- Email: [email protected]
The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs
Discrete Applied Mathematics 161 (2013) 3063–3071
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs✩ Weizhong Wang a,b,∗ , Dong Yang b , Yanfeng Luo b a
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, PR China
b
Department of Mathematics, Lanzhou University, Lanzhou 730000, PR China
article
info
Article history: Received 15 June 2012 Received in revised form 3 June 2013 Accepted 6 June 2013 Available online 29 June 2013 Keywords: Resistance distance Kirchhoff index Laplacian spectrum Bound
abstract Let R(G) be the graph obtained from G by adding a new vertex corresponding to each edge of G and by joining each new vertex to the end vertices of the corresponding edge, and Q (G) be the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. In this paper, we determine the Laplacian polynomials of R(G) and Q (G) of a regular graph G; on the other hand, we derive formulae and lower bounds of the Kirchhoff index of these graphs. Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved.
1. Introduction Let G be a simple graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E (G) = {e1 , e2 , . . . , em }. Denote by A(G) and D(G) the adjacency matrix and the diagonal matrix with the vertex degrees of G on the diagonal, respectively. The matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G, for details see [22,23]. Denote by PG (λ) and µG (λ) the adjacent characteristic polynomial det(λIn − A(G)) and the Laplacian characteristic polynomial det(λIn − L(G)) of G, respectively. The multiset of eigenvalues of A(G) (resp., L(G)) are called the adjacency (resp., Laplacian) spectrum of G. Since A(G) and L(G) are all real symmetric matrices, their eigenvalues are real numbers. So we can assume that λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G) (resp., µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn (G)) are the adjacency (resp., Laplacian) eigenvalues of G. Clearly, all Laplacian eigenvalues of G are non-negative. If the graph G is connected, then µi (G) > 0 for i = 1, 2, . . . , n − 1 and µn (G) = 0 [13,14,22]. In what follows, the Laplacian spectrum of G is denoted by S (G) = {µ1 , µ2 , . . . , µn }. Suppose G is a simple graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E (G) = {e1 , e2 , . . . , em }. Define two graph operators R and Q (see the definitions in p. 63 in [5]) as follows. Let R(G) = (V (R(G)), E (R(G))) be the graph obtained from G by adding a new vertex e′ corresponding to each edge e = (a, b) of G and by joining each new vertex e′ to the end vertices a and b of the corresponding edge e = (a, b), i.e., R(G) is obtained from G by ‘‘changing each edge e = (a, b) of G into a triangle ae′ b’’. Thus, V (R(G)) = V (G) ∪ {e′ | e ∈ E (G)} and E (R(G)) = E (G) ∪ {(vi , e′ ), (vj , e′ ) | e = (vi , vj ) ∈ E (G)} (see Fig. 1(a) and (b) for example). Let Q (G) = (V (Q (G)), E (Q (G))) be the graph obtained from G by inserting a new vertex e′i into every edge ei of G and by joining by edges those pairs of these new vertices e′i and e′j which lie on adjacent edges ei and ej of G, i, j = 1, 2, . . . , m. Denote by vi1 and vi2 the end-vertices of edge ei of G. Then V (Q (G)) = V (G) ∪ {e′i | ei ∈ E (G),
✩ This research was partially supported by the National Natural Science Foundation of China (No. 10971086 and No. 11201201).
∗ Corresponding author at: Department of mathematics, Lanzhou Jiaotong University, Lanzhou 730070, PR China. Tel.: +86 09314938625; fax: +86 09318834765. E-mail addresses: [email protected], [email protected] (W. Wang). 0166-218X/$ – see front matter Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.06.010
3064
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
a
b
c
Fig. 1. (a) The graph G. (b) The graph R(G). (c) The graph Q (G).
i = 1, 2, . . . , m} and E (Q (G)) = {(vi1 , e′i ), (vi2 , e′i )|i = 1, 2, . . . , m} ∪ {(e′i , ej ′ ) | ei and ej are adjacent edges of G} (see Fig. 1(a) and (c) for example). In some graph theory problems it is necessary to compute the spectrum (resp., Laplacian spectrum) of a compound graph, obtained from some operations from some simple graphs. In [5] there exist many relations connecting the spectrum (resp., Laplacian spectrum) of a compound graph with spectra (resp., Laplacian spectra [17,24]) of graphs from which that graph is derived. However, it is worth considering the corresponding problems of graphs derived from a single graph, such as line graph, subdivision graph, total graph, R(G), Q (G), etc. In [17], the Laplacian polynomial of line graph, the subdivision graph and total graph of the regular graph are obtained. In the present work, on one hand, we determine the Laplacian polynomials of R(G) and Q (G); on the other hand, for these graphs, we compute the topological indices based on the concept of resistance distance. Let G = (V , E ) be a connected graph with vertex set V = {v1 , v2 , . . . , vn }. The (ordinary) distance between vertices vi and vj , denoted by dij , is the length of a shortest path connecting them. The original index based on distance in a graph G is the Wiener index W (G) [28], which counts the sum of distances between pairs of vertices in G. In 1993, Klein and Randić [20] defined a new distance function named resistance distance framed in terms of electrical network theory. However, this concept has been discussed much earlier (1949) for another purpose by Foster [10] as recently pointed out by Palacios [27]. The resistance distance between vertices vi and vj of G, denoted by rij , is defined to be the effective resistance between nodes vi and vj as computed with Ohm’s law when all the edges of G are considered to be unit resistors. As an analogue to the Wiener index, the sum Kf (G) = i
det
M P
N Q
= detMdet(Q − PM −1 N ).
The line graph of a graph G, denoted by L(G), is the graph whose vertices correspond to the edges of G, with two vertices of L(G) being adjacent if and only if the corresponding edges of G share a common vertex. The following result is well known [5].
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3065
Lemma 2.2 ([5]). Let G be an r-regular graph with n vertices and m edges. Then PL(G) (λ) = (λ + 2)m−n PG (λ − r + 2). A.K. Kel’Mans [17] characterized the Laplacian polynomial of L(G) by means of the Laplacian polynomial of G. Lemma 2.3 ([17]). Let G be an r-regular simple graph with n vertices and m edges. Then
µL(G) (λ) = (λ − 2r )m−n µG (λ). Lemma 2.4 ([24]). Let G be the disjoint union of graphs G1 , G2 , . . . , Gk . Then
µG (λ) =
k
µGi (λ).
i=1
Now we consider the case for R(G). For a regular graph G, the next theorem gives the representation of the Laplacian polynomial of R(G) by means of the characteristic polynomial and the Laplacian polynomial of G, respectively. Theorem 2.5. Let G be an r-regular (r ̸= 2) graph with n vertices and m edges. Then (i) µR(G) (λ) = (λ − 2)m−n (3 − λ)n PG
(ii) µR(G) (λ) = (λ − 2)
m−n
(λ − 3) µG n
λ2 −2(r +1)λ+3r 3−λ λ(λ−r −2) λ−3
.
.
Proof. (i) Let A and I be the adjacency matrix and the incidence matrix of G, respectively. (In what follows, the unit matrix of order p will be denoted by Ip , and it should not be confused with the incidence matrix.) Then A(R(G)) =
0m I
IT A
2Im −I
−I T . 2rIn − A
D(R(G)) =
and
2Im 0
0 2rIn
.
