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A note on Bartholdi zeta function and graph invariants based on resistance distance
Discrete Mathematics 341 (2018) 786–792
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A note on Bartholdi zeta function and graph invariants based on resistance distance Deqiong Li a,b , Yaoping Hou a, * a
Key Laboratory of HPCSIP, Ministry of Education of China, College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China b College of Mathematics and Computation, Hunan Science and Technology University, Xiangtan, Hunan 411201, China
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Article history: Received 10 April 2017 Received in revised form 23 November 2017 Accepted 24 November 2017
Keywords: Bartholdi zeta function Resistance distance Complexity Kirchhoff index
a b s t r a c t Let G be a finite connected graph. In this note, we show that the complexity of G can be obtained from the partial derivatives at (1 − 1t , t) of a determinant in terms of the Bartholdi zeta function of G. Moreover, the second order partial derivatives at (1 − 1t , t) of this determinant can all be expressed as the linear combination of the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index of the graph G. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Let G = (V (G), E(G)) be a connected graph (possibly multiple edges and loops) of order n = |V (G)| and size m = |E(G)|. The resistance distance, a novel distance function defined on graphs, is put forward by Klein and Randić [7]. The resistance distance between a pair of vertices νi and νj of G, denoted by rij , is the net effective resistance measured across nodes νi and νj in the electrical network constructed from G by replacing every edge with a unit resistor. As an important branch of electrical circuit theory, the resistance distance has been much studied in Physics and Engineering. Consequently, analogous to distance-based graph invariants, various graph invariants based on the resistance distance have been defined and researched. The most famous graph invariant based on the resistance distance is the Kirchhoff index. The Kirchhoff index of a connected graph G is defined by Klein and Randić in [7], which is the sum of resistance distances between all pairs of vertices of G, that is K fG =
∑
rij .
1≤i
In recent years, two related descriptors that incorporate the degree di of the vertex νi ∈ V (G) are considered. The multiplicative degree-Kirchhoff index is one, which was introduced by Chen and Zhang in [4] and defined as follows: K fG∗ =
∑
di dj rij .
1≤i
*
Corresponding author. E-mail address: [email protected] (Y. Hou).
https://doi.org/10.1016/j.disc.2017.11.018 0012-365X/© 2017 Elsevier B.V. All rights reserved.
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
787
The other one named the additive degree-Kirchhoff index is defined [5] as follows:
∑
K fG+ =
(di + dj )rij .
1≤i
Three resistance distance-based graph invariants are currently researched in a good deal of literature. The zeta functions of finite graphs feature of Riemanns zeta functions and can be considered as an analogue of the Dedekind zeta functions of a number theory. The Ihara zeta function [15,16] of a finite graph G is defined to be the function of a complex number t with |t | sufficiently small, given by
ςG (t) =
∏
(1 − t |C | )−1 ,
[C ]
where [C ] runs over all equivalence classes of prime, reduced cycles of G and |C | denotes the length of C . We refer the reader to [16] for an in-depth treatment of the zeta functions of graphs. Bass [3] proved that the zeta function of a graph satisfies the following determinant formula:
ςG−1 (t) = (1 − t 2 )m−n det(In − tA + t 2 (D − In )),
(1)
where n and m denote the order and size of the graph G, and A and D are the adjacency and degree matrices of G, respectively. Furthermore, In is the identity matrix with order n. The complexity of a connected graph G, denoted by κ (G), is the number of spanning trees in G. By applying the idea of taking the derivative, the bridges between the complexity and the variations of graphs have been found in [6,10,13]. In particular, S. Northshield [13] found the complexity of a connected graph can be obtained as a derivative at t = 1 of a variation of the Ihara zeta function, that is f ′ (1) = 2(m − n)κ (G),
(2)
where f (t) = det(In − tA + t 2 (D − In )). Until recently, in [14] M. Somodi indicated that the second derivative of f (t) at the point 1 determines a linear combination of the Kirchhoff index, the multiplicative degree-Kirchhoff index, and the additive degree-Kirchhoff index of G, i.e., f ′′ (1) = 2(K fG∗ − 2K fG+ + 4K fG + 2mn − 2n2 + n)κ (G).
