n−1
n−1
j=0 r=0
δ2r,k+2+j−n˙ P σ j ,
where δ is the Kronecker function, σ is the permutation (12 . . . n), P σ j is the permutation matrix of σ j and n˙ runs over the multiples of n.
According to Bapat and Lal (cf. [1]), a graph G is called path-positive of order
m if Pk (G) ≥ 0, for k = 1, 2, . . . , m, and G is simply called path-positive if it is pathpositive of any order. In [3], Bapat and Lal have characterized all graphs that are
path-positive. The following corollary is immediate from the theorem above.
Corollary 1.4. The circuit Cn is path-positive.
We define the complete graph Kn , to be the graph with n vertices in which each
pair of vertices is adjacent. The adjacency matrix of a complete graph, which we
identify also by Kn , is the n × n matrix
(1.1)
0
1
1
..
.
Kn =
1
1
In this note, we evaluate Pk (Kn ).
1 1
0 1
1 0
.. ..
. .
1 1
1 1
··· 1 1
··· 1 1
··· 1 1
. . .
..
. .. ..
··· 0 1
··· 1 0
2. The polynomial Pk . Let us consider the tridiagonal matrix Ak whose entries
are given by
1 if |i − j| = 1
(Ak )ij =
0 otherwise.
The expansion of the determinant
det (xIk − Ak ) = Pk (x)
along the first row or column gives us the recurrence relation
(2.1)
Pk (x) = xPk−1 (x) − Pk−2 (x),
for any positive integer k, with the convention P−1 (x) = 0 and P0 (x) = 1.
It is well known that
x
(2.2)
, x ∈ C, (k = 0, 1, . . .) ,
Pk (x) = Uk
2
where Uk (x) are the Chebyshev polynomials of the second kind.
Electronic Journal of Linear Algebra ISSN 1081-3810
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157
From (2.2), it is straightforward to prove that
k−1
Pk (x) − Pk (y)
=
Pℓ (x) Pk−1−ℓ (y).
x−y
(2.3)
ℓ=0
Then, from (2.1) and (2.3), we may conclude the following lemma.
Lemma 2.1. For any positive integer k and square matrices A and B,
Pk (A) − Pk (B) =
k−1
Pℓ (A) (A − B) Pk−1−ℓ (B).
ℓ=0
As in Bapat and Lal [1], note that a connected graph is path-positive if it has
a spanning subgraph which is path-positive. Thus we have this immediate corollary
from Corollary 1.4 .
Corollary 2.2. The complete graph Kn is path-positive.
3. Evaluating Pk of a complete graph. If a matrix A = (aij ) satisfies the
relation
aij = a1σ1−i (j)
we say that A is a circulant matrix. Therefore, to define a circulant matrix A is
equivalent to presenting an n−tuple, say (a1 , . . . , an ). Then
A=
n−1
i=0
ai P σ i ,
and its eigenvalues are given by
(3.1)
λh =
n−1
ζ hℓ aℓ ,
ℓ=0
where ζ = exp i 2π
n . Given a polynomial p(x), the image of A is
p(A) = p
n−1
n−1
n−1
n−1
i
−1
−hj
hℓ
ai P σ
ζ
p
ζ aℓ P σ j .
=n
n−1
n−1
n−1
i
ai P σ
= n−1
ζ −hj Pk (λh ) P σ j ,
i=0
j=0 h=0
ℓ=0
Then,
Pk
i=0
where λh is defined as in (3.1).
j=0 h=0
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C. M. da Fonseca
The matrix Kn , defined in (1.1), is a circulant matrix and it can be written
Kn =
n−1
i=1
P σi .
By (3.1), Kn has the eigenvalues λ0 = n − 1 and λℓ = −1, for ℓ = 1, . . . , n − 1.
Therefore,
n−1
i
P σ
Pk (Kn ) = Pk
= n−1
i=1
n−1
n−1
j=0 h=0
=n
−1
n−1
j=0
ζ −hj Pk (λh ) P σ j
Pk (n − 1) + Pk (−1)
n−1
h=1
ζ
−hj
P σj
n−1
P σj .
= Pk (−1) P σ 0 + n−1 (Pk (n − 1) − Pk (−1))
j=0
Note that P σ 0 is the identity matrix.
We have thus proved the main result of this section:
Theorem 3.1. For any nonnegative integer k, the diagonal entries of Pk (Kn )
are the weighted average n1 Pk (n − 1) + n−1
n Pk (−1) and the off-diagonal entries are
1
1
P
(n
−
1)
−
P
(−1).
k
k
n
n
We can easily evaluate the different values of each term of the sum Pk (Kn ).
According to (2.2),
−1 if k ≡ 1 mod 3
0 if k ≡ 2 mod 3 .
Pk (−1) =
1 if k ≡ 0 mod 3
Another relation already known ([11, p.72]) for Pk (x) is
⌊k/2⌋
Pk (x) =
(−1)
ℓ
ℓ=0
k−ℓ
ℓ
xk−2ℓ ,
where ⌊z⌋ denotes the greatest integer less or equal to z. Therefore we have also
⌊k/2⌋
Pk (n − 1) − Pk (−1) =
(−1)
ℓ
ℓ=0
k−ℓ
ℓ
⌊k/2⌋ k−2ℓ
=n
ℓ=0 j=1
(−1)k−j+ℓ
(n − 1)k−2ℓ − (−1)k−2ℓ
(k − ℓ)!
nj−1 .