We thus have L(R(G)) =
It follows that
µR(G) (λ) = det
(λ − 2)Im
IT . (λ − 2r )In + A
I
(1)
Case 1. If r = 1, then G is a disjoint union of copies K2 and hence R(G) is a disjoint union of copies of K3 , i.e., G = 2n K2 and R(G) = 2n K3 , where n is even. It follows from Lemma 2.4, PK2 (λ) = λ2 − 1 and µK3 (λ) = λ3 − 6λ2 + 9λ that Theorem 2.5(i) holds. Case 2. If r > 2 and λ = 2, then m = nr > n. It follows from (1) that 2
0 µR(G) (λ) = det m I
IT . (2 − 2r )In + A
It is easy to see the first m lines of
0m I
IT
(2 − 2r )In + A
are linearly dependent since m > n. Thus µR(G) (λ) = 0.
If λ ̸= 2, it follows from Lemma 2.1 that
µR(G) (λ) = det
(λ − 2)Im I
IT (λ − 2r )In + A
Im T = (λ − 2)m det (λ − 2r )In + A − I I λ−2 1 = (λ − 2)m det (λ − 2r )In + A − (A + rIn ) λ−2 (λ − 2r )(λ − 2) − r m−n n In − A = (λ − 2) (3 − λ) det 3−λ 2 λ − 2(r + 1)λ + 3r = (λ − 2)m−n (3 − λ)n PG . 3−λ
(2)
3066
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
(ii) Recall that L(G) = rIn − A(G). It follows from (2) that
(λ − 2r )(λ − 2) − r In + A λ−3 2 λ − ( r + 2)λ In − (rIn − A) = (λ − 2)m−n (λ − 3)n det λ−3 λ(λ − r − 2) = (λ − 2)m−n (λ − 3)n µG . λ−3
µR(G) (λ) = (λ − 2)m−n (λ − 3)n det
The proof is thus completed.
Remark 1. Theorem 2.5(ii) has appeared in [8]. Here we just obtain the result in a different way. Lastly, we consider the case for Q (G). For a regular graph G, the next theorem gives the representation of the Laplacian polynomial of Q (G) by means of the characteristic polynomial and the Laplacian polynomial of G, respectively. Theorem 2.6. Let G be an r-regular graph with n vertices and m edges. Then (i) µQ (G) (λ) = (r + 1 − λ)n
m−n
λ2 −2(r +1)λ+r 2 +r . r +1−λ 2 m−n )λ+2r (r +1) −λ) λ)n λ −(3r +r2−λ µG λ(rr++12−λ .
λ2 −(3r +2)λ+2r 2 +2r λ−r
(ii) µQ (G) (λ) = (−1)m (r + 1 −
PG
Proof. (i) Let B and I be the adjacency matrix of L(G) and incidence matrix of G, respectively. Then A(Q (G)) =
B I
IT 0n
and D(Q (G)) =
2rIm rIn
.
We thus have L(Q (G)) =
2rIm − B −I
−I T
rIn
.
Hence
µQ (G) (λ) = det
I
= det
(λ − 2r )Im + B (λ − r )In
IT
(λ − r )In
I (λ − 2r )Im + B
IT
In I = (λ − r )n det (λ − 2r )Im + B − I T λ−r 1 n = (λ − r ) det (λ − 2r )Im + B − (B + 2Im ) λ−r 2 λ − 3r λ + 2r 2 − 2 n−m m = (λ − r ) (r + 1 − λ) det Im − B r +1−λ 2 λ − 3r λ + 2r 2 − 2 = (λ − r )n−m (r + 1 − λ)m PL(G) . r +1−λ
(3)
It follows from Lemma 2.2 that
PL(G)
λ2 − 3r λ + 2r 2 − 2 r +1−λ
=
λ2 − (3r + 2)λ + 2r 2 + 2r r +1−λ
m−n
PG
λ2 − 2(r + 1)λ + r 2 + r r +1−λ
Combine (3) with (4), we get
µQ (G) (λ) = (r + 1 − λ)n
λ2 − (3r + 2)λ + 2r 2 + 2r λ−r
m−n
PG
λ2 − 2(r + 1)λ + r 2 + r r +1−λ
.
.
(4)
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3067
(ii) Note that G is r-regular and the line graph L(G) of G is also a (2r − 2)-regular graph with m = that D(L(G)) = (2r − 2)Im and L(L(G)) = D(L(G)) − B. By (3) it follows that
nr 2
vertices. Notice also
λ2 − 3r λ + 2r 2 − 2 Im − B r +1−λ 2 −λ + r λ + 2λ = (−1)n (r − λ)n−m (r + 1 − λ)m det Im − ((2r − 2)Im − B) r +1−λ λ(r + 2 − λ) n n−m m . = (−1) (r − λ) (r + 1 − λ) µL(G) r +1−λ
µQ (G) (λ) = (λ − r )n−m (r + 1 − λ)m det
(5)
By Lemma 2.3 we deduce that
µL(G)
λ(r + 2 − λ) r +1−λ
=
−λ2 + (3r + 2)λ − 2r (r + 1) r +1−λ
m−n
µG
λ(r + 2 − λ) . r +1−λ
(6)
By virtue of (5) and (6) we have in fact already established the statement in the theorem. Thus the theorem is proved. 3. The Kirchhoff index of R (G ) and Q (G ) In this section, we will explore the Kirchhoff index of the R(G) and Q (G) of a regular graph G. Gutman and Mohar [15] and Zhu et al. [35] established a nice relationship between the Laplacian spectrum and Kirchhoff index as follows: Lemma 3.1 ([15,35]). Let G be a connected graph with n ≥ 2 vertices. Then Kf (G) = n
n −1 1 . µ i i=1
Denote by δi the degree of vertex vi ∈ V (G). Zhou and Trinajstić [34] proved that Lemma 3.2 ([34]). Let G be a connected graph with n ≥ 2 vertices. Then Kf (G) ≥ −1 + (n − 1)
1 δ v ∈V (G) i i
with equality attained if and only if G = Kn or G = Kt ,n−t for 1 ≤ t ≤ ⌊ 2n ⌋. The following lemma will be used in the remainder of this paper. Lemma 3.3 ([12]). Let G be a connected graph with n ≥ 2 vertices and µG (λ) = λn + a1 λn−1 + · · · + an−1 λ. Then Kf (G) n
=−
an − 2 an − 1
(an−2 = 1 whenever n = 2).
(7)
Let Kn denote the complete graph with n vertices. If G = K1 , we have nothing to discuss. So we assume that G ̸= K1 throughout this paper. The following Theorem 3.4 shows that Kf (R(G)) can be completely determined by the Kirchhoff index, the number of vertices and the vertex degree of regular graph G. Theorem 3.4. Let G be a connected r-regular (r ̸= 2) graph with n vertices. Then Kf (R(G)) =
(r + 2)2 6
Kf (G) +
(n2 − n)(r + 2) 6
+
n2 (r 2 − 4) 8
n
+ .