(3)
Motivated by above results, instead of considering a variation of the Ihara zeta function, we start from a variation of the Bartholdi zeta function, which is considered as a generalization of the determinant function f (t). 1 Let G be a connected graph. A path P = e1 e2 · · · ek is considered to have a bump at the vertex t(ej ) if ej+1 = e− j 1 means the reverse directed edge (1 ≤ j ≤ n − 1), where t(ej ) denotes the terminal vertex of the directed edge ej , and e− j to ej . The bump count bc(P) of a path P is the number of bumps in P. Furthermore, the cycle bump count cbc(C ) of a cycle 1 C = e1 e2 · · · ek is cbc(C ) = |{i|e− i+1 = ei , i = 1, 2, . . . , s}|, where es+1 = e1 . Bartholdi introduced a bi-variants zeta function,
called the Bartholdi zeta function [2],
ςG (u, t) =
∏
(1 − ucbc(C ) t |C | )−1 ,
[C ]
where [C ] runs over all equivalence classes of prime cycles of G, and u, t are two complex variables with |u|, |t | sufficiently small. If u = 0, then the Bartholdi zeta function of G is the Ihara zeta function ςG (t) of G. Bartholdi gave a determinant expression of the Bartholdi zeta function of a graph as follows: Theorem 1 ([2]). Let G be a connected graph with n vertices and m edges. Then the reciprocal of the Bartholdi zeta function of G is given by
ςG−1 (u, t) = (1 − (1 − u2 )t 2 )m−n det(In − tA + (1 − u)(D − (1 − u)In )t 2 ). Theorem 1 implies the determinant formula (1) of the Ihara zeta function for a graph if u = 0. As the generalization of the Ihara zeta function, the Bartholdi zeta function of a graph received more attention. For more details on the Bartholdi zeta function of a graph can refer to [8,9,11,12]. For a finite connected graph G with n vertices and m edges, let F (u, t) = det(In − tA + (1 − u)(D − (1 − u)In )t 2 )
(4)
be a variation of the Bartholdi zeta function of G. In this note, we give the relationships between its partial derivatives and the complexity and the Kirchhoff index, the multiplicative degree-Kirchhoff index, and the additive degree-Kirchhoff index of G.
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D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
2. Main results Let G be a finite connected graph. By L = D − A we denote the Laplacian matrix of G. By the famous Matrix-Tree Theorem [1], the complexity κ (G) of G can be expressed by
κ (G) = det L(i), where L(i) is the matrix obtained from L by deleting its ith row and ith column. Lemma 2 ([1]). The resistance distance of a connected graph G between any pairs of vertices νi and νj is rij =
det L(i, j) det L(i)
,
where LG (i, j) is the matrix obtained from the Laplacian matrix LG of G by deleting its i, jth rows and i, jth columns. According to the Matrix-Tree Theorem and Lemma 2, the Kirchhoff index of G can be represented as follows:
∑ det L(i, j) . κ (G)
K fG =
(5)
1≤i
Next, we will show the previous results (2) and (3) can be obtained from Theorems 3 and 4 as their special circumstances. Theorem 3. Let G be a connected graph with n vertices and m edges and F (u, t) be defined in (4). Then for any complex number t ̸ = 0, the complexity of G can be expressed as follows:
κ (G) =
1 2t n−2 (mt − n)
∂ F (u, t) |(1− 1 ,t) , t ∂t
(6)
∂ F (u, t) | 1 . ∂ u (1− t ,t)
(7)
and
κ (G) = −
1 2t n (mt − n)
In particular, we have [6]
∂ F (u, t) ∂ F (u, t) |(0,1) = − | = 2(m − n)κ (G). ∂t ∂ u (0,1) Proof. Let M(u, t) = In − tA + (1 − u)(D − (1 − u)In )t 2 . Then M(u, t) can be rewritten as M(u, t) = (1 − t 2 (1 − u)2 )In + t(t(1 − u) − 1)D + tL. So, we have F (u, t) = det M(u, t) = det((1 − t 2 (1 − u)2 )In + t(t(1 − u) − 1)D + tL). (t)
For i = 1, 2, . . . , n, let Mi (u, t) denote the matrix M(u, t) with each entry of the ith row replaced by the corresponding partial derivative with respect to t (the other rows remain unchanged). Thus n
∂ F (u, t) ∑ (t) = det Mi (u, t). ∂t
(8)
i=1
(t)
Note that the ith row of Mi (u, t) is (0, . . . , 0, (2t(1 − u) − 1)di − 2t(1 − u)2 , 0, . . . , 0) + Li ,
i−1
where Li denotes the ith row of L. In light of the Matrix-Tree Theorem det L(i) = κ (G) for each i = 1, 2, . . . , n and the fact det L = 0, for any complex number t ̸ = 0 we can see the expansion of the determinant (t)
det Mi (u, t)|(1− 1 ,t) = (di − t
2
)t n−1 det L(i) + t n−1 det L t = (tdi − 2)t n−2 κ (G).
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
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Thus, it is easy to obtain n ∑ ∂ F (u, t) (t) det Mi (u, t)|(1− 1 ,t) |(1− 1 ,t) = t t ∂t i=1
=
n ∑
(tdi − 2)t n−2 κ (G)
i=1
= 2(mt − n)t n−2 κ (G). This proves Eq. (6). And Eq. (7) can be proved with the same method.
□
S. Northshield [13] showed that lim ςG (t)(1 − t)m−n+1 = −
t →1−0
2n−m−1 (m − n)κ (G)
.
Then we can get the following result immediately: lim lim ςG−1 (u, t)(1 − t
√
1 − u2 )n−m−1 = −2m−n+1 (m − n)κ (G).
t →1−0 u→0
For convenience, let K fGc (t) = t 2 K fG∗ − 2tK fG+ + 4K fG for any complex number t ̸ = 0. By using the method similar to the proof of Theorem 3, we have the following theorem: Theorem 4. Let G be a connected graph with n vertices and m edges and F (u, t) be defined in (4). Then for any complex number t ̸ = 0, we have
∂ 2 F (u, t) |(1− 1 ,t) = 2t n−4 κ (G)[nt(2mt − 2n + 1) + KfGc (t)], t ∂t2
(9)
∂ 2 F (u, t) | 1 = 2t n−2 κ (G)[t(n + 1)(n − tm) − KfGc (t)], ∂ t ∂ u (1− t ,t)
(10)
∂ 2 F (u, t) |(1− 1 ,t) = 2t n κ (G)(KfGc (t) − nt). t ∂ u2
(11)
and
(tt)
Proof. For i = 1, 2, . . . , n, let Mii (u, t) be the matrix obtained by differentiating twice the ith row of M(u, t) with respect (tt) to t (the other rows remain unchanged), and the ith row of Mii (u, t) is (0, . . . , 0, 2(1 − u)(di − 1 + u), 0, . . . , 0).
