- Email: [email protected]
The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal Möbius graphs
Applied Mathematics and Computation 355 (2019) 33–46
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal Möbius graphs Xiaoling Ma a, Hong Bian b,∗ a b
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, PR China School of Mathematical Science, Xinjiang Normal University, Urumqi 830054, PR China
a r t i c l e
i n f o
a b s t r a c t
MSC: 05C99
Let HMn be a hexagonal Möbius graph of length n. In this paper, due to the normalized Laplacian polynomial decomposition theorem, we obtain that the normalized Laplacian spectrum of HMn consists of the eigenvalues of two symmetric quasi-tridiagonal matrices LA and LS of order 2n. Finally, by applying the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit closed formulas of the degree-Kirchhoff index and the number of spanning trees of HMn are given in terms of the index n.
Keywords: Hexagonal Möbius graph Normalized Laplacian Degree-Kirchhoff index Spanning trees
© 2019 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper, we only consider simple, finite and connected graphs. Let G = (V (G ), E (G )) be a simple undirected graph with vertex set V (G ) = {v1 , v2 , . . . , vn } and edge set E(G), where its order and size are |V (G )| = n(G ) = n and |E (G )| = m(G ) = m, respectively. We denote by di the degree of vi in G for 1 ≤ i ≤ n. The adjacency matrix A(G) of G is a n × n matrix whose (i, j)-entry is equal to 1 if vertices vi and v j are adjacent and 0 otherwise. The Laplacian matrix of G is L(G ) = D(G ) − A(G ), where D(G ) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix. Our notation is standard and taken mainly from [1]. The conventional distance between vertices vi and v j , denoted by dij , is the length of a shortest path between them. Klein and Randic´ [2] introduced a new distance function named resistance distance based on electrical network theory, the resistance distance between vertices vi and v j , denoted by rij , is defined to be the effective electrical resistance between them if each edge of G is replaced by a unit resistor [2]. Similar to the standard distance, the resistance distance is also intrinsic to the graph, not only with some fine purely mathematical properties, but also with a substantial potential for chemical applications (see [3,4] for details). It is well known that the resistance distance between two arbitrary vertices in an electronic network can be obtained in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the network. One famous resistance distance-based parameter called the Kirchhoff index, K(G), was given by K (G ) = i< j ri j ; see [2]. Later it is shown, independently, by Klein and Ivanciuc [4] and Lovász [5] that
K (G ) =
i< j
ri j = n
n 1 i=2
μi
,
where 0 = μ1 < μ2 ≤ · · · ≤ μn (n ≥ 2 ) are the eigenvalues of L(G). ∗
Corresponding author. E-mail address: [email protected] (H. Bian).
https://doi.org/10.1016/j.amc.2019.02.052 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
34
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
Fig. 1. The hexagonal Möbius graph.
In recent years, the normalized Laplacian, L(G ), which is consistent with the eigenvalues in spectral geometry and in random processes [6], has attracted more attention from researchers because many results which were only known for regular graphs can be generalized to all graphs. The normalized Laplacian is defined to be 1
1
1
1
L(G ) = I − D 2 (D−1 A )D− 2 = D− 2 LD− 2 with the convention that D(G )−1 (i, i ) = 0 if di = 0. Thus it is easy to know that the normalized Laplacian is a square matrix with rows and columns are indexed by vertices of G, and for any two vertices vi and v j of G, the (i, j)-entry of it is given by
(L(G ))i j =
⎧ ⎨1,
− √1 ,
if i = j and di = 0, if i = j and vi is adjacent to v j ,
0,
otherwise.
⎩
di d j
(1)
We denote the characteristic polynomial det(I − L(G )) of L(G ) by (L(G )). The roots of (L(G )) are known as the normalized Laplacian eigenvalues of G. The multiset of the normalized Laplacian eigenvalues of G is called the normalized Laplacian spectrum of G. The spectrum of L(G ) is denoted by S(G ) = {λ1 , λ2 , . . . , λn } with 0 = λ1 ≤ λ2 ≤ · · · ≤ λn . It is well known that G is connected if and only if λ2 > 0. Chen and Zhang [7] showed that the resistance distance can be expressed naturally in terms of the eigenvalues and eigenvectors of the normalized Laplacian and introduced another graph invariant, defined by DK (G ) = i< j di d j ri j , which is called the degree-Kirchhoff index (see [8,9]). It is closely related to the corresponding spectrum of the normalized Laplacian (see Lemma 1(a) in the next section). Like Kirchhoff index, degree-Kirchhoff index is a structured descriptor [7]. Unfortunately, it is difficult to compute resistant distance and degree-Kirchhoff index in a graph from their computational complexity. Hence, it is necessary to find closed-form formulae for the degree-Kirchhoff index. To this end, many researchers devote to find explicit closed-form formulae for some classes of graphs, such as linear hexagonal chains [10], linear polyomino chains [11], quadrilateral graphs [12], corona and edge corona of two graphs [13], and so on. Recently, according to the normalized Laplacian polynomial decomposition theorem, Huang et al. [10] gave an explicit closed form formula for degree-Kirchhoff index of linear hexagonal chains. In this paper, we concentrate our attention on hexagonal Möbius graphs and consider degree-Kirchhoff index of hexagonal Möbius graphs. Though we also make use of the decomposition of normalized Laplacian polynomial technique, the calculation process, especially the method of handling the matrix is very different to that in [10]. A hexagonal Möbius graph of length n, denoted by HMn , is defined as the graph obtained from P2 2P2n by removing the edges {(0, 2i + 1 ), (1, 2i + 1 )} with 0 ≤ i ≤ n − 1 and adding two new edges ((0, 0 )(1, 2n − 1 )) and ((1, 0 )(0, 2n − 1 )) depicted in Fig. 1(a), where P2 2P2n denotes the Cartesian product of P2 and P2n ,
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
35
i.e., the graph obtained from P2n and its copy by connecting all the corresponding vertices. Then it is routine to check that |V (HMn )| = 4n and |E (HMn )| = 5n. As an example, a hexagonal Möbius graph of length 5 is shown in Fig. 1(b). Hexagonal Möbius graphs are one type of molecular graphs embedded into the Möbius strip such that each face is a hexagon. Hence, hexagonal Möbius graphs have been of great interest and extensively studied. In 2011, Wang and Xu [14] considered the Kirchhoff index of hexagonal Möbius graphs. Later, the Wiener index of hexagonal Möbius graphs is determined in [15]. For more results on hexagonal Möbius graphs one may be referred to [16–21] and the references therein. The structure of the paper is as follows. In Section 2, we state again the decomposition theorem of the normalized Laplacian polynomial and obtain that the normalized Laplacian spectrum of HMn consists of the eigenvalues of two symmetric quasi-tridiagonal matrices LA and LS of order 2n. In Section 3, without calculating the eigenvalues of LA and LS , we obtain the sum of their reciprocals according to the relationship between roots and coefficients of the characteristic polynomial of LA and LS . In the last section, explicit closed-form formulas for the degree-Kirchhoff index and the number of spanning trees of HMn are derived in terms of the index n. 2. Some preliminary results An automorphism of G is a permutation π of V(G), which has the property that vi v j is an edge of G if and only if
π (vi )π (v j ) is an edge of G.
Suppose that G has an automorphism π , which can be written as the product of disjoint 1-cycles and transpositions, that
is
π = (10 )(20 ) . . . (m0 )(1, 1 )(2, 2 ) . . . (k, k ). Then we have |V (G )| = m + 2k. Let V0 = {10 , 20 , . . . , m0 }, V1 = {1, 2, . . . , k}, V2 = {1 , 2 , . . . , k }, then by a suitable arrangement of vertices in G, L(G ) can be written as the following block matrix
L (G ) =
LV0V0 LV1V0 LV2V0
LV0V1 LV1V1 LV2V1
LV0V2 V0 LV1V2 V1 LV2V2 V2
where LViV j is the submatrix formed by rows corresponding to vertices in Vi and columns corresponding to vertices in Vj for i, j = 0, 1, 2. Let
LA ( G ) =
√ 2LV0V1 , LV1V1 + LV1V2
√LV0V0 2LV1V0
LS (G ) = LV2V2 − LV1V2 . Huang et al. [10] obtained the following decomposition theorem of normalized Laplacian polynomial, which is similar to the decomposition theorem of the Laplacian polynomial in [22]. Theorem 1 [10]. Let L(G ), LA (G ), LS (G ) be defined as above. Then
(L(G )) = (LA (G )) · (LS (G )). In particular, if π can be written only as the product of disjoint transpositions (i.e. V0 = ∅), then
LA (G ) = LV1V1 + LV1V2 , LS (G ) = LV2V2 − LV1V2 , LV1V1 = LV2V2 . Kemeny’s constant Kc(G) of G is the expected number of steps required for the transition from a starting vertex vi to a destination vertex, which is chosen randomly according to a stationary distribution of unbiased random walks on G. Kemeny’s constant gives an interesting quantity for finite ergodic Markov chains, which is independent of the initial state of the Markov chain [23]. In terms of the normalized Laplacian spectrum of G, the special calculation formulae for the degree-Kirchhoff index, Kemeny’s constant and the number of spanning trees of graph G can be expressed as follows. Lemma 1. Let G be a connected graph with n vertices and m edges, and {0 = λ1 < λ2 ≤ · · · ≤ λn } be the spectrum on the normalized Laplacian L(G ) of G. Then each of the following holds. (a) [7] The degree-Kirchhoff index of G is
DK (G ) = 2m
n 1 i=2
λi
.
36
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
(b) [24] Kemeny’s constant of G is
Kc (G ) =
n 1 i=2
λi
.
(c) [6] The number τ (G) of spanning trees of G is n
di
i=1
n
λk = 2mτ (G ).
k=2
Apparently, from Lemma 1(a) and (b) the straightforward relation between the degree-Kirchhoff index and Kemeny’s constant is DK (G ) = 2mKc (G ).
3. The normalized Laplacian spectrum of the hexagonal Möbius graphs HMn In this section, we determine the normalized Laplacian eigenvalues of HMn according to Theorem 1. Given an n × n matrix M, we will use M[{i, j, . . . , k}] to denote the submatrix of M resulted from the deletion of the ith, jth,..., kth rows and columns. For convenience, we abbreviate LA (HMn ) and LS (HMn ) to LA and LS , respectively. We label the vertices of HMn as depicted in Fig. 1(a). Obviously, π = ((0, 0 )(1, 0 ))((0, 1 ), (1, 1 ))((0, 2 ), (1, 2 )) . . . ((0, 2n − 1 ), (1, 2n − 1 )) is an automorphism of HMn . Then V0 = ∅, V1 = {(0, 0 ), (0, 1 ), (0, 2 ), . . . , (0, 2n − 1 )} and V2 = {(1, 0 ), (1, 1 ), (1, 2 ), . . . , (1, 2n − 1 )}. Accordingly, LV1V1 (LV2V2 ) and LV1V2 are given as follows:
⎛
LV1V1
1 ⎜− √16 ⎜ ⎜ 0 ⎜ ⎜ · =⎜ ⎜ · ⎜ · ⎜ ⎝ 0 0
⎛
LV1V2
− 13 ⎜ 0 ⎜ 0 ⎜ ⎜ · =⎜ ⎜ · ⎜ ⎜ · ⎝ 0 − √16
− √16 1 − √16 · · · 0 0
0 0 0 · · · 0 0
0 − √16 1 · · · 0 0
0 0 − 13 · · · 0 0
0 0 0 · · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 0 · · · − 13 0
⎞
0 0 0 · · · 1 − √16
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
,
2n×2n
⎞
− √16 0 ⎟ ⎟ 0 ⎟ · ⎟ ⎟ · ⎟ ⎟ · ⎟ 0 ⎠ 0
.