(8)
2
Proof. Suppose first that r = 1, i.e. G ∼ = K2 . Then R(K2 ) = K3 . Since S (K3 ) = {3, 3, 0}. It follows from Lemma 3.1 that Kf (K3 ) = 2. It is easy to check that the Eq. (8) holds in this case. Suppose now that r > 2. Let µG (λ) = λn + a1 λn−1 + · · · + an−1 λ. It follows from Theorem 2.5(ii) that
µR(G) (λ) = (λ − 2)
m−n
(λ − 3)
n
λ
n
λ−r −2 λ−3
n
+ · · · + an − 2 λ
2
λ−r −2 λ−3
2
= (λ − 2)m−n λn (λ − r − 2)n + · · · + an−2 λ2 (λ − r − 2)2 (λ − 3)n−2 + an−1 λ (λ − r − 2) (λ − 3)n−1 ,
λ−r −2 + an − 1 λ λ−3
3068
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
where λ ̸= 3 and m =
> n. So the coefficient of λ2 in µR(G) (λ) is
rn 2
(−2)m−n [an−2 (r + 2)2 (−3)n−2 + an−1 (−(r + 2))(n − 1)(−3)n−2 + an−1 (−3)n−1 ] + an−1 (m − n)(−2)m−n−1 (−(r + 2))(−3)n−1
(9)
and the coefficient of λ in µR(G) (λ) is
(−2)m−n an−1 (−(r + 2))(−3)n−1 .
(10)
Notice that R(G) has m + n vertices. It follows from Lemma 3.3, (9) and (10) that Kf (R(G)) m+n
=−
an − 2 r + 2 an − 1
3
Substituting Eq. (7) and m = Kf (R(G)) =
=
nr 2
3n 6
3
m−n
+
2
+
1 r +2
.
into the above equation
(m + n)(r + 2) (r + 2)2
n−1
+
Kf (G) +
Kf (G) +
(m + n)(n − 1) 3
(n − 1)n(r + 2) 6
Summing up, we complete the proof.
+
+
(m + n)(m − n) 2
n2 (r 2 − 4) 8
m+n
+
r +2
n
+ . 2
In what follows, we give a lower bound for the Kirchhoff index for R(G) in terms of the number of vertices and the vertex degree of a connected regular graph. Corollary 3.5. Let G be a connected r-regular (r ̸= 2) graph with n vertices. Then Kf (R(G)) ≥
(n2 − n)(r + 2)2 6r
+
(n2 − n − r − 2)(r + 2) 6
+
n2 (r 2 − 4) 8
n
+ , 2
and the equality holds if and only if G ∼ = Kn or G ∼ = K n , n and n is even. 2 2
Proof. It follows from Lemma 3.2 and Theorem 3.4 that Kf (R(G)) ≥
=
(r + 2)2
6
(n − 1)n r
(n2 − n)(r + 2)2 6r
+
−1 +
(n − 1)n(r + 2) 6
(n2 − n − r − 2)(r + 2) 6
+ +
n2 (r 2 − 4) 8 n2 (r 2 − 4) 8
Clearly, the equality holds if and only if G ∼ = Kn or G ∼ = K 2n , 2n and n is even.
+
n 2 n
+ . 2
The following result is proved in a way that is certainly similar in spirit to the proof of Theorem 3.4, but is a little more complicated in detail. Theorem 3.6. Let G be a connected r-regular graph with n vertices. Then Kf (Q (G)) =
(r + 2)2 (r + 2)2 n2 − 4n Kf (G) + . 2(r + 1) 8(r + 1)
(11)
Proof. Suppose first that r = 1, i.e. G ∼ = K2 . Then Q (K2 ) ∼ = P3 , where P3 is a path with three vertices. Since S (P3 ) = {3, 1, 0}. It follows from Lemma 3.1 that Kf (P3 ) = 4. We can easily verify that the Eq. (11) holds in this case. Suppose now that r ≥ 2. Let µG (λ) = λn + a1 λn−1 + · · · + an−1 λ. It follows from Theorem 2.6(ii) that
µQ (G) (λ) = (−1)m (r − λ)n−m (r + 1 − λ)n (λ2 − (3r + 2)λ + 2r (r + 1))m−n n 2 r +2−λ r +2−λ r +2−λ n 2 + · · · + an−2 λ + an−1 λ × λ r +1−λ r +1−λ r +1−λ = (−1)m (r − λ)n−m (λ2 − (3r + 2)λ + 2r (r + 1))m−n [λn (r + 2 − λ)n + · · · + an−2 λ2 (r + 2 − λ)2 (r + 1 − λ)n−2 + an−1 λ(r + 2 − λ)(r + 1 − λ)n−1 ], where λ ̸= r + 1 and m = Let
nr 2
≥ n.
H (λ) = (r − λ)m−n µQ (G) (λ),
(12)
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3069
i.e., H (λ) = (−1)m (λ2 − (3r + 2)λ + 2r (r + 1))m−n [λn (r + 2 − λ)n + · · ·
+ an−2 λ2 (r + 2 − λ)2 (r + 1 − λ)n−2 + an−1 λ(r + 2 − λ)(r + 1 − λ)n−1 ]. Denote by Cµ2 and CH2 (resp., Cµ1 and CH1 ) the coefficient of λ2 (resp. λ) of µQ (G) (λ) and H (λ), respectively. Then CH2 = (−1)m [(2r (r + 1))m−n (an−2 (r + 2)2 (r + 1)n−2 − an−1 (r + 2)(n − 1)(r + 1)n−2
− an−1 (r + 1)n−1 ) − an−1 (m − n)(3r + 2)(2r (r + 1))m−n−1 (r + 2)(r + 1)n−1 ] and CH1 = (−1)m (2r (r + 1))m−n an−1 (r + 2)(r + 1)n−1 . Comparing the coefficients of λ2 and λ between the left hand and the right hand of Eq. (12), we have Cµ2 = (−1)m 2m−n−1 (r + 1)m−2 [2(r + 2)2 an−2 − 2(r + 2)(n − 1)an−1
− 2(r + 1)an−1 (m − n)(3r + 2)r −1 (r + 2)an−1 + 2(m − n)r −1 (r + 1)(r + 2)an−1 ]
(13)
and Cµ1 = (−1)m 2m−n (r + 1)m−1 (r + 2)an−1 .
(14)
Note that Q (G) has m + n vertices. It follows from Lemma 3.3, (13) and (14) that Kf (Q (G)) m+n
=− =−
Cµ2 Cµ1 an − 2 r + 2 an−1 r + 1
Substituting Eq. (7) and m =
nr 2
+
n−1 r +1
+
1 r +2
+
(m − n)(3r + 2) m − n − . 2r (r + 1) r
into the above equation,
(m + n)(n − 1) m + n (3r + 2)(m + n)(m − n) (m + n)(m − n) (m + n)(r + 2) Kf (G) + + + − (r + 1)n r +1 r +2 2r (r + 1) r (r + 2)2 (r + 2)2 n2 − 4n = Kf (G) + . 2(r + 1) 8(r + 1)
Kf (Q (G)) =
Thus the Theorem 3.6 holds in all cases.