ℓ!j!(k − 2ℓ − j)!
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The Path Polynomial of a Complete Graph
4. Evaluating Pk of some Laplacian matrices. Let G be a graph. Denote
D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix.
Then
L(G) = D(G) − A(G)
is the Laplacian matrix of G.
In this section, expressions for Pk (L(An )) and Pk (L(Cn )), the path polynomials
of the Laplacian matrices of a path and a circuit, respectively, with n vertices, are
determined.
Let us consider the following recurrence relation:
P0 (x) = 1,
Pk (x) = (x + 2)Pk−1 (x) − Pk−2 (x),
and
Therefore
and
P1 (x) = x + 1,
for 2 ≤ k ≤ n − 1,
Pn (x) = (x + 1)Pn−1 (x) − Pn−2 (x).
x
x
+ 1 − Uk−1
+1 ,
Pk (x) = Uk
2
2
for 2 ≤ k ≤ n − 1,
x
Pn (x) = xUn−1
+1 .
2
where Uk (x) are the Chebyshev polynomials of the second kind.
Then the zeroes of Pn (x) are
λj = 2 cos
jπ
− 2,
n
j = 0, . . . , n − 1.
The recurrence relation above can be written in the following matricial
P0 (x)
P0 (x)
−1 1
0
P1 (x)
P1 (x)
1 −2 1
.
.
.
.
.
.
..
.
.
.
(x)
+
P
x
=
n
.
.
.
.
1 −2 1
Pn−2 (x)
Pn−2 (x)
0
1 −1
Pn−1 (x)
Pn−1 (x)
Thus, for j = 0, . . . , n − 1, the vector
jπ
P0 (λj )
cos 2n
jπ
cos 3 2n
P1 (λj )
−1
..
..
= cos jπ
(4.1)
.
.
2n
cos(2n − 3) jπ
Pn−2 (λj )
2n
jπ
cos(2n − 1) 2n
Pn−1 (λj )
way:
0
0
.. .
.
0
1
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C. M. da Fonseca
is an eigenvector associated to the eigenvalue λj of −L(An ).
Therefore the matrix −L(An ) is diagonalizable and, for 0 ≤ k ≤ n, the (i, j)th
entry of Pk (L(An )) is given by
(Pk (L(An )))ij = (−1)k
n−1
ℓ=0
which is equal to
(−1)k cos
n
kπ
2
+
Pi−1 (λℓ ) Pk (λℓ ) Pj−1 (λℓ )
2
n
s=1 Ps−1 (λℓ )
n−1
(−1)k 2
ℓπ
ℓπ
ℓπ
− 1 cos(2j − 1) .
cos(2i − 1) Uk cos
n
2n
n
2n
ℓ=1
If we define
αpm
=
n−1
cos m
ℓ=1
ℓπ
ℓπ
cosp ,
n
n
then
αpm =
and
(4.2)
αpm
1 p−1
αm−1 + αp−1
m+1
2
p
1 p
= p
α0m+2ℓ−p ,
ℓ
2
ℓ=0
with
1
α0m = nδm,2n˙ − (1 + (−1)m ),
2
where n˙ represents a multiple of n.
Using the trigonometric transformation formula and the Taylor formula
k
(p)
Uk (−1)
ℓπ
ℓπ
−1 =
cosp ,
Uk cos
n
p!
n
p=0
we can state the following proposition.
Theorem 4.1. For 0 ≤ k ≤ n, n being a positive integer,
(−1)k cos
(Pk L(An ))ij =
n
kπ
where αpm is defined as in (4.2).
2
+
k
(p)
(−1)k 2 Uk (−1) p
αi−j + αpi+j−1 ,
n
p!
p=0
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The Path Polynomial of a Complete Graph
161
(p)
Note that Uk (−1) can be easily evaluated, since
⌊k/2⌋
Uk (x) =
(−1)ℓ
ℓ=0
k−ℓ
ℓ
(2x)k−2ℓ ,
and then
⌊k/2⌋
(p)
Uk (−1) =
(−1)k−ℓ−p 2k−2ℓ
ℓ=0
(k − ℓ)!
.
ℓ!(k − 2ℓ − p)!
Now, we can find the matrix Pk (L(Cn )) using the same techniques of the last
section. L(Cn ) is the circulant matrix
2 −1
−1
−1 2 −1 0
..
..
..
.
.
.
.
0 −1 2 −1
−1
−1 2
Hence
L(Cn ) = 2P σ 0 − P (σ) − P σ n−1 .
The eigenvalues of L(Cn ) are
2 − 2 cos
2ℓπ
,
n
for ℓ = 0, . . . , n − 1 and thus
Pk L(Cn ) = Pk 2P σ 0 − P (σ) − P σ n−1
n−1
n−1
2ℓjπ
2ℓπ
e−i n Uk 1 − cos
= n−1
P σj
n
j=0
ℓ=0
= (−1)k
p
n−1
k
j=0 p=0 ℓ=0
(p)
Uk (−1)
δj+2ℓ−p,n˙ P σ j .
p
ℓ!(p − ℓ)!2
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Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 10, pp. 155-162, June 2003
ELA
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162
C. M. da Fonseca
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