2n×2n
Hence
LA = LV1V1 + LV1V2 ⎛ 2 − √16 3 1 ⎜− √6 1 ⎜ ⎜ 0 − √16 ⎜ ⎜ · · = ⎜ · ⎜ · ⎜ · · ⎜ ⎝ 0 0 − √16 0
0 − √16 2 3
· · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · − √16 0
0 0 0 · · ·
2 3 − √16
⎞
− √16 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
2n×2n
,
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
LS = LV1V1 − LV1V2 ⎛ 4 − √16 3 1 ⎜− √6 1 ⎜ ⎜ 0 − √16 ⎜ ⎜ · · = ⎜ · ⎜ · ⎜ · · ⎜ ⎝ 0 0 √1 0 6
0 − √16
0 0 − √16 · · · 0 0
4 3
· · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
√1 6
0 0 0 · · ·
⎞
0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
4 3 − √16
37
.
2n×2n
Suppose that the eigenvalues of LA and LS are, respectively, denoted by α i (i = 1, 2, . . . , 2n ) and β j ( j = 1, 2, . . . , 2n ) with α 1 ≤ α 2 ≤ ≤ α 2n , β 1 ≤ β 2 ≤ ≤ β 2n . Then, by Theorem 1, the spectrum of HMn is just {α1 , α2 , . . . , α2n , β1 , β2 , . . . , β2n }, and it is straightforward to check that α1 = 0, α i > 0 (i = 2, 3, . . . , 2n ), and β j > 0 ( j = 1, 2, . . . , 2n ). In the next section, we will compute some indices of HMn which are related to the normalized Laplacian spectrum of HMn .
4. Degree-Kirchhoff index and the number of spanning trees of hexagonal Möbius graphs In this section, we first provide a complete description of the sum of the normalized Laplacian eigenvalues’ reciprocals and the product of the normalized Laplacian eigenvalues, then apply these results to compute the degree-Kirchhoff index, Kemeny’s constant and the number of spanning trees of HMn . n 1 2 n 1 Based on the relationship between the roots and coefficients of (LA ) and (LS ), the formulas of 2i=2 j=1 β α and i
j
are derived in the next two lemmas, respectively. Lemma 2. Let 0 = α1 < α2 ≤ · · · ≤ α2n be eigenvalues of LA . Then 2n 1 i=2
αi
=
25n2 − 7 . 30
Proof. Let
(LA ) = x2n + a1 x2n−1 + · · · + a2n−2 x2 + a2n−1 x = x(x2n−1 + a1 x2n−2 + · · · + a2n−2 x + a2n−1 ). where a2n−1 = 0. Then α2 , α3 , . . . , α2n are the roots of the following equation
x2n−1 + a1 x2n−2 + · · · + a2n−2 x + a2n−1 = 0. By Vieta’s theorem, we have 2n 1 i=2
αi
=
a2n−2 . −a2n−1
(2)
Before giving a2n−2 and −a2n−1 , we first consider ith order principal submatrices, Fi and Fi , formed by the first i rows and columns of the following matrices LA and LA , respectively, i = 1, 2, . . . , 2n. Let
⎛
2 3 − √16
LA =
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 · · · 0 0
− √16 1 − √16 · · · 0 0
0 − √16 2 3
· · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 0 · · ·
2 3 − √16
⎞
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
2n×2n
38
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
and
⎛
− √16
1 ⎜− √16 ⎜ ⎜ 0 ⎜ ⎜ · LA = ⎜ ⎜ · ⎜ · ⎜ ⎝ 0 0
0 − √16 1 · · · 0 0
2 3 − √16
· · · 0 0
0 0 − √16 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · 0 0
0 0 0 · · · − √16 0
⎞
0 0 0 · · · 1 − √16
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 2 3
.
2n×2n
Put fi = det Fi and fi = det Fi . We proceed by showing the following claims. Claim 1. For 1 ≤ i ≤ 2n,
⎧ i ⎪ 1 ⎪ ⎪ ( i + 1 ) , √ ⎨ 6 fi = i √ ⎪ ⎪ 6 1 ⎪ ⎩ (i + 1 ) √ , 3
6
if i is even; if i is odd.
Claim 2. For 1 ≤ i ≤ 2n,
fi =
⎧ i ⎪ 1 ⎪ ⎪ ⎨ (i + 1 ) √ ,
if i is even;
i √ ⎪ ⎪ 6 1 ⎪ ⎩ (i + 1 ) √ ,
if i is odd.
6
2
6
Proof of Claim 1. It is straightforward to check that f 1 = respect to its last row yields
fi =
⎧ ⎨ fi−1 − 1 fi−2 ,
if i is even;
⎩2f
if i is odd.
3
6
i−1
−
1 fi−2 , 6
2 3,
f2 =
1 2,
f3 =
For 1 ≤ i ≤ n, let bi = f2i and for 1 ≤ i ≤ n − 1, let ci = f2i+1 . Then b1 =
1 2,
2 9
and f4 =
c1 =
2 9
5 36 .
For 3 ≤ i ≤ 2n, expanding det Fi with
and, for i ≥ 2, we have
⎧ ⎨ bi =ci−1 − 1 bi−1 , 6
(3)
⎩ c =2b − 1c . i i i−1 3
6
From the first equation in (3), one has ci−1 = bi + 16 bi−1 . Hence, ci = bi+1 + 16 bi . Substituting ci−1 and ci into the sec1 1 ond equation in (3) yields bi+1 = 13 bi − 36 bi−1 , i ≥ 2. In a similar way, we can obtain ci+1 = 13 ci − 36 ci−1 , i ≥ 2. Therefore, fi satisfies the recurrence relation
fi =
1 1 fi−2 − fi−4 , 3 36
( i ≥ 5 ),
f1 =
2 , 3
f2 =
Then the characteristic equation of (4) is x4 =
1 2 3x
−
1 , 2 1 36 ,
f3 =
2 , 9
f4 =
5 . 36
whose roots are x1 = x2 =
(4) √1 , 6
x3 = x4 = − √1 . The general solu6
tion of (4) is
1 f i = ( y1 i + y2 ) √ 6
i
1 + ( y3 i + y4 ) − √ 6
i .
Combining with the initial conditions in (4) yields the system of equations
(5)
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
39
⎧
1 1 2 ⎪ ⎪ ( y1 + y2 ) √ − ( y3 + y4 ) √ = , ⎪ ⎪ 3 ⎪ 6 6 ⎪ ⎪ ⎪
2 2 ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎪ ⎨ (2y1 + y2 ) √6 + (2y3 + y4 ) √6 = 2 , 3 3 ⎪ 1 1 2 ⎪ ⎪ ( 3 y + y ) − ( 3 y + y ) = , √ √ ⎪ 1 2 3 4 ⎪ 9 ⎪ 6 6 ⎪ ⎪ ⎪ 4 4 ⎪ ⎪ 1 1 5 ⎪ ⎪ + ( 4y3 + y4 ) √ = . ⎩ ( 4y1 + y2 ) √ 6
6
36
The unique solution of this system of equations is y1 = y2 =
1 2
+
√1 , 6
y3 = y4 =
1 2
−
√1 . 6
Thus Claim 1 follows by substituting
y1 , y2 , y3 and y4 back into (5), as desired. The proof of Claim 2 is carried out in the same way as in the case of Claim 1. Now we determine a2n−1 and a2n−2 in (2) (based on Claim 1 and Claim 2), respectively. For the sake of convenience, in the following discussion we assume f0 = 1. Claim 3. −a2n−1 =
5 2 1 n−1 3n (6)
Proof of Claim 3. It is not difficult to see that the coefficient −a2n−1 (= (−1 )2n−1 a2n−1 ) of x in (LA ) = det(xI2n − LA ) is the sum of the constant terms of the determinants obtained by deleting the ith row and column for 1 ≤ i ≤ 2n. Noticing that these constant terms can be obtained by putting x = 0 in the resulting determinants, we have
−a2n−1 =
2n
2n
det LA [{i}] =
i=1
i=2,i
2 n−1
det LA [{i}] +
is even
i=1,i
det LA [{i}].
(6)
is odd
During computing determinant of LA [{i}], we can obtain
⎧ ⎨ fi−1 f2n−i − 1 fi−2 f2n−i−1 ,2 ≤ i ≤ 2n − 2 and i is even;
det LA [{i}] =
⎩f
i−1
6 1 f f2n−i−1 ,3 ≤ i ≤ 2n − 1 and i is odd. 2n−i − 6 i−2
f
By (6), Claims 1 and 2, we have
−a2n−1 =
2n i=1
n−1
2 1 = n2 3 6 =
2 n−2
det LA [{i}] =
n−1
5 2 1 n 3 6
i=2,i
1 3
·
is even
i=3,i
det LA [{i}] + f2 n−1
is odd
1 + n2 ( )n−1 6 .
This completes the proof of Claim 3. Claim 4. a2n−2 =
2 n−1
det LA [{i}] + f2n−1 +
(7)
n2 (25n2 −7 ) 6n
Proof of Claim 4. Since the coefficient a2n−2 (= (−1 )2n−2 a2n−2 ) of x2 in (LA ) = det(xI2n − LA ) is the sum of the constant terms of the determinants obtained by deleting the ith, jth row and column for 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2n, we have
a2n−2 =
det LA [{i, j}]
1≤i< j≤2n
=
2 n−1
2n
fi−1 Fj−1−i F2n− j −
i=1 j=i+1
where
Fj−1−i =
f j−1−i , f j−1 −i ,
if i is even; if i is odd.
1 f Fj−1−i F2n− j−1 , 6 i−2
(8)
(9)
40
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
F2n− j =
f 2n− j ,
if j is even;
f
2n− j ,
F2n− j−1 =
(10)
if j is odd.
f2n− j−1 ,
if j is even;
f2 n− j−1 ,
(11)
if j is odd.