Similarly to Corollary 3.5, a lower bound for Kirchhoff index of Q (G) is obtained as follows. Corollary 3.7. Let G be a connected r-regular graph with n vertices. Then
(r + 2)2 (n2 − n) (r + 2)2 (n2 − 4) − 4n + , 2r (r + 1) 8(r + 1) ∼ Kn or G ∼ the equality holds if and only if G = = K n , n and n is even. Kf (R(G)) ≥
2 2
Proof. Lemma 3.2 and Theorem 3.6 yield the result.
4. Some applications In this section, we discuss some special graphs and give formulae for their Kirchhoff index. 4.1. Complete graph Kn (n ≥ 2) It is well known that Kn is (n − 1)-regular and Kf (Kn ) = n − 1. It follows from Theorem 3.4 that Kf (R(Kn )) =
=
(n + 1)2 6 n4 8
+
n3 12
Kf (Kn ) +
−
5n2 24
+
(n − 1)n(n + 1) 6 n−1 6
.
+
n2 ((n − 1)2 − 4) 8
+
n 2
3070
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
Fig. 2. The gear graph GE16 .
It follows from Theorem 3.6 that
(n + 1)2
Kf (Q (Kn )) =
Kf (Kn ) +
n2 (n + 1)2 − 4n
2n 8n n4 + 6n3 + 5n2 − 8n − 4
=
.
8n
4.2. Cycle Cn For a r-regular graph G with n vertices, it is easy to see, but interesting, that R(G) ∼ = Q (G) if and only if r = 2. In this special situation, we call R(Cn ) or Q (Cn ) a gear graph with 2n vertices, denoted by GE2n . For example, GE16 is shown in Fig. 2. In what follows, we give the formula of Kirchhoff index of the gear graph. It was proved in [21] that Kf (Cn ) = Kf (GE2n ) =
=
2
4
Kf (Cn ) +
6 2n3 9
+
2n2 3
n3 −n . 12
4n(n − 1)
−
6 7n 18
+
Note that R(Cn ) ∼ = Q (Cn ) and Cn is 2-regular. By Eq. (11) n 2
.
4.3. Complete bipartite graph Kn,n Note that Kn,n is n-regular with 2n vertices. Recall from [12] that Kf (Kn,n ) = 4n − 3.
(15)
It follows from (15) and Theorem 3.4 that Kf (R(Kn,n )) =
(n + 1)2 6
Kf (Kn,n ) +
4n3
(2n − 1)2n(n + 2) 6
+
(2n)2 ((2n)2 − 4) 8
+
2n 2
7n2
+ + n − 2. 3 6 Analogously, it follows from (15) and Theorem 3.6 that = 2n4 +
(n + 2)2 (n + 2)2 (2n)2 − 4(2n) Kf (Kn,n ) + 2(n + 1) 8(n + 1) n4 + 8n3 + 17n2 + 2n − 12 = . 2(n + 1)
Kf (Q (Kn,n )) =
Acknowledgements The authors are grateful to the anonymous referees for many valuable comments and suggestions, which led to great improvements of the original manuscript. References [1] D. Babić, D.J. Klein, I. Lukovits, S. Nikolić, N. Trinajstić, Resistance-distance matrix: a computational algorithm and its applications, Int. J. Quantum Chem. 90 (2002) 166–176. [2] A.T. Balaban, X. Liu, D.J. Klein, D. Babić, T.G. Schmalz, W.A. Seitz, M. Randić, Graph invariants for fullerenes, J. Chem. Inf. Comput. Sci. 35 (1995) 396–404. [3] R.B. Bapat, I. Gutman, W.J. Xiao, A simple method for computing resistance distance, Z. Naturforsch. 58a (2003) 494–498.
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3071
[4] D. Bonchev, A.T. Balaban, X. Liu, D.J. Klein, Molecular cyclicity and centricity of polycyclic graphs. I: cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50 (1994) 1–20. [5] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York, 1980. [6] K.C. Das, A.D. Güngör, A.S. Cevik, On the Kirchhoff index and the resistance-distance energy of a graph, MATCH Commun. Math. Comput. Chem. 67 (2012) 541–556. [7] K.C. Das, K. Xu, I. Gutman, Comparison between Kirchhoff index and the Laplacian-energy-like invariant, Linear Algebra Appl. 436 (2012) 3661–3671. [8] W. Deng, A. Kelmans, J. Meng, Laplacian spectra of regular graph transformations, Discrete Appl. Math. 161 (1–2) (2013) 118–133. [9] E. Estrada, N. Hatano, Topological atomic displacements, Kirchhoff and Wiener indices of molecules, Chem. Phys. Lett. 486 (2010) 166–170. [10] R.M. Foster, The average impedance of an electrical network, in: Contributions to Applied Mechanics, Reissner Anniversary Volume, Edwards Brothers, Ann Arbor, MI, 1949, pp. 333–340. [11] P.W. Fowler, Resistance distances in fullerene graphs, Croat. Chem. Acta 75 (2002) 401–408. [12] X. Gao, Y.F. Luo, W.W. Liu, Kirchhoff index in line, subdivision and total graphs of a regular graph, Discrete Appl. Math. 160 (2012) 560–565. [13] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221–229. [14] R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218–238. [15] I. Gutman, B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996) 982–985. [16] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. [17] A.K. Kel’Mans, Properties of the characteristic polynomial of a graph, Kibernetiky-na sluzbu kommunizmu 4 Energija, Moskva-Leningrad, 1967, pp. 27–41 (in Russian). [18] D.J. Klein, Resistance-distance sum rules, Croat. Chem. Acta 75 (2002) 633–649. [19] D.J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1995) 50–52. [20] D.J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81–95. [21] I. Lukovits, S. Nikolić, N. Trinajstić, Resistance distance in regular graphs, Int. J. Quantum Chem. 71 (1999) 217–225. [22] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197–198 (1994) 143–176. [23] R. Merris, A survey of graph Laplacians, Linear Multilinear Algebra 39 (1995) 19–31. [24] B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk (Eds.), Graph Theory, Combinatorics & Applications, Wiley, New York, 1991, pp. 871–898. [25] J.L. Palacios, Resistance distance in graphs and random walks, Int. J. Quantum Chem. 81 (2001) 29–33. [26] J.L. Palacios, Closed form formulae for Kirchhoff index, Int. J. Quantum Chem. 81 (2001) 135–140. [27] J.L. Palacios, Foster’s formulas via probability and the Kirchhoff index, Methodol. Comput. Appl. Probab. 6 (2004) 381–387. [28] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. [29] W.J. Xiao, I. Gutman, Resistance distance and Laplacian spectrum, Theor. Chem. Acc. 110 (2003) 284–289. [30] H. Xu, The Laplacian spectrum and Kirchhoff index of product and lexicographic product of graphs, J. Xiamen Univ. (Nat. Sci.) 42 (2003) 552–554 (in Chinese). [31] Y. Yang, X. Jiang, Unicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 60 (1) (2008) 107–120. [32] H. Zhang, X. Jiang, Y. Yang, Bicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 61 (3) (2009) 697–712. [33] H. Zhang, Y. Yang, C. Li, Kirchhoff index of composite graphs, Discrete Appl. Math. 157 (2009) 2918–2927. [34] B. Zhou, N. Trinajstić, A note on Kirchhoff index, Chem. Phys. Lett. 455 (2008) 120–123. [35] H.-Y. Zhu, D.J. Klein, I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996) 420–428.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs✩ Weizhong Wang a,b,∗ , Dong Yang b , Yanfeng Luo b a
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, PR China
b
Department of Mathematics, Lanzhou University, Lanzhou 730000, PR China
article
info
Article history: Received 15 June 2012 Received in revised form 3 June 2013 Accepted 6 June 2013 Available online 29 June 2013 Keywords: Resistance distance Kirchhoff index Laplacian spectrum Bound
abstract Let R(G) be the graph obtained from G by adding a new vertex corresponding to each edge of G and by joining each new vertex to the end vertices of the corresponding edge, and Q (G) be the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. In this paper, we determine the Laplacian polynomials of R(G) and Q (G) of a regular graph G; on the other hand, we derive formulae and lower bounds of the Kirchhoff index of these graphs. Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved.