i−1
(tt)
Besides, suppose that Mij (u, t) denotes the matrix M(u, t) with each entry of the ith row and jth row replaced by the corresponding partial derivative with respect to t (the other rows remain unchanged) for 1 ≤ i, j ≤ n(i ̸ = j). Then (0, . . . , 0, (2t(1 − u) − 1)di − 2t(1 − u)2 , 0, . . . , 0) + Li
i−1
and (0, . . . , 0, (2t(1 − u) − 1)dj − 2t(1 − u)2 , 0, . . . , 0) + Lj
j−1
(tt)
are the ith row and the jth row of Mij (u, t), respectively. By Eq. (8), we have n ∑ ∂ 2 F (u, t) ∑ (tt) (tt) = det M (u , t) + 2 det Mij (u, t). ii ∂ 2t 1≤i
i=1
On the one hand, it follows that n ∑
(tt)
det Mii (u, t)|(1− 1 ,t) =
n ∑ 2
t
i=1
i=1
t
(di −
1 t
)t n−1 det L(i) = 2t n−3 (2mt − n)κ (G).
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D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
On the other hand, by using the Matrix-Tree Theorem and Lemma 2, we get (tt)
det Mij (u, t)|(1− 1 ,t) t
2
2
2
= (di − )(dj − )t n−2 det L(i, j) + (di − )t n−2 det L(i) t
t
2
+ (dj − )t
n−2
t
t
det L(j) + t
n−2
det L
2
2
4
t
t
t
= t n−2 κ (G)[(di − )(dj − )rij + (di + dj − )]. Therefore,
∑
(tt)
1≤i
det Mij (u, t)|(1− 1 ,t) t
= t n−2 κ (G)[
∑
(di −
1≤i
2 t
)(dj −
2
= t n−2 κ (G)[KfG∗ − KfG+ + t
= t n−4 κ (G)[t 2 KfG∗ − 2tKfG+
4
2 t
∑
)rij +
1≤i
4 t
)]
2
(n − 1)(mt − n)] t + 4KfG + 2t(n − 1)(mt − n)] t2
KfG +
(di + dj −
= t n−4 κ (G)[KfGc (t) + 2t(n − 1)(mt − n)]. Hence, we have n ∑ ∑ ∂ 2 F (u, t) (tt) (tt) | = det Mii (u, t)|(1− 1 ,t) + 2 det Mij (u, t)|( t −1 ,t) 1 (1− t ,t) 2 t t ∂t 1≤i
i=1
= 2t n−3 (2mt − n)κ (G) + 2t n−4 κ (G)[KfGc (t) + 2t(n − 1)(mt − n)] = 2t n−4 κ (G)[KfGc (t) + nt(2mt − 2n + 1)]. (uu)
Eq. (11) can be obtained by using the same method as Eq. (9). Let Mii (u, t) be the matrix obtained by differentiating (uu) twice the ith row of M(u, t) with respect to u for i = 1, 2, . . . , n. So the ith row of Mii (u, t) is (0, . . . , 0, −2t 2 , 0, . . . , 0).
i−1
(uu)
Besides, for i, j = 1, 2, . . . , n (i ̸ = j), suppose that Mij (u, t) denotes the matrix M(u, t) with each entry of the ith row and jth row replaced by the corresponding partial derivative with respect to u. Then (0, . . . , 0, 2t 2 (1 − u) − t 2 di , 0, . . . , 0)
i−1
and (0, . . . , 0, 2t 2 (1 − u) − t 2 dj , 0, . . . , 0)
j−1
(uu)
are the ith row and the jth row of Mij It follows that n ∑
(uu)
det Mii
(u, t)|(1− 1 ,t) =
n ∑
t
i=1
(u, t), respectively.
−2t n+1 det L(i) = −2nt n+1 κ (G),
i=1
and
∑
(uu)
det Mij
1≤i
=
∑
(u, t)|(1− 1 ,t) t
(2t − t 2 di )(2t − t 2 dj )t n−2 det L(i, j)
1≤i
= t n κ (G)KfGc (t).
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
791
Hence, we obtain n ∑ ∑ ∂ 2 F (u, t) (uu) (uu) | = det Mii (u, t)|(1− 1 ,t) + 2 det Mij (u, t)|( t −1 ,t) 1 (1− t ,t) t t ∂ u2 1≤i
i=1
= −2nt n+1 κ (G) + 2t n κ (G)KfGc (t) = 2t n κ (G)(KfGc (t) − nt). (tu)
For 1 ≤ i, j ≤ n (i ̸ = j), denote Mii (u, t) the modification of M(u, t), obtained by replacing the ith row of M(u, t) with (tu) its second order mixed derivative with respect to t and u, and Mij (u, t) denotes the matrix obtained by differentiating the ith row of M(u, t) with respect to t and differentiating the jth row of M(u, t) with respect to u. So we have n ∑ ∂ det Mi(t) (u, t) ∂ 2 F (u, t) = ∂t∂u ∂u i=1
=
n ∑
(tu)
det Mii (u, t) +
∑
(tu)
(tu)
(det Mij (u, t) + det Mji (u, t)).