By the expression of LA , we know that det LA [{i, j}] will change according to the different choosability of i and j. Hence, by (8–11), we have
a2n−2 =
det LA [{i, j}]
1≤i< j≤2n
=
2 n−1
2n
fi−1 Fj−1−i F2n− j −
i=1 j≥i+1 2 n−1
= i=1,i
2n
is odd
j≥i+1, j
i=2,i
i=1,i
is even
j≥i+1, j
i=2,i
is even
2n
is odd
j≥i+1, j
2 n−1
+
2n
2 n−1
+
fi−1 f j−1 −i f 2n− j −
is odd
2 n−1
+
1 f Fj−1−i F2n− j−1 6 i−2
is even 2n
is even
j≥i+1, j
1 f f f 6 i−2 j−1−i 2n− j−1
fi−1 f j−1−i f2n− j −
1 f f j−1−i f2n− j−1 6 i−2
fi−1 f j−1 −i f 2n− j −
1 f f f2n− j−1 6 i−2 j−1−i
fi−1 f j−1−i f2 n− j −
1 f f j−1−i f2 n− j−1 . 6 i−2
is odd
(12)
For the sake of convenience, let 2 n−1
α= i=1,i
is odd
2n j≥i+1, j
is odd
2n
β= i=2,i
is even
j≥i+1, j
i=1,i
is odd
j≥i+1, j
2 n−1
δ= i=2,i
is even
is even
2n
is even 2n
j≥i+1, j
fi−1 f j−1 −i f 2n− j −
2n
2 n−1
γ =
1 f f f ; 6 i−2 j−1−i 2n− j−1
fi−1 f j−1−i f2n− j −
1 f f j−1−i f2n− j−1 ; 6 i−2
fi−1 f j−1 −i f 2n− j −
1 f f f2n− j−1 ; 6 i−2 j−1−i
fi−1 f j−1−i f2 n− j −
1 f f j−1−i f2 n− j−1 . 6 i−2
is odd
Now, we proceed by distinguishing the following four cases to compute a2n−2 . Case 1. Both i and j are odd. 2 n−3
α= i=1,i
=
is odd 2 n−3
2 n−1 j≥i+1, j
is odd
2 n−1
fi−1 f j−1 −i f 2n− j −
1 f f f 6 i−2 j−1−i 2n− j−1
i−1 √ j−i−1 √ 2n− j 1 6 1 6 1 (i + 1 − 1 ) √ · ( j − 1 − i + 1) √ · ( 2n − j + 1 ) √
2 2 6 6 is odd j≥i+1, j is odd i−2 √ j−1−i 2n− j−1 √ 1 6 1 6 1 1 − (i − 2 + 1 ) √ · ( j − 1 − i + 1) √ · ( 2n − j − 1 + 1 ) √ 6 2 2 6 6 6
6
i=1,i
2 n−3
= i=1,i
is odd
2 n−1 j≥i+1, j
is odd
3 i( j − i )(2n + 1 − j ) 2
1 √ 6
2n−2
1 3 − (i − 1 )( j − i )(2n − j ) 6 2
1 √ 6
2n−4
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
1 3 · − n(−2n4 − 5n3 + 5n + 2 ) 2 30
=
1
=
2
n4 −
1 2n−2
1 2 n 2
√ 6
1 2n−2 √ 6
41
−
1 3 1 1 · · n ( 2n4 − 5n3 + 5n − 2 ) √ 6 2 30 6
2n−4
.
(13)
Through similar calculations, we can obtain Case 2. Both i and j are even. 2 n−2
β=
is even
i=2,i
2n j≥i+1, j
fi−1 f j−1−i f2n− j −
is even
2
1 = · − n(−2n4 − 5n3 + 5n + 2 ) 3 30 =
2 9
n4 −
2 2 n 9
1 2n−2 √ 6
1 f f j−1−i f2n− j−1 6 i−2
1 2n−2 √ 6
1 2 1 1 − · · n ( 2n4 − 5n3 + 5n − 2 ) √ 6 3 30 6
2n−4
(14)
.
Case 3. i is odd and j is even. 2 n−1
γ=
is odd
i=1,i
2n j≥i+1, j
is even
1 1 = n(2n4 + 5n3 + 10n2 + 10n + 3 ) √ 30 6 =
1 3
n4 +
1 3 1 2 1 n + n + n 3 6 6
1 f f f2n− j−1 6 i−2 j−1−i
fi−1 f j−1 −i f 2n− j −
1 2n−2 √ 6
2n−2
1 1 1 − · n ( 2n4 − 5n3 + 5n − 2 ) √ 6 30 6
2n−4
(15)
.
Case 4. i is even and j is odd. 2 n−2
δ= i=2,i
= − =
is even
2 n−1 j≥i+1, j
is odd
fi−1 f j−1−i f2 n− j −
1 1 n(−2n4 − 5n3 + 5n + 2 ) √ 30 6
1
1 1 1 n + n2 − n − n3 3 6 6 3 4
2n−2
1 2n−2 √ 6
1 f f j−1−i f2 n− j−1 6 i−2
−
1 1 1 · n(2n4 − 5n3 + 10n2 − 10n + 3 ) √ 6 30 6
2n−4
(16)
.
By (12), we know
a2n−2 = α + β + γ + δ =
1 n2 (25n2 − 7 ) · . 3 6n
Thus Claim 4 holds. Substituting Claims 3 and 4 into (2) yields 2n 1 i=2
αi
=
25n2 − 7 , 30
as desired. Lemma 3. Let β 1 < β 2 ≤ ≤ β 2n be eigenvalues of LS . Then 2n 1 j=1
βj
=−
√ √ √ 7 2n(( 2 − 1 )2n − ( 2 + 1 )2n ) · √ . √ 4 ( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2
Proof. Let
(LS ) = x2n + k1 x2n−1 + · · · + k2n−2 x2 + k2n−1 x + k2n , where k2n = 0. Then β1 , β2 , . . . , β2n are the roots of the following equation
x2n−1 + k1 x2n−2 + · · · + k2n−2 x2 + k2n−1 x + k2n = 0.
(17)
42
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
Applying Vieta’s theorem, we have 2n 1 j=1
βj
=−
k2n−1 k2n−1 =− . k2n det LS
(18)
In order to determine k2n−1 and det LS , we first consider ith order principal submatrices, Wi and Wi , formed by the first i rows and columns of the following matrices LS and LS , respectively, i = 1, 2, . . . , 2n. Let
⎛
4 3 − √16
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
LS =
and
0 · · · 0 0
⎛
1 ⎜− √16 ⎜ ⎜ 0 ⎜ ⎜ · ⎜ ⎜ · ⎜ · ⎜ ⎝ 0 0
LS =
− √16 1 − √16 · · · 0 0
0 − √16
− √16 4 3 − √16
· · · 0 0
· · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 − √16 1 · · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
4 3
⎞
0 0 0 · · ·
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
4 3 − √16
2n×2n
⎞
0 0 0 · · · 1 − √16
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 4 3
.
2n×2n
Put wi = det Wi and wi = det Wi . We proceed by showing the following facts. Fact 1. For 1 ≤ i ≤ 2n,
⎧
i
i √ √ √ √ ⎪ 1+ 2 2+1 1− 2 2−1 ⎪ ⎪ + , √ √ ⎨ 2 2 6 6 wi = √
i √
i √ √ √ √ ⎪ ⎪ 3+ 6 2+1 3− 6 2−1 ⎪ ⎩ + , √ √ 3
6
3
6
if i is even; if i is odd.
Fact 2. For 1 ≤ i ≤ 2n,
⎧
i
i √ √ √ √ ⎪ 2+1 1− 2 2−1 ⎪ 1+ 2 ⎪ + , √ √ ⎨ 2 2 6 6 wi = √
i √
i √ √ √ √ ⎪ ⎪ 3+ 6 2+1 3− 6 2−1 ⎪ ⎩ + , √ √ 4
6
4
Proof of Fact 1. By a direct calculation, we have w1 = respect to its last row yields
wi =
⎧ ⎨ wi−1 − 1 wi−2 ,
if i is even;
⎩ 4w
if i is odd.
3
6
i−1
−
1 wi−2 , 6
6
4 3,
w2 =
7 6,
w3 =
For 1 ≤ i ≤ n, let si = w2i and for 1 ≤ i ≤ n − 1, let ti = w2i+1 . Then s1 =
⎧ ⎨ si =ti−1 − 1 si−1 , 6
⎩ t =4s − 1t . i i i−1 3
if i is even; if i is odd. 4 3
and w4 =
7 6 , t1
=
4 3
41 36 .
For 3 ≤ i ≤ 2n, expanding det Wi with
and, for i ≥ 2, we have
(19)
6
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
43
From the first equation in (19), one has ti−1 = si + 16 si−1 . Hence, ti = si+1 + 16 si . Substituting ti−1 and ti into the second 1 1 equation in (19) yields si+1 = si − 36 si−1 , i ≥ 2. Similarly, we can obtain ti+1 = ti − 36 ti−1 , i ≥ 2. Therefore, wi satisfies the recurrence relation
wi = wi−2 −
1 wi−4 , 36
( i ≥ 5 ),
w1 =
4 , 3
7 , 6
w2 =
Then the characteristic equation of (20) is x4 = x2 −
1 36 ,
w3 =
4 , 3
w4 =
whose roots are x1 =
41 . 36 √ 2+1 √ , 6
(20) x2 = −
√ 2+1 √ , 6
x3 =
√ 2−1 √ , 6
x4 = −
√ 2−1 √ . 6
The general solution of (20) is
√
2+1 √ 6
wi =
i
√ y1 +
2+1 − √ 6
i
√
2−1 √ 6
y2 +
i
√
2−1 − √ 6
y3 +
i y4 .