1. Introduction Let G be a simple graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E (G) = {e1 , e2 , . . . , em }. Denote by A(G) and D(G) the adjacency matrix and the diagonal matrix with the vertex degrees of G on the diagonal, respectively. The matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G, for details see [22,23]. Denote by PG (λ) and µG (λ) the adjacent characteristic polynomial det(λIn − A(G)) and the Laplacian characteristic polynomial det(λIn − L(G)) of G, respectively. The multiset of eigenvalues of A(G) (resp., L(G)) are called the adjacency (resp., Laplacian) spectrum of G. Since A(G) and L(G) are all real symmetric matrices, their eigenvalues are real numbers. So we can assume that λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G) (resp., µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn (G)) are the adjacency (resp., Laplacian) eigenvalues of G. Clearly, all Laplacian eigenvalues of G are non-negative. If the graph G is connected, then µi (G) > 0 for i = 1, 2, . . . , n − 1 and µn (G) = 0 [13,14,22]. In what follows, the Laplacian spectrum of G is denoted by S (G) = {µ1 , µ2 , . . . , µn }. Suppose G is a simple graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E (G) = {e1 , e2 , . . . , em }. Define two graph operators R and Q (see the definitions in p. 63 in [5]) as follows. Let R(G) = (V (R(G)), E (R(G))) be the graph obtained from G by adding a new vertex e′ corresponding to each edge e = (a, b) of G and by joining each new vertex e′ to the end vertices a and b of the corresponding edge e = (a, b), i.e., R(G) is obtained from G by ‘‘changing each edge e = (a, b) of G into a triangle ae′ b’’. Thus, V (R(G)) = V (G) ∪ {e′ | e ∈ E (G)} and E (R(G)) = E (G) ∪ {(vi , e′ ), (vj , e′ ) | e = (vi , vj ) ∈ E (G)} (see Fig. 1(a) and (b) for example). Let Q (G) = (V (Q (G)), E (Q (G))) be the graph obtained from G by inserting a new vertex e′i into every edge ei of G and by joining by edges those pairs of these new vertices e′i and e′j which lie on adjacent edges ei and ej of G, i, j = 1, 2, . . . , m. Denote by vi1 and vi2 the end-vertices of edge ei of G. Then V (Q (G)) = V (G) ∪ {e′i | ei ∈ E (G),
✩ This research was partially supported by the National Natural Science Foundation of China (No. 10971086 and No. 11201201).
∗ Corresponding author at: Department of mathematics, Lanzhou Jiaotong University, Lanzhou 730070, PR China. Tel.: +86 09314938625; fax: +86 09318834765. E-mail addresses: [email protected], [email protected] (W. Wang). 0166-218X/$ – see front matter Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.06.010
3064
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
a
b
c
Fig. 1. (a) The graph G. (b) The graph R(G). (c) The graph Q (G).
i = 1, 2, . . . , m} and E (Q (G)) = {(vi1 , e′i ), (vi2 , e′i )|i = 1, 2, . . . , m} ∪ {(e′i , ej ′ ) | ei and ej are adjacent edges of G} (see Fig. 1(a) and (c) for example). In some graph theory problems it is necessary to compute the spectrum (resp., Laplacian spectrum) of a compound graph, obtained from some operations from some simple graphs. In [5] there exist many relations connecting the spectrum (resp., Laplacian spectrum) of a compound graph with spectra (resp., Laplacian spectra [17,24]) of graphs from which that graph is derived. However, it is worth considering the corresponding problems of graphs derived from a single graph, such as line graph, subdivision graph, total graph, R(G), Q (G), etc. In [17], the Laplacian polynomial of line graph, the subdivision graph and total graph of the regular graph are obtained. In the present work, on one hand, we determine the Laplacian polynomials of R(G) and Q (G); on the other hand, for these graphs, we compute the topological indices based on the concept of resistance distance. Let G = (V , E ) be a connected graph with vertex set V = {v1 , v2 , . . . , vn }. The (ordinary) distance between vertices vi and vj , denoted by dij , is the length of a shortest path connecting them. The original index based on distance in a graph G is the Wiener index W (G) [28], which counts the sum of distances between pairs of vertices in G. In 1993, Klein and Randić [20] defined a new distance function named resistance distance framed in terms of electrical network theory. However, this concept has been discussed much earlier (1949) for another purpose by Foster [10] as recently pointed out by Palacios [27]. The resistance distance between vertices vi and vj of G, denoted by rij , is defined to be the effective resistance between nodes vi and vj as computed with Ohm’s law when all the edges of G are considered to be unit resistors. As an analogue to the Wiener index, the sum Kf (G) = i
det
M P
N Q
= detMdet(Q − PM −1 N ).
The line graph of a graph G, denoted by L(G), is the graph whose vertices correspond to the edges of G, with two vertices of L(G) being adjacent if and only if the corresponding edges of G share a common vertex. The following result is well known [5].
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3065
Lemma 2.2 ([5]). Let G be an r-regular graph with n vertices and m edges. Then PL(G) (λ) = (λ + 2)m−n PG (λ − r + 2). A.K. Kel’Mans [17] characterized the Laplacian polynomial of L(G) by means of the Laplacian polynomial of G. Lemma 2.3 ([17]). Let G be an r-regular simple graph with n vertices and m edges. Then
µL(G) (λ) = (λ − 2r )m−n µG (λ). Lemma 2.4 ([24]). Let G be the disjoint union of graphs G1 , G2 , . . . , Gk . Then
µG (λ) =
k
µGi (λ).
i=1
Now we consider the case for R(G). For a regular graph G, the next theorem gives the representation of the Laplacian polynomial of R(G) by means of the characteristic polynomial and the Laplacian polynomial of G, respectively. Theorem 2.5. Let G be an r-regular (r ̸= 2) graph with n vertices and m edges. Then (i) µR(G) (λ) = (λ − 2)m−n (3 − λ)n PG
(ii) µR(G) (λ) = (λ − 2)
m−n
(λ − 3) µG n
λ2 −2(r +1)λ+3r 3−λ λ(λ−r −2) λ−3
.
.
Proof. (i) Let A and I be the adjacency matrix and the incidence matrix of G, respectively. (In what follows, the unit matrix of order p will be denoted by Ip , and it should not be confused with the incidence matrix.) Then A(R(G)) =
0m I
IT A
2Im −I
−I T . 2rIn − A
D(R(G)) =
and
2Im 0
0 2rIn
.