1≤i
i=1 (tu)
Since the ith row of Mii (u, t) is (0, . . . , 0, 4t(1 − u) − 2tdi , 0, . . . , 0), it follows that
i−1
n ∑
(tu)
det Mii (u, t)|(1− 1 ,t) =
n ∑
(4 − 2tdi )t n−1 det L(i) = 4t n−1 κ (G)(n − tm).
t
i=1
i=1
Moreover, (0, . . . , 0, (2t(1 − u) − 1)di − 2t(1 − u)2 , 0, . . . , 0) + Li
i−1
and (0, . . . , 0, 2t 2 (1 − u) − t 2 dj , 0, . . . , 0)
j−1
(tu)
are the ith row and the jth row of Mij (u, t), respectively. Thus we have (tu)
det Mij (u, t)|(1− 1 ,t) = (2t − t 2 dj )[t n−2 (di − t
2 t
) det L(i, j) + t n−2 det L(j)]
4
= t n−1 κ (G)(2di rij − rij + 2 − tdi dj rij + 2dj rij − tdj ). t
(tu)
(tu)
Noting the symmetry of Mij (u, t) and Mji (u, t), it is obvious that (tu)
det Mji (u, t)|(1− 1 ,t) = t n−1 κ (G)(2dj rij − t
4 t
rij + 2 − tdi dj rij + 2di rij − tdi ).
So, we get
∑
(tu)
(det Mij (1 −
1≤i
= t n−1 κ (G)
∑
1 t
1
, t) + det Mji(tu) (1 − , t)) t
8
[4(di + dj )rij − rij + 4 − 2tdi dj rij − t(di + dj )] t
1≤i
8
= t n−1 κ (G)[4KfG+ − KfG − 2tKfG∗ + 2n(n − 1) − 2tm(n − 1)] t
= 2t n−2 κ (G)[−KfGc (t) + t(n − 1)(n − tm)].
792
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
Therefore, we obtain
∂ 2 F (u, t) | 1 ∂ t ∂ u (1− t ,t) n ∑ ∑ 1 1 1 (tu) = det Mii (1 − , t) + [det Mij(tu) (1 − , t) + det Mji(tu) (1 − , t)] i=1
t
t
1≤i
= 4t κ (G)(n − tm) + 2t κ (G)[− = 2t n−2 κ (G)[t(n + 1)(n − tm) − KfGc (t)]. n−1
n−2
KfGc (t)
t
+ t(n − 1)(n − tm)] □
Specially, the following results can be obtained easily: Corollary 5. Let F (u, t) be a variation of the Bartholdi zeta function defined in (4). Then
∂ 2 F (u, t) |(0,1) = 2κ (G)(KfG∗ − 2KfG+ + 4KfG + 2nm − 2n2 + n). ∂t2 Also, we have
∂ 2 F (u, t) |(0,1) = 2κ (G)(KfG∗ − 2KfG+ + 4KfG − n), ∂ u2 and
∂ 2 F (u, t) | = 2κ (G)[−KfG∗ + 2KfG+ − 4KfG + (n + 1)(n − m)]. ∂ t ∂ u (0,1) According to limt →1−0
ςG−1 (t)(1−t)n−m−1 +2m−n+1 (m−n)κ (G) 1−t
= 2m−n (KfGc (1) + m2 − n2 + n)κ (G) in [14], it is obvious that
√
ςG−1 (u, t)(1 − t 1 − u2 )n−m−1 + 2m−n+1 (m − n)κ (G) √ t →1−0 u→0 1 − t 1 − u2 = 2m−n (KfGc (1) + m2 − n2 + n)κ (G) = 2m−n (KfG∗ − 2KfG+ + 4KfG + m2 − n2 + n)κ (G). lim lim
Acknowledgments This project was supported by the National Natural Science Foundation of China (No. 11571101) and the Construct Program of the Key Discipline in Hunan Province. The authors are grateful to the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
R.B. Bapat, Graph and Matrices, in: Universitext, Springer/Hindustan Book Agency, London/New Delhi, 2010. L. Bartholdi, Counting paths in graphs, Enseign. Math. 45 (1999) 83–131. H. Bass, The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992) 717–797. H.Y. Chen, F.J. Zhang, Resistance distance and normalized Laplacian spectrum, Discrete Appl. Math. 155 (2007) 654–661. I. Gutman, L. Feng, G. Yu, Degree resistance distance of unicyclic graphs, Trans. Combin. 1 (2) (2012) 27–40. D. Kim, Y.S. Kwon, J. Lee, The weighted complexity and the determinant functions of graphs, Linear Algebra Appl. 433 (2010) 348–355. D.J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81–95. H. Mizuno, I. Sato, Bartholdi zeta funtion of digraphs, European J. Combin. 24 (8) (2003) 947–954. H. Mizuno, I. Sato, Bartholdi zeta funtion of graphs coverings, J. Combin. Theory Ser. B 89 (1) (2003) 27–41. H. Mizuno, I. Sato, On the weighted complexity of a regular covering of a graph, J. Combin. Theory Ser. B 89 (2003) 17–26. H. Mizuno, I. Sato, Bartholdi zeta funtion of graphs of line graphs and middle graphs ofgraph coverings, Discrete Math. 292 (2005) 143–157. H. Mizuno, I. Sato, Bartholdi zeta funtion of some graphs, Discrete Math. 306 (2006) 220–230. S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408–410. M. Somodi, On the Ihara zeta function and resistance distance-based indices, Linear Algebra Appl. 513 (2017) 201–209. H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996) 124–165. A. Terras, Zeta Funtions of Graphs: A Stroll Through the Garden, Cambridge U. Press, 2011.