(21)
Together with the initial conditions in (20) yields the system of equations
⎧ √
√
2+1 2−1 4 ⎪ ⎪ ( y − y ) + ( y − y ) = , √ √ ⎪ 1 2 3 4 ⎪ 3 ⎪ 6 6 ⎪ ⎪ ⎪ √
2 √
2 ⎪ ⎪ 2+1 2−1 7 ⎪ ⎪ + ( y3 + y4 ) = , √ √ ⎪ ⎨ ( y1 + y2 ) 6 6 6 √
3 √
3 ⎪ 2+1 2−1 4 ⎪ ⎪ ( y1 − y2 ) + ( y3 − y4 ) = , √ √ ⎪ ⎪ 3 ⎪ 6 6 ⎪ ⎪ ⎪ √
4 √
4 ⎪ ⎪ 2−1 41 ⎪ ⎪ (y1 + y2 ) 2√+ 1 + (y3 + y4 ) = . √ ⎩ 6
36
6
The unique solution of this system of equations is y1 = √
√
√
2 6−2 3−3 2+3 . 12
√ √ √ 3 2+2 3+2 6+3 , 12
y2 =
√ √ √ 3 2−2 3−2 6+3 , 12
Thus Fact 1 follows by substituting y1 , y2 , y3 and y4 back into (21), as desired. The proof of Fact 2 is carried out in the same way as in the case of Fact 1. By exploiting the property of determinant, we may obtain
4 3 1 − √6 0 · det LS = · · 0 √1 6
− √16 1 − √16 · · · 0 0
0 − √16 4 3
· · · 0 0
0 0 − √16 · · · 0 0
4 3 − √16 0 0 0 1 − √6 1 − √16 0 0 1 4 1 0 √ √ − 6 − 6 0 3 · · · · · = · · · · · · · · · · 0 0 0 0 0 √1 0 0 0 0 6 4 1 3 − √6 0 0 1 − √6 1 − √16 0 4 0 √1 − √16 − 3 6 · · · · + · · · · · · · · 0 0 0 0 √1 0 0 0 6
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ··· 0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 0 · · · − √16 0 ··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 · · · − √16 1 √1 6
0 0 0 · · ·
4 3 − √16
0 0 0 · · · − √16 1
0 0 0 · · ·
4 3 − √16
0 0 0 · · ·
4 3 − √16
0 0 · · · 0 0
√1 6
2n×2n
y3 =
√ √ √ 2 3−3 2−2 6+3 , y4 12
=
44
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
4 3 − √16 0 0 0 ··· 0 0 0 1 − √6 1 − √16 0 0 ··· 0 0 0 4 0 − √16 − √16 0 ··· 0 0 0 3 · · · · · ··· · · · = · · · · ··· · · · · · · · · · ··· · · · 4 0 √1 √1 0 0 0 0 · · · − − 3 6 6 1 0 √ 0 0 0 0 ··· 0 − 6 1 0 − √16 0 0 0 ··· 0 0 0 0 1 − √16 0 0 ··· 0 0 0 4 0 − √1 − √16 0 ··· 0 0 0 3 6 · · · · · ··· · · · + · · · · ··· · · · · · · · · · ··· · · · 1 4 0 0 0 0 0 · · · − √6 − √16 3 1 √1 0 0 0 0 ··· 0 − √6 1 6 4 √1 3 − √16 0 0 0 ··· 0 0 6 1 − √6 1 − √16 0 0 ··· 0 0 0 1 4 0 √1 − √6 − 0 · · · 0 0 0 3 6 · · · · · ··· · · · + · · · · ··· · · · · · · · · · ··· · · · 4 0 √1 0 0 0 0 · · · − 0 3 6 √1 √1 0 0 0 0 · · · 0 − 0 6 6 2n×2n 1 n 1 n 1 = w 2n +
+
6
−
6
6
w2n−2 .
Together with Fact 1, we obtain the following fact immediately. Fact 3.
√
det LS =
√
( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 6n
.
(22)
Now we are ready to determine k2n−1 in (18). For convenience, suppose w0 = 1. Fact 4. Since −k2n−1 (= (−1 )2n−1 k2n−1 ) is the sum of all those principal minors of LS which have 2n − 1 rows and columns, for 1 ≤ i ≤ 2n, we have
−k2n−1 =
2n
2n
det LS [{i}] =
i=1
i=2,i
2 n−1
det LS [{i}] +
is even
i=1,i
det LS [{i}].
(23)
is odd
During computing determinant of LS [{i}], we can obtain
det LS [{i}] =
⎧ ⎨ wi−1 w2n−i − 1 wi−2 w2n−i−1 ,2 ≤ i ≤ 2n − 2 and i is even; ⎩w
i−1
6 1 w w2n−i−1 ,3 ≤ i ≤ 2n − 1 and i is odd. 2n−i − 6 i−2
w
By (23), Facts 1 and 2, we have
−k2n−1 =
2n
det LS [{i}] =
2 n−2
det LS [{i}] + w2n−1 +
2 n−1
det LS [{i}] + w2n−1
i=1
i=2,i is even i=3,i is odd √ √ √ √ √ √ − 2n(( 2 − 1 )2n − ( 2 + 1 )2n ) 3 2n(( 2 − 1 )2n − ( 2 + 1 )2n ) = − · 6n 4 6n √ √ √ 2n 2n 7 2n(( 2 − 1 ) − ( 2 + 1 ) ) =− · . 4 6n
Thus, Fact 4 follows immediately. In view of (18), Facts 3 and 4, Lemma 3 follows directly.
(24)
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
45
Table 1 Degree-Kirchhoff indices of hexagonal Möbius graph from HM1 to HM30 . G
DK(G)
G
DK(G)
G
DK(G)
HM1 HM2 HM3 HM4 HM5 HM6 HM7 HM8 HM9 HM10
23.50 155.33 438.50 919.29 1648.53 2676.91 4054.68 5831.92 8058.65 10784.87
HM11 HM12 HM13 HM14 HM15 HM16 HM17 HM18 HM19 HM20
14060.60 17935.82 22460.54 27684.75 33658.47 40431.68 48054.39 56576.60 66048.29 76519.49
HM21 HM22 HM23 HM24 HM25 HM26 HM27 HM28 HM29 HM30
88040.19 100660.39 114430.08 129399.27 145617.96 163136.15 182003.83 202271.01 223987.69 247203.86
Note that |E (HMn )| = 5n. Based on Lemmas 2 and 3, the following Theorems 2 and 3 are an immediate consequence of Lemma 1(a) and (b). Theorem 2. Let HMn denote a hexagonal Möbius graph of length n. Then
DK (HMn )
2n 1
=10n
+
βj √2n((√2 − 1 )2n − (√2 + 1 )2n )
7 25n2 − 7 =10n + − · √ √ 30 4 ( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 √ √ √ √ (50n3 + 105 2n2 − 14n )(3 + 2 2 )n + (50n3 − 105 2n2 − 14n )(3 − 2 2 )n + 100n3 − 28n = . √ √ 6((3 + 2 2 )n + (3 − 2 2 )n + 2 )
i=2
αi
2n 1 j=1
Theorem 3. Let HMn denote a hexagonal Möbius graph of length n. Then
Kc (HMn ) =
2n 1
2n 1
βj √2n((√2 − 1 )2n − (√2 + 1 )2n ) 7 25n2 − 7 = + − · √ . √ 30 4 ( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 i=2
αi
+
j=1
Degree-Kirchhoff indices of hexagonal Möbius graph from HM1 to HM30 are listed in Table 1. The explicit closed formula for the number of spanning trees of HMn is in the following. Theorem 4. Let HMn denote a hexagonal Möbius graph of length n. Then
√
√
τ (HMn ) = n((3 + 2 2 )n + (3 − 2 2 )n + 2 ). Proof. From the proof of Lemma 2, we know that α2 , α3 , . . . , α2n are the roots of the equation x2n−1 + a1 x2n−2 + · · · + a2n−2 x + a2n−1 = 0. Then we have 2n
αi = −a2n−1 .
i=2
By Claim 3, we have 2n
5 3
αi = n2
i=2
Similarly, 2n
βj =
j=1
1 n−1 6
.
√
√
( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 6n
.
Note that 4n i=1
di (HMn ) = 22n · 32n ,
|E (HMn )| = 5n.
46
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46 Table 2 The number of spanning trees of hexagonal Möbius graphs from HM1 to HM12 . G
τ (G)
G
τ (G)
G
τ (G)
G
τ (G)
HM1 HM2 HM3
8 72 600
HM4 HM5 HM6
4624 33640 235224
HM7 HM8 HM9
1,599,416 10,653,728 69,856,200
HM10 HM11 HM12
452,390,760 2,900,399,128 18,441,561,648
Together with Lemma 1(c), Theorem 4 follows immediately. The numbers of spanning trees of hexagonal Möbius graphs from HM1 to HM12 are listed in Table 2. Acknowledgment The authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions which lead to a considerably improved presentation. This work is supported by the Natural Science Foundation of Xinjiang Province (No. 2016D01C072). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, MCMillan, London, 1976. D.J. Klein, M. Randic´ , Resistance distance, J. Math. Chem. 12 (1993) 81–95. D.J. Klein, Resistance-distance sum rules, Croat. Chem. Acta. 75 (2002) 633–649. D.J. Klein, O. Ivanciuc, Graph cyclicity, excess conductance, and resistance deficit, J. Math. Chem. 30 (2001) 271–287. L. Lovász, Random walks on graphs: a survey, Bolyai Soc. Math. Stud., 2, János Bolyai Math. Soc., Budapest, 1996, pp. 353–397. F.R.K. Chung, in: Spectral Graph Theory, American Mathematical Society, Providence, RI, 1997. H.Y. Chen, F.J. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math. 155 (2007) 654–661. L.H. Feng, I. Gutman, G.H. Yu, Degree Kirchhoff index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 629–648. J. Huang, S.C. Li, On the normalised Laplacian spectrum, degree-Kirchhoff index and spanning trees of graphs, Bull. Aust. Math. Soc. 91 (3) (2015) 353–367. J. Huang, S.C. Li, L.Q. Sun, The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains, Discrete Appl. Math. 207 (2016a) 67–79. J. Huang, S.C. Li, X.C. Li, The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains, Appl. Math. Comput. 289 (2016b) 324–334. D.Q. Li, Y.P. Hou, The normalized Laplacian spectrum of quadrilateral graphs and its applications, Appl. Math. Comput. 297 (2017) 180–188. H.Y. Chen, L.W. Liao, The normalized Laplacian spectra of the corona and edge corona of two graphs, Linear Multilinear Algebra 65 (2017) 582–592. G.F. Wang, B.G. Xu, Kirchhoff index of hexagonal Möbius graphs, in: Proceedings of the 2011 International Conference on Multimedia Technology, 2011, pp. 5912–5915. G.F. Wang, R.X. Xie, Wiener index of hexagonal Möbius graph, J. Henan Univ. Sci. Technol. Nat. Sci. 33 (2012) 60–66. D.M. Walba, R.M. Richards, R.C. Haltiwanger, Total synthesis of the first molecular Möbius strip, J. Am. Chem. Soc. 104 (1982) 3219–3221. D.M. Walba, T.C. Homan, R.M. Richards, Topological stereochemistry.9. Synthesis and cutting in half of a molecular Möbius strip, New J. Chem. 17 (1993) 661–681. R. Herges, Topology in chemistry: designing Möbius molecules, Chem. Rev. 106 (2006) 4820–4842. M.C. Stantos, F. Alvarez, Spin current in the Möbius cyclacene belts, Chem. Phys. Lett. 471 (2009) 276–279. H.L. Xu, Z.R. Li, F.F. Wang, et al., What is the shape effect on the (hyper) polarizabilities? A comparison study on the Möbius, normal cyclacene, and linear nitrogen-substituted strip polyacenes, Chem. Phys. Lett. 454 (2008) 323–326. J.M. André, B. Champagne, E.A. Perpète, et al., Linear, cyclic, and Möbius strip polyacenes: the influence of the topology on the size-dependent HOMO-LUMO energy gap, Int. J. Quant. Chem. 84 (2001) 607–616. Y.J. Yang, H.P. Zhang, Kirchhoff index of linear hexagonal chains, Int. J. Quant. Chem. 108 (2008) 503–512. M. Levene, G. Loizou, Kemeny’s constant and the random surfer, Am. Math. Mon. 109 (2002) 741–745. S. Butler, Algebraic aspects of the normalized Laplacian, in: A. Beveridge, J. Griggs, L. Hogben, G. Musiker, e. P. Tetali (Eds.), Recent Trends in Combinatorics, The IMA Volumes in Mathematics and its Applications, IMA, 2016.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal Möbius graphs Xiaoling Ma a, Hong Bian b,∗ a b
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, PR China School of Mathematical Science, Xinjiang Normal University, Urumqi 830054, PR China
a r t i c l e
i n f o
a b s t r a c t
MSC: 05C99
Let HMn be a hexagonal Möbius graph of length n. In this paper, due to the normalized Laplacian polynomial decomposition theorem, we obtain that the normalized Laplacian spectrum of HMn consists of the eigenvalues of two symmetric quasi-tridiagonal matrices LA and LS of order 2n. Finally, by applying the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit closed formulas of the degree-Kirchhoff index and the number of spanning trees of HMn are given in terms of the index n.