We thus have L(R(G)) =
It follows that
µR(G) (λ) = det
(λ − 2)Im
IT . (λ − 2r )In + A
I
(1)
Case 1. If r = 1, then G is a disjoint union of copies K2 and hence R(G) is a disjoint union of copies of K3 , i.e., G = 2n K2 and R(G) = 2n K3 , where n is even. It follows from Lemma 2.4, PK2 (λ) = λ2 − 1 and µK3 (λ) = λ3 − 6λ2 + 9λ that Theorem 2.5(i) holds. Case 2. If r > 2 and λ = 2, then m = nr > n. It follows from (1) that 2
0 µR(G) (λ) = det m I
IT . (2 − 2r )In + A
It is easy to see the first m lines of
0m I
IT
(2 − 2r )In + A
are linearly dependent since m > n. Thus µR(G) (λ) = 0.
If λ ̸= 2, it follows from Lemma 2.1 that
µR(G) (λ) = det
(λ − 2)Im I
IT (λ − 2r )In + A
Im T = (λ − 2)m det (λ − 2r )In + A − I I λ−2 1 = (λ − 2)m det (λ − 2r )In + A − (A + rIn ) λ−2 (λ − 2r )(λ − 2) − r m−n n In − A = (λ − 2) (3 − λ) det 3−λ 2 λ − 2(r + 1)λ + 3r = (λ − 2)m−n (3 − λ)n PG . 3−λ
(2)
3066
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
(ii) Recall that L(G) = rIn − A(G). It follows from (2) that
(λ − 2r )(λ − 2) − r In + A λ−3 2 λ − ( r + 2)λ In − (rIn − A) = (λ − 2)m−n (λ − 3)n det λ−3 λ(λ − r − 2) = (λ − 2)m−n (λ − 3)n µG . λ−3
µR(G) (λ) = (λ − 2)m−n (λ − 3)n det
The proof is thus completed.
Remark 1. Theorem 2.5(ii) has appeared in [8]. Here we just obtain the result in a different way. Lastly, we consider the case for Q (G). For a regular graph G, the next theorem gives the representation of the Laplacian polynomial of Q (G) by means of the characteristic polynomial and the Laplacian polynomial of G, respectively. Theorem 2.6. Let G be an r-regular graph with n vertices and m edges. Then (i) µQ (G) (λ) = (r + 1 − λ)n
m−n
λ2 −2(r +1)λ+r 2 +r . r +1−λ 2 m−n )λ+2r (r +1) −λ) λ)n λ −(3r +r2−λ µG λ(rr++12−λ .
λ2 −(3r +2)λ+2r 2 +2r λ−r
(ii) µQ (G) (λ) = (−1)m (r + 1 −
PG
Proof. (i) Let B and I be the adjacency matrix of L(G) and incidence matrix of G, respectively. Then A(Q (G)) =
B I
IT 0n
and D(Q (G)) =
2rIm rIn
.
We thus have L(Q (G)) =
2rIm − B −I
−I T
rIn
.
Hence
µQ (G) (λ) = det
I
= det
(λ − 2r )Im + B (λ − r )In
IT
(λ − r )In
I (λ − 2r )Im + B
IT
In I = (λ − r )n det (λ − 2r )Im + B − I T λ−r 1 n = (λ − r ) det (λ − 2r )Im + B − (B + 2Im ) λ−r 2 λ − 3r λ + 2r 2 − 2 n−m m = (λ − r ) (r + 1 − λ) det Im − B r +1−λ 2 λ − 3r λ + 2r 2 − 2 = (λ − r )n−m (r + 1 − λ)m PL(G) . r +1−λ
(3)
It follows from Lemma 2.2 that
PL(G)
λ2 − 3r λ + 2r 2 − 2 r +1−λ
=
λ2 − (3r + 2)λ + 2r 2 + 2r r +1−λ
m−n
PG
λ2 − 2(r + 1)λ + r 2 + r r +1−λ
Combine (3) with (4), we get
µQ (G) (λ) = (r + 1 − λ)n
λ2 − (3r + 2)λ + 2r 2 + 2r λ−r
m−n
PG
λ2 − 2(r + 1)λ + r 2 + r r +1−λ
.
.
(4)
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3067
(ii) Note that G is r-regular and the line graph L(G) of G is also a (2r − 2)-regular graph with m = that D(L(G)) = (2r − 2)Im and L(L(G)) = D(L(G)) − B. By (3) it follows that
nr 2
vertices. Notice also
λ2 − 3r λ + 2r 2 − 2 Im − B r +1−λ 2 −λ + r λ + 2λ = (−1)n (r − λ)n−m (r + 1 − λ)m det Im − ((2r − 2)Im − B) r +1−λ λ(r + 2 − λ) n n−m m . = (−1) (r − λ) (r + 1 − λ) µL(G) r +1−λ
µQ (G) (λ) = (λ − r )n−m (r + 1 − λ)m det
(5)
By Lemma 2.3 we deduce that
µL(G)
λ(r + 2 − λ) r +1−λ
=
−λ2 + (3r + 2)λ − 2r (r + 1) r +1−λ
m−n
µG
λ(r + 2 − λ) . r +1−λ
(6)
By virtue of (5) and (6) we have in fact already established the statement in the theorem. Thus the theorem is proved. 3. The Kirchhoff index of R (G ) and Q (G ) In this section, we will explore the Kirchhoff index of the R(G) and Q (G) of a regular graph G. Gutman and Mohar [15] and Zhu et al. [35] established a nice relationship between the Laplacian spectrum and Kirchhoff index as follows: Lemma 3.1 ([15,35]). Let G be a connected graph with n ≥ 2 vertices. Then Kf (G) = n
n −1 1 . µ i i=1
Denote by δi the degree of vertex vi ∈ V (G). Zhou and Trinajstić [34] proved that Lemma 3.2 ([34]). Let G be a connected graph with n ≥ 2 vertices. Then Kf (G) ≥ −1 + (n − 1)
1 δ v ∈V (G) i i
with equality attained if and only if G = Kn or G = Kt ,n−t for 1 ≤ t ≤ ⌊ 2n ⌋. The following lemma will be used in the remainder of this paper. Lemma 3.3 ([12]). Let G be a connected graph with n ≥ 2 vertices and µG (λ) = λn + a1 λn−1 + · · · + an−1 λ. Then Kf (G) n
=−
an − 2 an − 1
(an−2 = 1 whenever n = 2).
(7)
Let Kn denote the complete graph with n vertices. If G = K1 , we have nothing to discuss. So we assume that G ̸= K1 throughout this paper. The following Theorem 3.4 shows that Kf (R(G)) can be completely determined by the Kirchhoff index, the number of vertices and the vertex degree of regular graph G. Theorem 3.4. Let G be a connected r-regular (r ̸= 2) graph with n vertices. Then Kf (R(G)) =
(r + 2)2 6
Kf (G) +
(n2 − n)(r + 2) 6
+
n2 (r 2 − 4) 8
n
+ .