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Note
A note on Bartholdi zeta function and graph invariants based on resistance distance Deqiong Li a,b , Yaoping Hou a, * a
Key Laboratory of HPCSIP, Ministry of Education of China, College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China b College of Mathematics and Computation, Hunan Science and Technology University, Xiangtan, Hunan 411201, China
article
info
Article history: Received 10 April 2017 Received in revised form 23 November 2017 Accepted 24 November 2017
Keywords: Bartholdi zeta function Resistance distance Complexity Kirchhoff index
a b s t r a c t Let G be a finite connected graph. In this note, we show that the complexity of G can be obtained from the partial derivatives at (1 − 1t , t) of a determinant in terms of the Bartholdi zeta function of G. Moreover, the second order partial derivatives at (1 − 1t , t) of this determinant can all be expressed as the linear combination of the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index of the graph G. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Let G = (V (G), E(G)) be a connected graph (possibly multiple edges and loops) of order n = |V (G)| and size m = |E(G)|. The resistance distance, a novel distance function defined on graphs, is put forward by Klein and Randić [7]. The resistance distance between a pair of vertices νi and νj of G, denoted by rij , is the net effective resistance measured across nodes νi and νj in the electrical network constructed from G by replacing every edge with a unit resistor. As an important branch of electrical circuit theory, the resistance distance has been much studied in Physics and Engineering. Consequently, analogous to distance-based graph invariants, various graph invariants based on the resistance distance have been defined and researched. The most famous graph invariant based on the resistance distance is the Kirchhoff index. The Kirchhoff index of a connected graph G is defined by Klein and Randić in [7], which is the sum of resistance distances between all pairs of vertices of G, that is K fG =
∑
rij .
1≤i
In recent years, two related descriptors that incorporate the degree di of the vertex νi ∈ V (G) are considered. The multiplicative degree-Kirchhoff index is one, which was introduced by Chen and Zhang in [4] and defined as follows: K fG∗ =
∑
di dj rij .
1≤i
*
Corresponding author. E-mail address: [email protected] (Y. Hou).
https://doi.org/10.1016/j.disc.2017.11.018 0012-365X/© 2017 Elsevier B.V. All rights reserved.
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
787
The other one named the additive degree-Kirchhoff index is defined [5] as follows:
∑
K fG+ =
(di + dj )rij .
1≤i
Three resistance distance-based graph invariants are currently researched in a good deal of literature. The zeta functions of finite graphs feature of Riemanns zeta functions and can be considered as an analogue of the Dedekind zeta functions of a number theory. The Ihara zeta function [15,16] of a finite graph G is defined to be the function of a complex number t with |t | sufficiently small, given by
ςG (t) =
∏
(1 − t |C | )−1 ,
[C ]
where [C ] runs over all equivalence classes of prime, reduced cycles of G and |C | denotes the length of C . We refer the reader to [16] for an in-depth treatment of the zeta functions of graphs. Bass [3] proved that the zeta function of a graph satisfies the following determinant formula:
ςG−1 (t) = (1 − t 2 )m−n det(In − tA + t 2 (D − In )),
(1)
where n and m denote the order and size of the graph G, and A and D are the adjacency and degree matrices of G, respectively. Furthermore, In is the identity matrix with order n. The complexity of a connected graph G, denoted by κ (G), is the number of spanning trees in G. By applying the idea of taking the derivative, the bridges between the complexity and the variations of graphs have been found in [6,10,13]. In particular, S. Northshield [13] found the complexity of a connected graph can be obtained as a derivative at t = 1 of a variation of the Ihara zeta function, that is f ′ (1) = 2(m − n)κ (G),
(2)
where f (t) = det(In − tA + t 2 (D − In )). Until recently, in [14] M. Somodi indicated that the second derivative of f (t) at the point 1 determines a linear combination of the Kirchhoff index, the multiplicative degree-Kirchhoff index, and the additive degree-Kirchhoff index of G, i.e., f ′′ (1) = 2(K fG∗ − 2K fG+ + 4K fG + 2mn − 2n2 + n)κ (G).