Keywords: Hexagonal Möbius graph Normalized Laplacian Degree-Kirchhoff index Spanning trees
© 2019 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper, we only consider simple, finite and connected graphs. Let G = (V (G ), E (G )) be a simple undirected graph with vertex set V (G ) = {v1 , v2 , . . . , vn } and edge set E(G), where its order and size are |V (G )| = n(G ) = n and |E (G )| = m(G ) = m, respectively. We denote by di the degree of vi in G for 1 ≤ i ≤ n. The adjacency matrix A(G) of G is a n × n matrix whose (i, j)-entry is equal to 1 if vertices vi and v j are adjacent and 0 otherwise. The Laplacian matrix of G is L(G ) = D(G ) − A(G ), where D(G ) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix. Our notation is standard and taken mainly from [1]. The conventional distance between vertices vi and v j , denoted by dij , is the length of a shortest path between them. Klein and Randic´ [2] introduced a new distance function named resistance distance based on electrical network theory, the resistance distance between vertices vi and v j , denoted by rij , is defined to be the effective electrical resistance between them if each edge of G is replaced by a unit resistor [2]. Similar to the standard distance, the resistance distance is also intrinsic to the graph, not only with some fine purely mathematical properties, but also with a substantial potential for chemical applications (see [3,4] for details). It is well known that the resistance distance between two arbitrary vertices in an electronic network can be obtained in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the network. One famous resistance distance-based parameter called the Kirchhoff index, K(G), was given by K (G ) = i< j ri j ; see [2]. Later it is shown, independently, by Klein and Ivanciuc [4] and Lovász [5] that
K (G ) =
i< j
ri j = n
n 1 i=2
μi
,
where 0 = μ1 < μ2 ≤ · · · ≤ μn (n ≥ 2 ) are the eigenvalues of L(G). ∗
Corresponding author. E-mail address: [email protected] (H. Bian).
https://doi.org/10.1016/j.amc.2019.02.052 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
34
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
Fig. 1. The hexagonal Möbius graph.
In recent years, the normalized Laplacian, L(G ), which is consistent with the eigenvalues in spectral geometry and in random processes [6], has attracted more attention from researchers because many results which were only known for regular graphs can be generalized to all graphs. The normalized Laplacian is defined to be 1
1
1
1
L(G ) = I − D 2 (D−1 A )D− 2 = D− 2 LD− 2 with the convention that D(G )−1 (i, i ) = 0 if di = 0. Thus it is easy to know that the normalized Laplacian is a square matrix with rows and columns are indexed by vertices of G, and for any two vertices vi and v j of G, the (i, j)-entry of it is given by
(L(G ))i j =
⎧ ⎨1,
− √1 ,
if i = j and di = 0, if i = j and vi is adjacent to v j ,
0,
otherwise.
⎩
di d j
(1)
We denote the characteristic polynomial det(I − L(G )) of L(G ) by (L(G )). The roots of (L(G )) are known as the normalized Laplacian eigenvalues of G. The multiset of the normalized Laplacian eigenvalues of G is called the normalized Laplacian spectrum of G. The spectrum of L(G ) is denoted by S(G ) = {λ1 , λ2 , . . . , λn } with 0 = λ1 ≤ λ2 ≤ · · · ≤ λn . It is well known that G is connected if and only if λ2 > 0. Chen and Zhang [7] showed that the resistance distance can be expressed naturally in terms of the eigenvalues and eigenvectors of the normalized Laplacian and introduced another graph invariant, defined by DK (G ) = i< j di d j ri j , which is called the degree-Kirchhoff index (see [8,9]). It is closely related to the corresponding spectrum of the normalized Laplacian (see Lemma 1(a) in the next section). Like Kirchhoff index, degree-Kirchhoff index is a structured descriptor [7]. Unfortunately, it is difficult to compute resistant distance and degree-Kirchhoff index in a graph from their computational complexity. Hence, it is necessary to find closed-form formulae for the degree-Kirchhoff index. To this end, many researchers devote to find explicit closed-form formulae for some classes of graphs, such as linear hexagonal chains [10], linear polyomino chains [11], quadrilateral graphs [12], corona and edge corona of two graphs [13], and so on. Recently, according to the normalized Laplacian polynomial decomposition theorem, Huang et al. [10] gave an explicit closed form formula for degree-Kirchhoff index of linear hexagonal chains. In this paper, we concentrate our attention on hexagonal Möbius graphs and consider degree-Kirchhoff index of hexagonal Möbius graphs. Though we also make use of the decomposition of normalized Laplacian polynomial technique, the calculation process, especially the method of handling the matrix is very different to that in [10]. A hexagonal Möbius graph of length n, denoted by HMn , is defined as the graph obtained from P2 2P2n by removing the edges {(0, 2i + 1 ), (1, 2i + 1 )} with 0 ≤ i ≤ n − 1 and adding two new edges ((0, 0 )(1, 2n − 1 )) and ((1, 0 )(0, 2n − 1 )) depicted in Fig. 1(a), where P2 2P2n denotes the Cartesian product of P2 and P2n ,
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
35
i.e., the graph obtained from P2n and its copy by connecting all the corresponding vertices. Then it is routine to check that |V (HMn )| = 4n and |E (HMn )| = 5n. As an example, a hexagonal Möbius graph of length 5 is shown in Fig. 1(b). Hexagonal Möbius graphs are one type of molecular graphs embedded into the Möbius strip such that each face is a hexagon. Hence, hexagonal Möbius graphs have been of great interest and extensively studied. In 2011, Wang and Xu [14] considered the Kirchhoff index of hexagonal Möbius graphs. Later, the Wiener index of hexagonal Möbius graphs is determined in [15]. For more results on hexagonal Möbius graphs one may be referred to [16–21] and the references therein. The structure of the paper is as follows. In Section 2, we state again the decomposition theorem of the normalized Laplacian polynomial and obtain that the normalized Laplacian spectrum of HMn consists of the eigenvalues of two symmetric quasi-tridiagonal matrices LA and LS of order 2n. In Section 3, without calculating the eigenvalues of LA and LS , we obtain the sum of their reciprocals according to the relationship between roots and coefficients of the characteristic polynomial of LA and LS . In the last section, explicit closed-form formulas for the degree-Kirchhoff index and the number of spanning trees of HMn are derived in terms of the index n. 2. Some preliminary results An automorphism of G is a permutation π of V(G), which has the property that vi v j is an edge of G if and only if
π (vi )π (v j ) is an edge of G.
Suppose that G has an automorphism π , which can be written as the product of disjoint 1-cycles and transpositions, that
is
π = (10 )(20 ) . . . (m0 )(1, 1 )(2, 2 ) . . . (k, k ). Then we have |V (G )| = m + 2k. Let V0 = {10 , 20 , . . . , m0 }, V1 = {1, 2, . . . , k}, V2 = {1 , 2 , . . . , k }, then by a suitable arrangement of vertices in G, L(G ) can be written as the following block matrix
L (G ) =
LV0V0 LV1V0 LV2V0
LV0V1 LV1V1 LV2V1
LV0V2 V0 LV1V2 V1 LV2V2 V2
where LViV j is the submatrix formed by rows corresponding to vertices in Vi and columns corresponding to vertices in Vj for i, j = 0, 1, 2. Let
LA ( G ) =
√ 2LV0V1 , LV1V1 + LV1V2
√LV0V0 2LV1V0
LS (G ) = LV2V2 − LV1V2 . Huang et al. [10] obtained the following decomposition theorem of normalized Laplacian polynomial, which is similar to the decomposition theorem of the Laplacian polynomial in [22]. Theorem 1 [10]. Let L(G ), LA (G ), LS (G ) be defined as above. Then
(L(G )) = (LA (G )) · (LS (G )). In particular, if π can be written only as the product of disjoint transpositions (i.e. V0 = ∅), then
LA (G ) = LV1V1 + LV1V2 , LS (G ) = LV2V2 − LV1V2 , LV1V1 = LV2V2 . Kemeny’s constant Kc(G) of G is the expected number of steps required for the transition from a starting vertex vi to a destination vertex, which is chosen randomly according to a stationary distribution of unbiased random walks on G. Kemeny’s constant gives an interesting quantity for finite ergodic Markov chains, which is independent of the initial state of the Markov chain [23]. In terms of the normalized Laplacian spectrum of G, the special calculation formulae for the degree-Kirchhoff index, Kemeny’s constant and the number of spanning trees of graph G can be expressed as follows. Lemma 1. Let G be a connected graph with n vertices and m edges, and {0 = λ1 < λ2 ≤ · · · ≤ λn } be the spectrum on the normalized Laplacian L(G ) of G. Then each of the following holds. (a) [7] The degree-Kirchhoff index of G is
DK (G ) = 2m
n 1 i=2
λi
.
36
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
(b) [24] Kemeny’s constant of G is
Kc (G ) =
n 1 i=2
λi
.
(c) [6] The number τ (G) of spanning trees of G is n
di
i=1
n
λk = 2mτ (G ).
k=2
Apparently, from Lemma 1(a) and (b) the straightforward relation between the degree-Kirchhoff index and Kemeny’s constant is DK (G ) = 2mKc (G ).
3. The normalized Laplacian spectrum of the hexagonal Möbius graphs HMn In this section, we determine the normalized Laplacian eigenvalues of HMn according to Theorem 1. Given an n × n matrix M, we will use M[{i, j, . . . , k}] to denote the submatrix of M resulted from the deletion of the ith, jth,..., kth rows and columns. For convenience, we abbreviate LA (HMn ) and LS (HMn ) to LA and LS , respectively. We label the vertices of HMn as depicted in Fig. 1(a). Obviously, π = ((0, 0 )(1, 0 ))((0, 1 ), (1, 1 ))((0, 2 ), (1, 2 )) . . . ((0, 2n − 1 ), (1, 2n − 1 )) is an automorphism of HMn . Then V0 = ∅, V1 = {(0, 0 ), (0, 1 ), (0, 2 ), . . . , (0, 2n − 1 )} and V2 = {(1, 0 ), (1, 1 ), (1, 2 ), . . . , (1, 2n − 1 )}. Accordingly, LV1V1 (LV2V2 ) and LV1V2 are given as follows:
⎛
LV1V1
1 ⎜− √16 ⎜ ⎜ 0 ⎜ ⎜ · =⎜ ⎜ · ⎜ · ⎜ ⎝ 0 0
⎛
LV1V2
− 13 ⎜ 0 ⎜ 0 ⎜ ⎜ · =⎜ ⎜ · ⎜ ⎜ · ⎝ 0 − √16
− √16 1 − √16 · · · 0 0
0 0 0 · · · 0 0
0 − √16 1 · · · 0 0
0 0 − 13 · · · 0 0
0 0 0 · · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 0 · · · − 13 0
⎞
0 0 0 · · · 1 − √16
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
,
2n×2n
⎞
− √16 0 ⎟ ⎟ 0 ⎟ · ⎟ ⎟ · ⎟ ⎟ · ⎟ 0 ⎠ 0
.