(8)
2
Proof. Suppose first that r = 1, i.e. G ∼ = K2 . Then R(K2 ) = K3 . Since S (K3 ) = {3, 3, 0}. It follows from Lemma 3.1 that Kf (K3 ) = 2. It is easy to check that the Eq. (8) holds in this case. Suppose now that r > 2. Let µG (λ) = λn + a1 λn−1 + · · · + an−1 λ. It follows from Theorem 2.5(ii) that
µR(G) (λ) = (λ − 2)
m−n
(λ − 3)
n
λ
n
λ−r −2 λ−3
n
+ · · · + an − 2 λ
2
λ−r −2 λ−3
2
= (λ − 2)m−n λn (λ − r − 2)n + · · · + an−2 λ2 (λ − r − 2)2 (λ − 3)n−2 + an−1 λ (λ − r − 2) (λ − 3)n−1 ,
λ−r −2 + an − 1 λ λ−3
3068
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
where λ ̸= 3 and m =
> n. So the coefficient of λ2 in µR(G) (λ) is
rn 2
(−2)m−n [an−2 (r + 2)2 (−3)n−2 + an−1 (−(r + 2))(n − 1)(−3)n−2 + an−1 (−3)n−1 ] + an−1 (m − n)(−2)m−n−1 (−(r + 2))(−3)n−1
(9)
and the coefficient of λ in µR(G) (λ) is
(−2)m−n an−1 (−(r + 2))(−3)n−1 .
(10)
Notice that R(G) has m + n vertices. It follows from Lemma 3.3, (9) and (10) that Kf (R(G)) m+n
=−
an − 2 r + 2 an − 1
3
Substituting Eq. (7) and m = Kf (R(G)) =
=
nr 2
3n 6
3
m−n
+
2
+
1 r +2
.
into the above equation
(m + n)(r + 2) (r + 2)2
n−1
+
Kf (G) +
Kf (G) +
(m + n)(n − 1) 3
(n − 1)n(r + 2) 6
Summing up, we complete the proof.
+
+
(m + n)(m − n) 2
n2 (r 2 − 4) 8
m+n
+
r +2
n
+ . 2
In what follows, we give a lower bound for the Kirchhoff index for R(G) in terms of the number of vertices and the vertex degree of a connected regular graph. Corollary 3.5. Let G be a connected r-regular (r ̸= 2) graph with n vertices. Then Kf (R(G)) ≥
(n2 − n)(r + 2)2 6r
+
(n2 − n − r − 2)(r + 2) 6
+
n2 (r 2 − 4) 8
n
+ , 2
and the equality holds if and only if G ∼ = Kn or G ∼ = K n , n and n is even. 2 2
Proof. It follows from Lemma 3.2 and Theorem 3.4 that Kf (R(G)) ≥
=
(r + 2)2
6
(n − 1)n r
(n2 − n)(r + 2)2 6r
+
−1 +
(n − 1)n(r + 2) 6
(n2 − n − r − 2)(r + 2) 6
+ +
n2 (r 2 − 4) 8 n2 (r 2 − 4) 8
Clearly, the equality holds if and only if G ∼ = Kn or G ∼ = K 2n , 2n and n is even.
+
n 2 n
+ . 2
The following result is proved in a way that is certainly similar in spirit to the proof of Theorem 3.4, but is a little more complicated in detail. Theorem 3.6. Let G be a connected r-regular graph with n vertices. Then Kf (Q (G)) =
(r + 2)2 (r + 2)2 n2 − 4n Kf (G) + . 2(r + 1) 8(r + 1)
(11)
Proof. Suppose first that r = 1, i.e. G ∼ = K2 . Then Q (K2 ) ∼ = P3 , where P3 is a path with three vertices. Since S (P3 ) = {3, 1, 0}. It follows from Lemma 3.1 that Kf (P3 ) = 4. We can easily verify that the Eq. (11) holds in this case. Suppose now that r ≥ 2. Let µG (λ) = λn + a1 λn−1 + · · · + an−1 λ. It follows from Theorem 2.6(ii) that
µQ (G) (λ) = (−1)m (r − λ)n−m (r + 1 − λ)n (λ2 − (3r + 2)λ + 2r (r + 1))m−n n 2 r +2−λ r +2−λ r +2−λ n 2 + · · · + an−2 λ + an−1 λ × λ r +1−λ r +1−λ r +1−λ = (−1)m (r − λ)n−m (λ2 − (3r + 2)λ + 2r (r + 1))m−n [λn (r + 2 − λ)n + · · · + an−2 λ2 (r + 2 − λ)2 (r + 1 − λ)n−2 + an−1 λ(r + 2 − λ)(r + 1 − λ)n−1 ], where λ ̸= r + 1 and m = Let
nr 2
≥ n.
H (λ) = (r − λ)m−n µQ (G) (λ),
(12)
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3069
i.e., H (λ) = (−1)m (λ2 − (3r + 2)λ + 2r (r + 1))m−n [λn (r + 2 − λ)n + · · ·
+ an−2 λ2 (r + 2 − λ)2 (r + 1 − λ)n−2 + an−1 λ(r + 2 − λ)(r + 1 − λ)n−1 ]. Denote by Cµ2 and CH2 (resp., Cµ1 and CH1 ) the coefficient of λ2 (resp. λ) of µQ (G) (λ) and H (λ), respectively. Then CH2 = (−1)m [(2r (r + 1))m−n (an−2 (r + 2)2 (r + 1)n−2 − an−1 (r + 2)(n − 1)(r + 1)n−2
− an−1 (r + 1)n−1 ) − an−1 (m − n)(3r + 2)(2r (r + 1))m−n−1 (r + 2)(r + 1)n−1 ] and CH1 = (−1)m (2r (r + 1))m−n an−1 (r + 2)(r + 1)n−1 . Comparing the coefficients of λ2 and λ between the left hand and the right hand of Eq. (12), we have Cµ2 = (−1)m 2m−n−1 (r + 1)m−2 [2(r + 2)2 an−2 − 2(r + 2)(n − 1)an−1
− 2(r + 1)an−1 (m − n)(3r + 2)r −1 (r + 2)an−1 + 2(m − n)r −1 (r + 1)(r + 2)an−1 ]
(13)
and Cµ1 = (−1)m 2m−n (r + 1)m−1 (r + 2)an−1 .
(14)
Note that Q (G) has m + n vertices. It follows from Lemma 3.3, (13) and (14) that Kf (Q (G)) m+n
=− =−
Cµ2 Cµ1 an − 2 r + 2 an−1 r + 1
Substituting Eq. (7) and m =
nr 2
+
n−1 r +1
+
1 r +2
+
(m − n)(3r + 2) m − n − . 2r (r + 1) r
into the above equation,
(m + n)(n − 1) m + n (3r + 2)(m + n)(m − n) (m + n)(m − n) (m + n)(r + 2) Kf (G) + + + − (r + 1)n r +1 r +2 2r (r + 1) r (r + 2)2 (r + 2)2 n2 − 4n = Kf (G) + . 2(r + 1) 8(r + 1)
Kf (Q (G)) =
Thus the Theorem 3.6 holds in all cases.
Similarly to Corollary 3.5, a lower bound for Kirchhoff index of Q (G) is obtained as follows. Corollary 3.7. Let G be a connected r-regular graph with n vertices. Then
(r + 2)2 (n2 − n) (r + 2)2 (n2 − 4) − 4n + , 2r (r + 1) 8(r + 1) ∼ Kn or G ∼ the equality holds if and only if G = = K n , n and n is even. Kf (R(G)) ≥
2 2
Proof. Lemma 3.2 and Theorem 3.6 yield the result.