(3)
Motivated by above results, instead of considering a variation of the Ihara zeta function, we start from a variation of the Bartholdi zeta function, which is considered as a generalization of the determinant function f (t). 1 Let G be a connected graph. A path P = e1 e2 · · · ek is considered to have a bump at the vertex t(ej ) if ej+1 = e− j 1 means the reverse directed edge (1 ≤ j ≤ n − 1), where t(ej ) denotes the terminal vertex of the directed edge ej , and e− j to ej . The bump count bc(P) of a path P is the number of bumps in P. Furthermore, the cycle bump count cbc(C ) of a cycle 1 C = e1 e2 · · · ek is cbc(C ) = |{i|e− i+1 = ei , i = 1, 2, . . . , s}|, where es+1 = e1 . Bartholdi introduced a bi-variants zeta function,
called the Bartholdi zeta function [2],
ςG (u, t) =
∏
(1 − ucbc(C ) t |C | )−1 ,
[C ]
where [C ] runs over all equivalence classes of prime cycles of G, and u, t are two complex variables with |u|, |t | sufficiently small. If u = 0, then the Bartholdi zeta function of G is the Ihara zeta function ςG (t) of G. Bartholdi gave a determinant expression of the Bartholdi zeta function of a graph as follows: Theorem 1 ([2]). Let G be a connected graph with n vertices and m edges. Then the reciprocal of the Bartholdi zeta function of G is given by
ςG−1 (u, t) = (1 − (1 − u2 )t 2 )m−n det(In − tA + (1 − u)(D − (1 − u)In )t 2 ). Theorem 1 implies the determinant formula (1) of the Ihara zeta function for a graph if u = 0. As the generalization of the Ihara zeta function, the Bartholdi zeta function of a graph received more attention. For more details on the Bartholdi zeta function of a graph can refer to [8,9,11,12]. For a finite connected graph G with n vertices and m edges, let F (u, t) = det(In − tA + (1 − u)(D − (1 − u)In )t 2 )
(4)
be a variation of the Bartholdi zeta function of G. In this note, we give the relationships between its partial derivatives and the complexity and the Kirchhoff index, the multiplicative degree-Kirchhoff index, and the additive degree-Kirchhoff index of G.
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D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
2. Main results Let G be a finite connected graph. By L = D − A we denote the Laplacian matrix of G. By the famous Matrix-Tree Theorem [1], the complexity κ (G) of G can be expressed by
κ (G) = det L(i), where L(i) is the matrix obtained from L by deleting its ith row and ith column. Lemma 2 ([1]). The resistance distance of a connected graph G between any pairs of vertices νi and νj is rij =
det L(i, j) det L(i)
,
where LG (i, j) is the matrix obtained from the Laplacian matrix LG of G by deleting its i, jth rows and i, jth columns. According to the Matrix-Tree Theorem and Lemma 2, the Kirchhoff index of G can be represented as follows:
∑ det L(i, j) . κ (G)
K fG =
(5)
1≤i
Next, we will show the previous results (2) and (3) can be obtained from Theorems 3 and 4 as their special circumstances. Theorem 3. Let G be a connected graph with n vertices and m edges and F (u, t) be defined in (4). Then for any complex number t ̸ = 0, the complexity of G can be expressed as follows:
κ (G) =
1 2t n−2 (mt − n)
∂ F (u, t) |(1− 1 ,t) , t ∂t
(6)
∂ F (u, t) | 1 . ∂ u (1− t ,t)
(7)
and
κ (G) = −
1 2t n (mt − n)
In particular, we have [6]
∂ F (u, t) ∂ F (u, t) |(0,1) = − | = 2(m − n)κ (G). ∂t ∂ u (0,1) Proof. Let M(u, t) = In − tA + (1 − u)(D − (1 − u)In )t 2 . Then M(u, t) can be rewritten as M(u, t) = (1 − t 2 (1 − u)2 )In + t(t(1 − u) − 1)D + tL. So, we have F (u, t) = det M(u, t) = det((1 − t 2 (1 − u)2 )In + t(t(1 − u) − 1)D + tL). (t)
For i = 1, 2, . . . , n, let Mi (u, t) denote the matrix M(u, t) with each entry of the ith row replaced by the corresponding partial derivative with respect to t (the other rows remain unchanged). Thus n
∂ F (u, t) ∑ (t) = det Mi (u, t). ∂t
(8)
i=1
(t)
Note that the ith row of Mi (u, t) is (0, . . . , 0, (2t(1 − u) − 1)di − 2t(1 − u)2 , 0, . . . , 0) + Li ,
i−1
where Li denotes the ith row of L. In light of the Matrix-Tree Theorem det L(i) = κ (G) for each i = 1, 2, . . . , n and the fact det L = 0, for any complex number t ̸ = 0 we can see the expansion of the determinant (t)
det Mi (u, t)|(1− 1 ,t) = (di − t
2
)t n−1 det L(i) + t n−1 det L t = (tdi − 2)t n−2 κ (G).
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
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Thus, it is easy to obtain n ∑ ∂ F (u, t) (t) det Mi (u, t)|(1− 1 ,t) |(1− 1 ,t) = t t ∂t i=1
=
n ∑
(tdi − 2)t n−2 κ (G)
i=1
= 2(mt − n)t n−2 κ (G). This proves Eq. (6). And Eq. (7) can be proved with the same method.
□
S. Northshield [13] showed that lim ςG (t)(1 − t)m−n+1 = −
t →1−0
2n−m−1 (m − n)κ (G)
.