2n×2n
Hence
LA = LV1V1 + LV1V2 ⎛ 2 − √16 3 1 ⎜− √6 1 ⎜ ⎜ 0 − √16 ⎜ ⎜ · · = ⎜ · ⎜ · ⎜ · · ⎜ ⎝ 0 0 − √16 0
0 − √16 2 3
· · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · − √16 0
0 0 0 · · ·
2 3 − √16
⎞
− √16 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
2n×2n
,
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
LS = LV1V1 − LV1V2 ⎛ 4 − √16 3 1 ⎜− √6 1 ⎜ ⎜ 0 − √16 ⎜ ⎜ · · = ⎜ · ⎜ · ⎜ · · ⎜ ⎝ 0 0 √1 0 6
0 − √16
0 0 − √16 · · · 0 0
4 3
· · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
√1 6
0 0 0 · · ·
⎞
0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
4 3 − √16
37
.
2n×2n
Suppose that the eigenvalues of LA and LS are, respectively, denoted by α i (i = 1, 2, . . . , 2n ) and β j ( j = 1, 2, . . . , 2n ) with α 1 ≤ α 2 ≤ ≤ α 2n , β 1 ≤ β 2 ≤ ≤ β 2n . Then, by Theorem 1, the spectrum of HMn is just {α1 , α2 , . . . , α2n , β1 , β2 , . . . , β2n }, and it is straightforward to check that α1 = 0, α i > 0 (i = 2, 3, . . . , 2n ), and β j > 0 ( j = 1, 2, . . . , 2n ). In the next section, we will compute some indices of HMn which are related to the normalized Laplacian spectrum of HMn .
4. Degree-Kirchhoff index and the number of spanning trees of hexagonal Möbius graphs In this section, we first provide a complete description of the sum of the normalized Laplacian eigenvalues’ reciprocals and the product of the normalized Laplacian eigenvalues, then apply these results to compute the degree-Kirchhoff index, Kemeny’s constant and the number of spanning trees of HMn . n 1 2 n 1 Based on the relationship between the roots and coefficients of (LA ) and (LS ), the formulas of 2i=2 j=1 β α and i
j
are derived in the next two lemmas, respectively. Lemma 2. Let 0 = α1 < α2 ≤ · · · ≤ α2n be eigenvalues of LA . Then 2n 1 i=2
αi
=
25n2 − 7 . 30
Proof. Let
(LA ) = x2n + a1 x2n−1 + · · · + a2n−2 x2 + a2n−1 x = x(x2n−1 + a1 x2n−2 + · · · + a2n−2 x + a2n−1 ). where a2n−1 = 0. Then α2 , α3 , . . . , α2n are the roots of the following equation
x2n−1 + a1 x2n−2 + · · · + a2n−2 x + a2n−1 = 0. By Vieta’s theorem, we have 2n 1 i=2
αi
=
a2n−2 . −a2n−1
(2)
Before giving a2n−2 and −a2n−1 , we first consider ith order principal submatrices, Fi and Fi , formed by the first i rows and columns of the following matrices LA and LA , respectively, i = 1, 2, . . . , 2n. Let
⎛
2 3 − √16
LA =
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 · · · 0 0
− √16 1 − √16 · · · 0 0
0 − √16 2 3
· · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 0 · · ·
2 3 − √16
⎞
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
2n×2n
38
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
and
⎛
− √16
1 ⎜− √16 ⎜ ⎜ 0 ⎜ ⎜ · LA = ⎜ ⎜ · ⎜ · ⎜ ⎝ 0 0
0 − √16 1 · · · 0 0
2 3 − √16
· · · 0 0
0 0 − √16 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · 0 0
0 0 0 · · · − √16 0
⎞
0 0 0 · · · 1 − √16
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 2 3
.
2n×2n
Put fi = det Fi and fi = det Fi . We proceed by showing the following claims. Claim 1. For 1 ≤ i ≤ 2n,
⎧ i ⎪ 1 ⎪ ⎪ ( i + 1 ) , √ ⎨ 6 fi = i √ ⎪ ⎪ 6 1 ⎪ ⎩ (i + 1 ) √ , 3
6
if i is even; if i is odd.
Claim 2. For 1 ≤ i ≤ 2n,
fi =
⎧ i ⎪ 1 ⎪ ⎪ ⎨ (i + 1 ) √ ,
if i is even;
i √ ⎪ ⎪ 6 1 ⎪ ⎩ (i + 1 ) √ ,
if i is odd.
6
2
6
Proof of Claim 1. It is straightforward to check that f 1 = respect to its last row yields
fi =
⎧ ⎨ fi−1 − 1 fi−2 ,
if i is even;
⎩2f
if i is odd.
3
6
i−1
−
1 fi−2 , 6
2 3,
f2 =
1 2,
f3 =
For 1 ≤ i ≤ n, let bi = f2i and for 1 ≤ i ≤ n − 1, let ci = f2i+1 . Then b1 =
1 2,
2 9
and f4 =
c1 =
2 9
5 36 .
For 3 ≤ i ≤ 2n, expanding det Fi with
and, for i ≥ 2, we have
⎧ ⎨ bi =ci−1 − 1 bi−1 , 6
(3)
⎩ c =2b − 1c . i i i−1 3
6
From the first equation in (3), one has ci−1 = bi + 16 bi−1 . Hence, ci = bi+1 + 16 bi . Substituting ci−1 and ci into the sec1 1 ond equation in (3) yields bi+1 = 13 bi − 36 bi−1 , i ≥ 2. In a similar way, we can obtain ci+1 = 13 ci − 36 ci−1 , i ≥ 2. Therefore, fi satisfies the recurrence relation
fi =
1 1 fi−2 − fi−4 , 3 36
( i ≥ 5 ),
f1 =
2 , 3
f2 =
Then the characteristic equation of (4) is x4 =
1 2 3x
−
1 , 2 1 36 ,
f3 =
2 , 9
f4 =
5 . 36
whose roots are x1 = x2 =
(4) √1 , 6
x3 = x4 = − √1 . The general solu6
tion of (4) is
1 f i = ( y1 i + y2 ) √ 6
i
1 + ( y3 i + y4 ) − √ 6
i .
Combining with the initial conditions in (4) yields the system of equations
(5)
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
39
⎧
1 1 2 ⎪ ⎪ ( y1 + y2 ) √ − ( y3 + y4 ) √ = , ⎪ ⎪ 3 ⎪ 6 6 ⎪ ⎪ ⎪
2 2 ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎪ ⎨ (2y1 + y2 ) √6 + (2y3 + y4 ) √6 = 2 , 3 3 ⎪ 1 1 2 ⎪ ⎪ ( 3 y + y ) − ( 3 y + y ) = , √ √ ⎪ 1 2 3 4 ⎪ 9 ⎪ 6 6 ⎪ ⎪ ⎪ 4 4 ⎪ ⎪ 1 1 5 ⎪ ⎪ + ( 4y3 + y4 ) √ = . ⎩ ( 4y1 + y2 ) √ 6
6
36
The unique solution of this system of equations is y1 = y2 =
1 2
+
√1 , 6
y3 = y4 =
1 2
−
√1 . 6
Thus Claim 1 follows by substituting
y1 , y2 , y3 and y4 back into (5), as desired. The proof of Claim 2 is carried out in the same way as in the case of Claim 1. Now we determine a2n−1 and a2n−2 in (2) (based on Claim 1 and Claim 2), respectively. For the sake of convenience, in the following discussion we assume f0 = 1. Claim 3. −a2n−1 =
5 2 1 n−1 3n (6)
Proof of Claim 3. It is not difficult to see that the coefficient −a2n−1 (= (−1 )2n−1 a2n−1 ) of x in (LA ) = det(xI2n − LA ) is the sum of the constant terms of the determinants obtained by deleting the ith row and column for 1 ≤ i ≤ 2n. Noticing that these constant terms can be obtained by putting x = 0 in the resulting determinants, we have
−a2n−1 =
2n
2n
det LA [{i}] =
i=1
i=2,i
2 n−1
det LA [{i}] +
is even
i=1,i
det LA [{i}].
(6)
is odd
During computing determinant of LA [{i}], we can obtain
⎧ ⎨ fi−1 f2n−i − 1 fi−2 f2n−i−1 ,2 ≤ i ≤ 2n − 2 and i is even;
det LA [{i}] =
⎩f
i−1
6 1 f f2n−i−1 ,3 ≤ i ≤ 2n − 1 and i is odd. 2n−i − 6 i−2
f
By (6), Claims 1 and 2, we have
−a2n−1 =
2n i=1
n−1
2 1 = n2 3 6 =
2 n−2
det LA [{i}] =
n−1
5 2 1 n 3 6
i=2,i
1 3
·
is even
i=3,i
det LA [{i}] + f2 n−1
is odd
1 + n2 ( )n−1 6 .
This completes the proof of Claim 3. Claim 4. a2n−2 =
2 n−1
det LA [{i}] + f2n−1 +
(7)
n2 (25n2 −7 ) 6n
Proof of Claim 4. Since the coefficient a2n−2 (= (−1 )2n−2 a2n−2 ) of x2 in (LA ) = det(xI2n − LA ) is the sum of the constant terms of the determinants obtained by deleting the ith, jth row and column for 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2n, we have
a2n−2 =
det LA [{i, j}]
1≤i< j≤2n
=
2 n−1
2n
fi−1 Fj−1−i F2n− j −
i=1 j=i+1
where
Fj−1−i =
f j−1−i , f j−1 −i ,
if i is even; if i is odd.
1 f Fj−1−i F2n− j−1 , 6 i−2
(8)
(9)
40
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
F2n− j =
f 2n− j ,
if j is even;
f
2n− j ,
F2n− j−1 =
(10)
if j is odd.
f2n− j−1 ,
if j is even;
f2 n− j−1 ,
(11)
if j is odd.