4. Some applications In this section, we discuss some special graphs and give formulae for their Kirchhoff index. 4.1. Complete graph Kn (n ≥ 2) It is well known that Kn is (n − 1)-regular and Kf (Kn ) = n − 1. It follows from Theorem 3.4 that Kf (R(Kn )) =
=
(n + 1)2 6 n4 8
+
n3 12
Kf (Kn ) +
−
5n2 24
+
(n − 1)n(n + 1) 6 n−1 6
.
+
n2 ((n − 1)2 − 4) 8
+
n 2
3070
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
Fig. 2. The gear graph GE16 .
It follows from Theorem 3.6 that
(n + 1)2
Kf (Q (Kn )) =
Kf (Kn ) +
n2 (n + 1)2 − 4n
2n 8n n4 + 6n3 + 5n2 − 8n − 4
=
.
8n
4.2. Cycle Cn For a r-regular graph G with n vertices, it is easy to see, but interesting, that R(G) ∼ = Q (G) if and only if r = 2. In this special situation, we call R(Cn ) or Q (Cn ) a gear graph with 2n vertices, denoted by GE2n . For example, GE16 is shown in Fig. 2. In what follows, we give the formula of Kirchhoff index of the gear graph. It was proved in [21] that Kf (Cn ) = Kf (GE2n ) =
=
2
4
Kf (Cn ) +
6 2n3 9
+
2n2 3
n3 −n . 12
4n(n − 1)
−
6 7n 18
+
Note that R(Cn ) ∼ = Q (Cn ) and Cn is 2-regular. By Eq. (11) n 2
.
4.3. Complete bipartite graph Kn,n Note that Kn,n is n-regular with 2n vertices. Recall from [12] that Kf (Kn,n ) = 4n − 3.
(15)
It follows from (15) and Theorem 3.4 that Kf (R(Kn,n )) =
(n + 1)2 6
Kf (Kn,n ) +
4n3
(2n − 1)2n(n + 2) 6
+
(2n)2 ((2n)2 − 4) 8
+
2n 2
7n2
+ + n − 2. 3 6 Analogously, it follows from (15) and Theorem 3.6 that = 2n4 +
(n + 2)2 (n + 2)2 (2n)2 − 4(2n) Kf (Kn,n ) + 2(n + 1) 8(n + 1) n4 + 8n3 + 17n2 + 2n − 12 = . 2(n + 1)
Kf (Q (Kn,n )) =
Acknowledgements The authors are grateful to the anonymous referees for many valuable comments and suggestions, which led to great improvements of the original manuscript. References [1] D. Babić, D.J. Klein, I. Lukovits, S. Nikolić, N. Trinajstić, Resistance-distance matrix: a computational algorithm and its applications, Int. J. Quantum Chem. 90 (2002) 166–176. [2] A.T. Balaban, X. Liu, D.J. Klein, D. Babić, T.G. Schmalz, W.A. Seitz, M. Randić, Graph invariants for fullerenes, J. Chem. Inf. Comput. Sci. 35 (1995) 396–404. [3] R.B. Bapat, I. Gutman, W.J. Xiao, A simple method for computing resistance distance, Z. Naturforsch. 58a (2003) 494–498.
W. Wang et al. / Discrete Applied Mathematics 161 (2013) 3063–3071
3071
[4] D. Bonchev, A.T. Balaban, X. Liu, D.J. Klein, Molecular cyclicity and centricity of polycyclic graphs. I: cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50 (1994) 1–20. [5] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York, 1980. [6] K.C. Das, A.D. Güngör, A.S. Cevik, On the Kirchhoff index and the resistance-distance energy of a graph, MATCH Commun. Math. Comput. Chem. 67 (2012) 541–556. [7] K.C. Das, K. Xu, I. Gutman, Comparison between Kirchhoff index and the Laplacian-energy-like invariant, Linear Algebra Appl. 436 (2012) 3661–3671. [8] W. Deng, A. Kelmans, J. Meng, Laplacian spectra of regular graph transformations, Discrete Appl. Math. 161 (1–2) (2013) 118–133. [9] E. Estrada, N. Hatano, Topological atomic displacements, Kirchhoff and Wiener indices of molecules, Chem. Phys. Lett. 486 (2010) 166–170. [10] R.M. Foster, The average impedance of an electrical network, in: Contributions to Applied Mechanics, Reissner Anniversary Volume, Edwards Brothers, Ann Arbor, MI, 1949, pp. 333–340. [11] P.W. Fowler, Resistance distances in fullerene graphs, Croat. Chem. Acta 75 (2002) 401–408. [12] X. Gao, Y.F. Luo, W.W. Liu, Kirchhoff index in line, subdivision and total graphs of a regular graph, Discrete Appl. Math. 160 (2012) 560–565. [13] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221–229. [14] R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218–238. [15] I. Gutman, B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996) 982–985. [16] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. [17] A.K. Kel’Mans, Properties of the characteristic polynomial of a graph, Kibernetiky-na sluzbu kommunizmu 4 Energija, Moskva-Leningrad, 1967, pp. 27–41 (in Russian). [18] D.J. Klein, Resistance-distance sum rules, Croat. Chem. Acta 75 (2002) 633–649. [19] D.J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1995) 50–52. [20] D.J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81–95. [21] I. Lukovits, S. Nikolić, N. Trinajstić, Resistance distance in regular graphs, Int. J. Quantum Chem. 71 (1999) 217–225. [22] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197–198 (1994) 143–176. [23] R. Merris, A survey of graph Laplacians, Linear Multilinear Algebra 39 (1995) 19–31. [24] B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk (Eds.), Graph Theory, Combinatorics & Applications, Wiley, New York, 1991, pp. 871–898. [25] J.L. Palacios, Resistance distance in graphs and random walks, Int. J. Quantum Chem. 81 (2001) 29–33. [26] J.L. Palacios, Closed form formulae for Kirchhoff index, Int. J. Quantum Chem. 81 (2001) 135–140. [27] J.L. Palacios, Foster’s formulas via probability and the Kirchhoff index, Methodol. Comput. Appl. Probab. 6 (2004) 381–387. [28] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. [29] W.J. Xiao, I. Gutman, Resistance distance and Laplacian spectrum, Theor. Chem. Acc. 110 (2003) 284–289. [30] H. Xu, The Laplacian spectrum and Kirchhoff index of product and lexicographic product of graphs, J. Xiamen Univ. (Nat. Sci.) 42 (2003) 552–554 (in Chinese). [31] Y. Yang, X. Jiang, Unicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 60 (1) (2008) 107–120. [32] H. Zhang, X. Jiang, Y. Yang, Bicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 61 (3) (2009) 697–712. [33] H. Zhang, Y. Yang, C. Li, Kirchhoff index of composite graphs, Discrete Appl. Math. 157 (2009) 2918–2927. [34] B. Zhou, N. Trinajstić, A note on Kirchhoff index, Chem. Phys. Lett. 455 (2008) 120–123. [35] H.-Y. Zhu, D.J. Klein, I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996) 420–428.