Then we can get the following result immediately: lim lim ςG−1 (u, t)(1 − t
√
1 − u2 )n−m−1 = −2m−n+1 (m − n)κ (G).
t →1−0 u→0
For convenience, let K fGc (t) = t 2 K fG∗ − 2tK fG+ + 4K fG for any complex number t ̸ = 0. By using the method similar to the proof of Theorem 3, we have the following theorem: Theorem 4. Let G be a connected graph with n vertices and m edges and F (u, t) be defined in (4). Then for any complex number t ̸ = 0, we have
∂ 2 F (u, t) |(1− 1 ,t) = 2t n−4 κ (G)[nt(2mt − 2n + 1) + KfGc (t)], t ∂t2
(9)
∂ 2 F (u, t) | 1 = 2t n−2 κ (G)[t(n + 1)(n − tm) − KfGc (t)], ∂ t ∂ u (1− t ,t)
(10)
∂ 2 F (u, t) |(1− 1 ,t) = 2t n κ (G)(KfGc (t) − nt). t ∂ u2
(11)
and
(tt)
Proof. For i = 1, 2, . . . , n, let Mii (u, t) be the matrix obtained by differentiating twice the ith row of M(u, t) with respect (tt) to t (the other rows remain unchanged), and the ith row of Mii (u, t) is (0, . . . , 0, 2(1 − u)(di − 1 + u), 0, . . . , 0).
i−1
(tt)
Besides, suppose that Mij (u, t) denotes the matrix M(u, t) with each entry of the ith row and jth row replaced by the corresponding partial derivative with respect to t (the other rows remain unchanged) for 1 ≤ i, j ≤ n(i ̸ = j). Then (0, . . . , 0, (2t(1 − u) − 1)di − 2t(1 − u)2 , 0, . . . , 0) + Li
i−1
and (0, . . . , 0, (2t(1 − u) − 1)dj − 2t(1 − u)2 , 0, . . . , 0) + Lj
j−1
(tt)
are the ith row and the jth row of Mij (u, t), respectively. By Eq. (8), we have n ∑ ∂ 2 F (u, t) ∑ (tt) (tt) = det M (u , t) + 2 det Mij (u, t). ii ∂ 2t 1≤i
i=1
On the one hand, it follows that n ∑
(tt)
det Mii (u, t)|(1− 1 ,t) =
n ∑ 2
t
i=1
i=1
t
(di −
1 t
)t n−1 det L(i) = 2t n−3 (2mt − n)κ (G).
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D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
On the other hand, by using the Matrix-Tree Theorem and Lemma 2, we get (tt)
det Mij (u, t)|(1− 1 ,t) t
2
2
2
= (di − )(dj − )t n−2 det L(i, j) + (di − )t n−2 det L(i) t
t
2
+ (dj − )t
n−2
t
t
det L(j) + t
n−2
det L
2
2
4
t
t
t
= t n−2 κ (G)[(di − )(dj − )rij + (di + dj − )]. Therefore,
∑
(tt)
1≤i
det Mij (u, t)|(1− 1 ,t) t
= t n−2 κ (G)[
∑
(di −
1≤i
2 t
)(dj −
2
= t n−2 κ (G)[KfG∗ − KfG+ + t
= t n−4 κ (G)[t 2 KfG∗ − 2tKfG+
4
2 t
∑
)rij +
1≤i
4 t
)]
2
(n − 1)(mt − n)] t + 4KfG + 2t(n − 1)(mt − n)] t2
KfG +
(di + dj −
= t n−4 κ (G)[KfGc (t) + 2t(n − 1)(mt − n)]. Hence, we have n ∑ ∑ ∂ 2 F (u, t) (tt) (tt) | = det Mii (u, t)|(1− 1 ,t) + 2 det Mij (u, t)|( t −1 ,t) 1 (1− t ,t) 2 t t ∂t 1≤i
i=1
= 2t n−3 (2mt − n)κ (G) + 2t n−4 κ (G)[KfGc (t) + 2t(n − 1)(mt − n)] = 2t n−4 κ (G)[KfGc (t) + nt(2mt − 2n + 1)]. (uu)
Eq. (11) can be obtained by using the same method as Eq. (9). Let Mii (u, t) be the matrix obtained by differentiating (uu) twice the ith row of M(u, t) with respect to u for i = 1, 2, . . . , n. So the ith row of Mii (u, t) is (0, . . . , 0, −2t 2 , 0, . . . , 0).
i−1
(uu)
Besides, for i, j = 1, 2, . . . , n (i ̸ = j), suppose that Mij (u, t) denotes the matrix M(u, t) with each entry of the ith row and jth row replaced by the corresponding partial derivative with respect to u. Then (0, . . . , 0, 2t 2 (1 − u) − t 2 di , 0, . . . , 0)
i−1
and (0, . . . , 0, 2t 2 (1 − u) − t 2 dj , 0, . . . , 0)
j−1
(uu)
are the ith row and the jth row of Mij It follows that n ∑
(uu)
det Mii
(u, t)|(1− 1 ,t) =
n ∑
t
i=1
(u, t), respectively.