By the expression of LA , we know that det LA [{i, j}] will change according to the different choosability of i and j. Hence, by (8–11), we have
a2n−2 =
det LA [{i, j}]
1≤i< j≤2n
=
2 n−1
2n
fi−1 Fj−1−i F2n− j −
i=1 j≥i+1 2 n−1
= i=1,i
2n
is odd
j≥i+1, j
i=2,i
i=1,i
is even
j≥i+1, j
i=2,i
is even
2n
is odd
j≥i+1, j
2 n−1
+
2n
2 n−1
+
fi−1 f j−1 −i f 2n− j −
is odd
2 n−1
+
1 f Fj−1−i F2n− j−1 6 i−2
is even 2n
is even
j≥i+1, j
1 f f f 6 i−2 j−1−i 2n− j−1
fi−1 f j−1−i f2n− j −
1 f f j−1−i f2n− j−1 6 i−2
fi−1 f j−1 −i f 2n− j −
1 f f f2n− j−1 6 i−2 j−1−i
fi−1 f j−1−i f2 n− j −
1 f f j−1−i f2 n− j−1 . 6 i−2
is odd
(12)
For the sake of convenience, let 2 n−1
α= i=1,i
is odd
2n j≥i+1, j
is odd
2n
β= i=2,i
is even
j≥i+1, j
i=1,i
is odd
j≥i+1, j
2 n−1
δ= i=2,i
is even
is even
2n
is even 2n
j≥i+1, j
fi−1 f j−1 −i f 2n− j −
2n
2 n−1
γ =
1 f f f ; 6 i−2 j−1−i 2n− j−1
fi−1 f j−1−i f2n− j −
1 f f j−1−i f2n− j−1 ; 6 i−2
fi−1 f j−1 −i f 2n− j −
1 f f f2n− j−1 ; 6 i−2 j−1−i
fi−1 f j−1−i f2 n− j −
1 f f j−1−i f2 n− j−1 . 6 i−2
is odd
Now, we proceed by distinguishing the following four cases to compute a2n−2 . Case 1. Both i and j are odd. 2 n−3
α= i=1,i
=
is odd 2 n−3
2 n−1 j≥i+1, j
is odd
2 n−1
fi−1 f j−1 −i f 2n− j −
1 f f f 6 i−2 j−1−i 2n− j−1
i−1 √ j−i−1 √ 2n− j 1 6 1 6 1 (i + 1 − 1 ) √ · ( j − 1 − i + 1) √ · ( 2n − j + 1 ) √
2 2 6 6 is odd j≥i+1, j is odd i−2 √ j−1−i 2n− j−1 √ 1 6 1 6 1 1 − (i − 2 + 1 ) √ · ( j − 1 − i + 1) √ · ( 2n − j − 1 + 1 ) √ 6 2 2 6 6 6
6
i=1,i
2 n−3
= i=1,i
is odd
2 n−1 j≥i+1, j
is odd
3 i( j − i )(2n + 1 − j ) 2
1 √ 6
2n−2
1 3 − (i − 1 )( j − i )(2n − j ) 6 2
1 √ 6
2n−4
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
1 3 · − n(−2n4 − 5n3 + 5n + 2 ) 2 30
=
1
=
2
n4 −
1 2n−2
1 2 n 2
√ 6
1 2n−2 √ 6
41
−
1 3 1 1 · · n ( 2n4 − 5n3 + 5n − 2 ) √ 6 2 30 6
2n−4
.
(13)
Through similar calculations, we can obtain Case 2. Both i and j are even. 2 n−2
β=
is even
i=2,i
2n j≥i+1, j
fi−1 f j−1−i f2n− j −
is even
2
1 = · − n(−2n4 − 5n3 + 5n + 2 ) 3 30 =
2 9
n4 −
2 2 n 9
1 2n−2 √ 6
1 f f j−1−i f2n− j−1 6 i−2
1 2n−2 √ 6
1 2 1 1 − · · n ( 2n4 − 5n3 + 5n − 2 ) √ 6 3 30 6
2n−4
(14)
.
Case 3. i is odd and j is even. 2 n−1
γ=
is odd
i=1,i
2n j≥i+1, j
is even
1 1 = n(2n4 + 5n3 + 10n2 + 10n + 3 ) √ 30 6 =
1 3
n4 +
1 3 1 2 1 n + n + n 3 6 6
1 f f f2n− j−1 6 i−2 j−1−i
fi−1 f j−1 −i f 2n− j −
1 2n−2 √ 6
2n−2
1 1 1 − · n ( 2n4 − 5n3 + 5n − 2 ) √ 6 30 6
2n−4
(15)
.
Case 4. i is even and j is odd. 2 n−2
δ= i=2,i
= − =
is even
2 n−1 j≥i+1, j
is odd
fi−1 f j−1−i f2 n− j −
1 1 n(−2n4 − 5n3 + 5n + 2 ) √ 30 6
1
1 1 1 n + n2 − n − n3 3 6 6 3 4
2n−2
1 2n−2 √ 6
1 f f j−1−i f2 n− j−1 6 i−2
−
1 1 1 · n(2n4 − 5n3 + 10n2 − 10n + 3 ) √ 6 30 6
2n−4
(16)
.
By (12), we know
a2n−2 = α + β + γ + δ =
1 n2 (25n2 − 7 ) · . 3 6n
Thus Claim 4 holds. Substituting Claims 3 and 4 into (2) yields 2n 1 i=2
αi
=
25n2 − 7 , 30
as desired. Lemma 3. Let β 1 < β 2 ≤ ≤ β 2n be eigenvalues of LS . Then 2n 1 j=1
βj
=−
√ √ √ 7 2n(( 2 − 1 )2n − ( 2 + 1 )2n ) · √ . √ 4 ( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2
Proof. Let
(LS ) = x2n + k1 x2n−1 + · · · + k2n−2 x2 + k2n−1 x + k2n , where k2n = 0. Then β1 , β2 , . . . , β2n are the roots of the following equation
x2n−1 + k1 x2n−2 + · · · + k2n−2 x2 + k2n−1 x + k2n = 0.
(17)
42
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
Applying Vieta’s theorem, we have 2n 1 j=1
βj
=−
k2n−1 k2n−1 =− . k2n det LS
(18)
In order to determine k2n−1 and det LS , we first consider ith order principal submatrices, Wi and Wi , formed by the first i rows and columns of the following matrices LS and LS , respectively, i = 1, 2, . . . , 2n. Let
⎛
4 3 − √16
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
LS =
and
0 · · · 0 0
⎛
1 ⎜− √16 ⎜ ⎜ 0 ⎜ ⎜ · ⎜ ⎜ · ⎜ · ⎜ ⎝ 0 0
LS =
− √16 1 − √16 · · · 0 0
0 − √16
− √16 4 3 − √16
· · · 0 0
· · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 − √16 1 · · · 0 0
0 0 − √16 · · · 0 0
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
4 3
⎞
0 0 0 · · ·
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 1
4 3 − √16
2n×2n
⎞
0 0 0 · · · 1 − √16
0 0 ⎟ ⎟ 0 ⎟ ⎟ · ⎟ ⎟ · ⎟ · ⎟ ⎟ − √16 ⎠ 4 3
.
2n×2n
Put wi = det Wi and wi = det Wi . We proceed by showing the following facts. Fact 1. For 1 ≤ i ≤ 2n,
⎧
i
i √ √ √ √ ⎪ 1+ 2 2+1 1− 2 2−1 ⎪ ⎪ + , √ √ ⎨ 2 2 6 6 wi = √
i √
i √ √ √ √ ⎪ ⎪ 3+ 6 2+1 3− 6 2−1 ⎪ ⎩ + , √ √ 3
6
3
6
if i is even; if i is odd.
Fact 2. For 1 ≤ i ≤ 2n,
⎧
i
i √ √ √ √ ⎪ 2+1 1− 2 2−1 ⎪ 1+ 2 ⎪ + , √ √ ⎨ 2 2 6 6 wi = √
i √
i √ √ √ √ ⎪ ⎪ 3+ 6 2+1 3− 6 2−1 ⎪ ⎩ + , √ √ 4
6
4
Proof of Fact 1. By a direct calculation, we have w1 = respect to its last row yields
wi =
⎧ ⎨ wi−1 − 1 wi−2 ,
if i is even;
⎩ 4w
if i is odd.
3
6
i−1
−
1 wi−2 , 6
6
4 3,
w2 =
7 6,
w3 =
For 1 ≤ i ≤ n, let si = w2i and for 1 ≤ i ≤ n − 1, let ti = w2i+1 . Then s1 =
⎧ ⎨ si =ti−1 − 1 si−1 , 6
⎩ t =4s − 1t . i i i−1 3
if i is even; if i is odd. 4 3
and w4 =
7 6 , t1
=
4 3
41 36 .
For 3 ≤ i ≤ 2n, expanding det Wi with
and, for i ≥ 2, we have
(19)
6
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
43
From the first equation in (19), one has ti−1 = si + 16 si−1 . Hence, ti = si+1 + 16 si . Substituting ti−1 and ti into the second 1 1 equation in (19) yields si+1 = si − 36 si−1 , i ≥ 2. Similarly, we can obtain ti+1 = ti − 36 ti−1 , i ≥ 2. Therefore, wi satisfies the recurrence relation
wi = wi−2 −
1 wi−4 , 36
( i ≥ 5 ),
w1 =
4 , 3
7 , 6
w2 =
Then the characteristic equation of (20) is x4 = x2 −
1 36 ,
w3 =
4 , 3
w4 =
whose roots are x1 =
41 . 36 √ 2+1 √ , 6
(20) x2 = −
√ 2+1 √ , 6
x3 =
√ 2−1 √ , 6
x4 = −
√ 2−1 √ . 6
The general solution of (20) is
√
2+1 √ 6
wi =
i
√ y1 +
2+1 − √ 6
i
√
2−1 √ 6
y2 +
i
√
2−1 − √ 6
y3 +
i y4 .
(21)
Together with the initial conditions in (20) yields the system of equations
⎧ √
√
2+1 2−1 4 ⎪ ⎪ ( y − y ) + ( y − y ) = , √ √ ⎪ 1 2 3 4 ⎪ 3 ⎪ 6 6 ⎪ ⎪ ⎪ √
2 √
2 ⎪ ⎪ 2+1 2−1 7 ⎪ ⎪ + ( y3 + y4 ) = , √ √ ⎪ ⎨ ( y1 + y2 ) 6 6 6 √
3 √
3 ⎪ 2+1 2−1 4 ⎪ ⎪ ( y1 − y2 ) + ( y3 − y4 ) = , √ √ ⎪ ⎪ 3 ⎪ 6 6 ⎪ ⎪ ⎪ √
4 √
4 ⎪ ⎪ 2−1 41 ⎪ ⎪ (y1 + y2 ) 2√+ 1 + (y3 + y4 ) = . √ ⎩ 6
36
6
The unique solution of this system of equations is y1 = √
√
√
2 6−2 3−3 2+3 . 12
√ √ √ 3 2+2 3+2 6+3 , 12
y2 =
√ √ √ 3 2−2 3−2 6+3 , 12
Thus Fact 1 follows by substituting y1 , y2 , y3 and y4 back into (21), as desired. The proof of Fact 2 is carried out in the same way as in the case of Fact 1. By exploiting the property of determinant, we may obtain
4 3 1 − √6 0 · det LS = · · 0 √1 6
− √16 1 − √16 · · · 0 0
0 − √16 4 3
· · · 0 0
0 0 − √16 · · · 0 0
4 3 − √16 0 0 0 1 − √6 1 − √16 0 0 1 4 1 0 √ √ − 6 − 6 0 3 · · · · · = · · · · · · · · · · 0 0 0 0 0 √1 0 0 0 0 6 4 1 3 − √6 0 0 1 − √6 1 − √16 0 4 0 √1 − √16 − 3 6 · · · · + · · · · · · · · 0 0 0 0 √1 0 0 0 6
0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ··· 0 0 0 · · · 0 0
··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 0 · · · − √16 0 ··· ··· ··· ··· ··· ··· ··· ···
0 0 0 · · · − √16 0
0 0 · · · − √16 1 √1 6
0 0 0 · · ·
4 3 − √16
0 0 0 · · · − √16 1
0 0 0 · · ·
4 3 − √16
0 0 0 · · ·
4 3 − √16
0 0 · · · 0 0
√1 6
2n×2n
y3 =
√ √ √ 2 3−3 2−2 6+3 , y4 12
=
44
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
4 3 − √16 0 0 0 ··· 0 0 0 1 − √6 1 − √16 0 0 ··· 0 0 0 4 0 − √16 − √16 0 ··· 0 0 0 3 · · · · · ··· · · · = · · · · ··· · · · · · · · · · ··· · · · 4 0 √1 √1 0 0 0 0 · · · − − 3 6 6 1 0 √ 0 0 0 0 ··· 0 − 6 1 0 − √16 0 0 0 ··· 0 0 0 0 1 − √16 0 0 ··· 0 0 0 4 0 − √1 − √16 0 ··· 0 0 0 3 6 · · · · · ··· · · · + · · · · ··· · · · · · · · · · ··· · · · 1 4 0 0 0 0 0 · · · − √6 − √16 3 1 √1 0 0 0 0 ··· 0 − √6 1 6 4 √1 3 − √16 0 0 0 ··· 0 0 6 1 − √6 1 − √16 0 0 ··· 0 0 0 1 4 0 √1 − √6 − 0 · · · 0 0 0 3 6 · · · · · ··· · · · + · · · · ··· · · · · · · · · · ··· · · · 4 0 √1 0 0 0 0 · · · − 0 3 6 √1 √1 0 0 0 0 · · · 0 − 0 6 6 2n×2n 1 n 1 n 1 = w 2n +
+
6
−
6
6
w2n−2 .
Together with Fact 1, we obtain the following fact immediately. Fact 3.
√
det LS =
√
( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 6n
.
(22)
Now we are ready to determine k2n−1 in (18). For convenience, suppose w0 = 1. Fact 4. Since −k2n−1 (= (−1 )2n−1 k2n−1 ) is the sum of all those principal minors of LS which have 2n − 1 rows and columns, for 1 ≤ i ≤ 2n, we have
−k2n−1 =
2n
2n
det LS [{i}] =
i=1
i=2,i
2 n−1
det LS [{i}] +
is even
i=1,i
det LS [{i}].
(23)
is odd
During computing determinant of LS [{i}], we can obtain
det LS [{i}] =
⎧ ⎨ wi−1 w2n−i − 1 wi−2 w2n−i−1 ,2 ≤ i ≤ 2n − 2 and i is even; ⎩w
i−1
6 1 w w2n−i−1 ,3 ≤ i ≤ 2n − 1 and i is odd. 2n−i − 6 i−2
w
By (23), Facts 1 and 2, we have
−k2n−1 =
2n
det LS [{i}] =
2 n−2
det LS [{i}] + w2n−1 +
2 n−1
det LS [{i}] + w2n−1
i=1
i=2,i is even i=3,i is odd √ √ √ √ √ √ − 2n(( 2 − 1 )2n − ( 2 + 1 )2n ) 3 2n(( 2 − 1 )2n − ( 2 + 1 )2n ) = − · 6n 4 6n √ √ √ 2n 2n 7 2n(( 2 − 1 ) − ( 2 + 1 ) ) =− · . 4 6n
Thus, Fact 4 follows immediately. In view of (18), Facts 3 and 4, Lemma 3 follows directly.
(24)
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46
45
Table 1 Degree-Kirchhoff indices of hexagonal Möbius graph from HM1 to HM30 . G
DK(G)
G
DK(G)
G
DK(G)
HM1 HM2 HM3 HM4 HM5 HM6 HM7 HM8 HM9 HM10
23.50 155.33 438.50 919.29 1648.53 2676.91 4054.68 5831.92 8058.65 10784.87
HM11 HM12 HM13 HM14 HM15 HM16 HM17 HM18 HM19 HM20
14060.60 17935.82 22460.54 27684.75 33658.47 40431.68 48054.39 56576.60 66048.29 76519.49
HM21 HM22 HM23 HM24 HM25 HM26 HM27 HM28 HM29 HM30
88040.19 100660.39 114430.08 129399.27 145617.96 163136.15 182003.83 202271.01 223987.69 247203.86
Note that |E (HMn )| = 5n. Based on Lemmas 2 and 3, the following Theorems 2 and 3 are an immediate consequence of Lemma 1(a) and (b). Theorem 2. Let HMn denote a hexagonal Möbius graph of length n. Then
DK (HMn )
2n 1
=10n
+
βj √2n((√2 − 1 )2n − (√2 + 1 )2n )
7 25n2 − 7 =10n + − · √ √ 30 4 ( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 √ √ √ √ (50n3 + 105 2n2 − 14n )(3 + 2 2 )n + (50n3 − 105 2n2 − 14n )(3 − 2 2 )n + 100n3 − 28n = . √ √ 6((3 + 2 2 )n + (3 − 2 2 )n + 2 )
i=2
αi
2n 1 j=1
Theorem 3. Let HMn denote a hexagonal Möbius graph of length n. Then
Kc (HMn ) =
2n 1
2n 1
βj √2n((√2 − 1 )2n − (√2 + 1 )2n ) 7 25n2 − 7 = + − · √ . √ 30 4 ( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 i=2
αi
+
j=1
Degree-Kirchhoff indices of hexagonal Möbius graph from HM1 to HM30 are listed in Table 1. The explicit closed formula for the number of spanning trees of HMn is in the following. Theorem 4. Let HMn denote a hexagonal Möbius graph of length n. Then
√
√
τ (HMn ) = n((3 + 2 2 )n + (3 − 2 2 )n + 2 ). Proof. From the proof of Lemma 2, we know that α2 , α3 , . . . , α2n are the roots of the equation x2n−1 + a1 x2n−2 + · · · + a2n−2 x + a2n−1 = 0. Then we have 2n
αi = −a2n−1 .
i=2
By Claim 3, we have 2n
5 3
αi = n2
i=2
Similarly, 2n
βj =
j=1
1 n−1 6
.
√
√
( 2 + 1 ) 2n + ( 1 − 2 ) 2n + 2 6n
.
Note that 4n i=1
di (HMn ) = 22n · 32n ,
|E (HMn )| = 5n.
46
X. Ma and H. Bian / Applied Mathematics and Computation 355 (2019) 33–46 Table 2 The number of spanning trees of hexagonal Möbius graphs from HM1 to HM12 . G
τ (G)
G
τ (G)
G
τ (G)
G
τ (G)
HM1 HM2 HM3
8 72 600
HM4 HM5 HM6
4624 33640 235224
HM7 HM8 HM9
1,599,416 10,653,728 69,856,200
HM10 HM11 HM12
452,390,760 2,900,399,128 18,441,561,648
Together with Lemma 1(c), Theorem 4 follows immediately. The numbers of spanning trees of hexagonal Möbius graphs from HM1 to HM12 are listed in Table 2. Acknowledgment The authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions which lead to a considerably improved presentation. This work is supported by the Natural Science Foundation of Xinjiang Province (No. 2016D01C072). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, MCMillan, London, 1976. D.J. Klein, M. Randic´ , Resistance distance, J. Math. Chem. 12 (1993) 81–95. D.J. Klein, Resistance-distance sum rules, Croat. Chem. Acta. 75 (2002) 633–649. D.J. Klein, O. Ivanciuc, Graph cyclicity, excess conductance, and resistance deficit, J. Math. Chem. 30 (2001) 271–287. L. Lovász, Random walks on graphs: a survey, Bolyai Soc. Math. Stud., 2, János Bolyai Math. Soc., Budapest, 1996, pp. 353–397. F.R.K. Chung, in: Spectral Graph Theory, American Mathematical Society, Providence, RI, 1997. H.Y. Chen, F.J. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math. 155 (2007) 654–661. L.H. Feng, I. Gutman, G.H. Yu, Degree Kirchhoff index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 629–648. J. Huang, S.C. Li, On the normalised Laplacian spectrum, degree-Kirchhoff index and spanning trees of graphs, Bull. Aust. Math. Soc. 91 (3) (2015) 353–367. J. Huang, S.C. Li, L.Q. Sun, The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains, Discrete Appl. Math. 207 (2016a) 67–79. J. Huang, S.C. Li, X.C. Li, The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains, Appl. Math. Comput. 289 (2016b) 324–334. D.Q. Li, Y.P. Hou, The normalized Laplacian spectrum of quadrilateral graphs and its applications, Appl. Math. Comput. 297 (2017) 180–188. H.Y. Chen, L.W. Liao, The normalized Laplacian spectra of the corona and edge corona of two graphs, Linear Multilinear Algebra 65 (2017) 582–592. G.F. Wang, B.G. Xu, Kirchhoff index of hexagonal Möbius graphs, in: Proceedings of the 2011 International Conference on Multimedia Technology, 2011, pp. 5912–5915. G.F. Wang, R.X. Xie, Wiener index of hexagonal Möbius graph, J. Henan Univ. Sci. Technol. Nat. Sci. 33 (2012) 60–66. D.M. Walba, R.M. Richards, R.C. Haltiwanger, Total synthesis of the first molecular Möbius strip, J. Am. Chem. Soc. 104 (1982) 3219–3221. D.M. Walba, T.C. Homan, R.M. Richards, Topological stereochemistry.9. Synthesis and cutting in half of a molecular Möbius strip, New J. Chem. 17 (1993) 661–681. R. Herges, Topology in chemistry: designing Möbius molecules, Chem. Rev. 106 (2006) 4820–4842. M.C. Stantos, F. Alvarez, Spin current in the Möbius cyclacene belts, Chem. Phys. Lett. 471 (2009) 276–279. H.L. Xu, Z.R. Li, F.F. Wang, et al., What is the shape effect on the (hyper) polarizabilities? A comparison study on the Möbius, normal cyclacene, and linear nitrogen-substituted strip polyacenes, Chem. Phys. Lett. 454 (2008) 323–326. J.M. André, B. Champagne, E.A. Perpète, et al., Linear, cyclic, and Möbius strip polyacenes: the influence of the topology on the size-dependent HOMO-LUMO energy gap, Int. J. Quant. Chem. 84 (2001) 607–616. Y.J. Yang, H.P. Zhang, Kirchhoff index of linear hexagonal chains, Int. J. Quant. Chem. 108 (2008) 503–512. M. Levene, G. Loizou, Kemeny’s constant and the random surfer, Am. Math. Mon. 109 (2002) 741–745. S. Butler, Algebraic aspects of the normalized Laplacian, in: A. Beveridge, J. Griggs, L. Hogben, G. Musiker, e. P. Tetali (Eds.), Recent Trends in Combinatorics, The IMA Volumes in Mathematics and its Applications, IMA, 2016.