−2t n+1 det L(i) = −2nt n+1 κ (G),
i=1
and
∑
(uu)
det Mij
1≤i
=
∑
(u, t)|(1− 1 ,t) t
(2t − t 2 di )(2t − t 2 dj )t n−2 det L(i, j)
1≤i
= t n κ (G)KfGc (t).
D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
791
Hence, we obtain n ∑ ∑ ∂ 2 F (u, t) (uu) (uu) | = det Mii (u, t)|(1− 1 ,t) + 2 det Mij (u, t)|( t −1 ,t) 1 (1− t ,t) t t ∂ u2 1≤i
i=1
= −2nt n+1 κ (G) + 2t n κ (G)KfGc (t) = 2t n κ (G)(KfGc (t) − nt). (tu)
For 1 ≤ i, j ≤ n (i ̸ = j), denote Mii (u, t) the modification of M(u, t), obtained by replacing the ith row of M(u, t) with (tu) its second order mixed derivative with respect to t and u, and Mij (u, t) denotes the matrix obtained by differentiating the ith row of M(u, t) with respect to t and differentiating the jth row of M(u, t) with respect to u. So we have n ∑ ∂ det Mi(t) (u, t) ∂ 2 F (u, t) = ∂t∂u ∂u i=1
=
n ∑
(tu)
det Mii (u, t) +
∑
(tu)
(tu)
(det Mij (u, t) + det Mji (u, t)).
1≤i
i=1 (tu)
Since the ith row of Mii (u, t) is (0, . . . , 0, 4t(1 − u) − 2tdi , 0, . . . , 0), it follows that
i−1
n ∑
(tu)
det Mii (u, t)|(1− 1 ,t) =
n ∑
(4 − 2tdi )t n−1 det L(i) = 4t n−1 κ (G)(n − tm).
t
i=1
i=1
Moreover, (0, . . . , 0, (2t(1 − u) − 1)di − 2t(1 − u)2 , 0, . . . , 0) + Li
i−1
and (0, . . . , 0, 2t 2 (1 − u) − t 2 dj , 0, . . . , 0)
j−1
(tu)
are the ith row and the jth row of Mij (u, t), respectively. Thus we have (tu)
det Mij (u, t)|(1− 1 ,t) = (2t − t 2 dj )[t n−2 (di − t
2 t
) det L(i, j) + t n−2 det L(j)]
4
= t n−1 κ (G)(2di rij − rij + 2 − tdi dj rij + 2dj rij − tdj ). t
(tu)
(tu)
Noting the symmetry of Mij (u, t) and Mji (u, t), it is obvious that (tu)
det Mji (u, t)|(1− 1 ,t) = t n−1 κ (G)(2dj rij − t
4 t
rij + 2 − tdi dj rij + 2di rij − tdi ).
So, we get
∑
(tu)
(det Mij (1 −
1≤i
= t n−1 κ (G)
∑
1 t
1
, t) + det Mji(tu) (1 − , t)) t
8
[4(di + dj )rij − rij + 4 − 2tdi dj rij − t(di + dj )] t
1≤i
8
= t n−1 κ (G)[4KfG+ − KfG − 2tKfG∗ + 2n(n − 1) − 2tm(n − 1)] t
= 2t n−2 κ (G)[−KfGc (t) + t(n − 1)(n − tm)].
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D. Li, Y. Hou / Discrete Mathematics 341 (2018) 786–792
Therefore, we obtain
∂ 2 F (u, t) | 1 ∂ t ∂ u (1− t ,t) n ∑ ∑ 1 1 1 (tu) = det Mii (1 − , t) + [det Mij(tu) (1 − , t) + det Mji(tu) (1 − , t)] i=1
t
t
1≤i
= 4t κ (G)(n − tm) + 2t κ (G)[− = 2t n−2 κ (G)[t(n + 1)(n − tm) − KfGc (t)]. n−1
n−2
KfGc (t)
t
+ t(n − 1)(n − tm)] □
Specially, the following results can be obtained easily: Corollary 5. Let F (u, t) be a variation of the Bartholdi zeta function defined in (4). Then
∂ 2 F (u, t) |(0,1) = 2κ (G)(KfG∗ − 2KfG+ + 4KfG + 2nm − 2n2 + n). ∂t2 Also, we have
∂ 2 F (u, t) |(0,1) = 2κ (G)(KfG∗ − 2KfG+ + 4KfG − n), ∂ u2 and
∂ 2 F (u, t) | = 2κ (G)[−KfG∗ + 2KfG+ − 4KfG + (n + 1)(n − m)]. ∂ t ∂ u (0,1) According to limt →1−0
ςG−1 (t)(1−t)n−m−1 +2m−n+1 (m−n)κ (G) 1−t
= 2m−n (KfGc (1) + m2 − n2 + n)κ (G) in [14], it is obvious that
√
ςG−1 (u, t)(1 − t 1 − u2 )n−m−1 + 2m−n+1 (m − n)κ (G) √ t →1−0 u→0 1 − t 1 − u2 = 2m−n (KfGc (1) + m2 − n2 + n)κ (G) = 2m−n (KfG∗ − 2KfG+ + 4KfG + m2 − n2 + n)κ (G). lim lim
Acknowledgments This project was supported by the National Natural Science Foundation of China (No. 11571101) and the Construct Program of the Key Discipline in Hunan Province. The authors are grateful to the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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