Kirchhoff index and degree Kirchhoff index of complete multipartite graphs

Discrete Applied Mathematics (

)

–

Contents lists available at ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

Kirchhoff index and degree Kirchhoff index of complete multipartite graphs Ravindra B. Bapat a , Masoud Karimi b,c, *, Jia-Bao Liu d a

Indian Statistical Institute, Delhi Centre, 7 S.J.S.S. Marg, New Delhi 110 016, India School of Mathematical Sciences, Anhui University, Hefei, PR China Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran d School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, PR China b c

article

info

Article history: Received 29 November 2016 Received in revised form 25 July 2017 Accepted 28 July 2017 Available online xxxx Keywords: Resistance distance Kirchhoff index Complete multipartite graph

a b s t r a c t Given a complete multipartite graph G, we explicitly formulate the effective resistance distance between any two vertices of G, using methods from linear algebra. The Kirchhoff index and the degree Kirchhoff index are defined in terms of the effective resistance distances for any simple graph. In this paper, for G, the exact formulae for the two indices are provided. Along with these results, we obtain the optimal Kirchhoff index and the optimal degree Kirchhoff index. © 2017 Elsevier B.V. All rights reserved.

1. Introduction We consider simple graphs, that is, graphs without loops or parallel edges. For basic notions in graph theory we refer to [13], whereas for preliminaries on graphs and matrices, see [1]. A graph G is said to be a regular graph if all its vertices have the same degree. The complete graph of order p is denoted by Kp . The disjoint union of graphs G and H is denoted by G ∪ H. The complement of G is denoted by G, which is a simple graph such that G ∪ G is a complete graph. The complete join of graphs G and H is denoted by G ∨ H which is a graph with vertex set V (G ∨ H) := V (G) ∪ V (H) and edge set E(G ∨ H) := E(G) ∪ E(H) ∪ {uv | u ∈ V (G), v ∈ V (H)}. A complete r-partite graph is a complete join of r empty graphs. We denote Kp1 ∨· · ·∨ Kpr by Kp1 ,...,pr . Also for convenience, by Kpa1 ,pa2 ,...,pas we denote a complete r-partite graph with ai parts of size pi with pi < pi+1 , for i = 1, . . . , s. Thus, a1 +· · ·+as = r 1

2

s

and a1 p1 + · · · + as ps = n is the number of vertices. Thus Kp1 ,...,pr and Kpa1 ,pa2 ,...,pas denote the same complete r-partite graph. 1

2

s

The Laplacian matrix of the graph G, denoted by L(G), is defined as D − A, where D is the diagonal matrix with degrees of vertices of G on the diagonal and A is the adjacency matrix of G. We will use I and J if the order is clear from the context, otherwise by In and Jn , respectively, the n × n identity matrix and the n × nmatrix with all ones. The column vector 1 has all entries equal to one. Let A be an n × n matrix. If T , S ⊆ {1, . . . , n}, then A[S |T ] will denote the submatrix of A indexed by the rows corresponding to S and the columns corresponding to T . The submatrix obtained by deleting the rows in S and the columns in T will be denoted by A(S |T ). When S = {i} and T = {j}, A(S |T ) is denoted by A(i|j). Direct sum of two matrices A and B is written as A ⊕ B which is a diagonal block matrix with A and B as block-diagonal entries. The sum of all the diagonal entries of a

*

Corresponding author at: Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran. E-mail addresses: [email protected] (R.B. Bapat), [email protected] (M. Karimi), [email protected] (J.-B. Liu).

http://dx.doi.org/10.1016/j.dam.2017.07.040 0166-218X/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

2

R.B. Bapat et al. / Discrete Applied Mathematics (

)

–

matrix A is denoted by trace(A). If A is an invertible matrix, by A−1 we denote its inverse matrix. Also, by A+ we denote the Moore–Penrose inverse of A. The transpose of any matrix or vector A is denoted by A′ . We recall the notion of equitable partition. Suppose A is a symmetric matrix whose rows and columns are indexed by {1, . . . , n}. Let X = {X1 , . . . , Xs } be a partition of {1, . . . , n}. By definition, X is an equitable partition if A[Xi |Xj ]1 = bij 1 for i, j = 1, . . . , s. Let H denote the n × s matrix whose jth column is the characteristic vector of Xj , for j = 1, . . . , s. The s × s matrix B = ((bij )) is called the quotient matrix of A with respect to the partition X . Then we have AH = HB, and so eigenvalues of A consist of the eigenvalues of B, together with the eigenvalues belonging to the eigenvectors orthogonal to the columns of H (i.e., summing to zero on each part of the partition); see [3, page 24].

2. Resistance distance Gervacio in [5], by using methods and principles in electrical circuits such as the principle of elimination and the principle of substitution, explicitly expressed the effective resistance distance between any pair of vertices in the complete multipartite graph. In this section, we employ methods from linear algebra to obtain the same expression. In the process we prove a result (Theorem 2.2) which is of independent interest. Lemma 2.1. Suppose that τn′ = [a1 , . . . , an ] and σn′ = [b1 , . . . , bn ] are vectors. If ci ̸ = 0 for i = 1, . . . , n, we have det(τn σn′ + diag(c1 , . . . , cn )) = c1 c2 . . . cn

( 1+

a1 b 1 c1

+ ··· +

an bn cn

)

.

Proof. Put B := diag(c1 , . . . , cn ). Then it is not hard to see that

⏐ ⏐ 1 ⏐−b ⏐ 1 det(τn σn′ + B) = ⏐⏐ . ⏐ .. ⏐−b n

...

a1

an ⏐

⏐ ⏐ ( ) ⏐ ⏐ = det(B) 1 + a1 b1 + · · · + an bn , ⏐ c1 cn ⏐ ⏐

B

(1)

and the proof is complete. ■ Theorem 2.2. Let Kp1 ,p2 ,...,pr be a complete r-partite graph with n vertices and with V1 , . . . , Vr as its parts. Let A ⊂ {1, . . . , n}. Then the eigenvalues of L(A|A) are the roots of the polynomial

( 1+

r ∑ k=1

)

tk − pk dk + pk − tk − x

r ∏

(dk + pk − tk − x)(dk − x)pk −tk −1 ,

(2)

k=1

where ti := |A ∩ Vi |, and di is the degree of a vertex in Vi for i = 1, . . . , r. Proof. Let L denote the Laplacian of Kp1 ,p2 ,...,pr . We can verify that L = (d1 Ip1 + Jp1 ) ⊕ · · · ⊕ (dr Ipr + Jpr ) − Jn . Thus, L(A|A) = (d1 Ip1 + Jp1 )(T1 |T1 ) ⊕ · · · ⊕ (dr Ipr + Jpr )(Tr |Tr ) − Jn (A|A) where Ti := A ∩ Vi . We can see that π = {V1 − T1 , . . . , Vr − Tr } is an equitable partition with respect to L(A|A) with the quotient matrix

⎡

d1

⎢t1 − p1 ⎢t − p 1 1 B := ⎢ ⎢ . ⎣ .. t1 − p1

t2 − p2 d2 t2 − p2

.. . t2 − p2

t3 − p3 t3 − p3 d3

.. . t3 − p3

... ... ... .. . ...

tr − pr tr − pr ⎥ tr − pr ⎥ ⎥. ⎥

⎤

.. .

⎦

dr

Now, the eigenvalues of L(A|A) consist of [di ]pi −ti −1 for i = 1, . . . , r and the roots of det(B − xI) = 0. Since B − xI = diag(d1 + p1 − t1 − x, . . . , dr + pr − tr − x) + 1[t1 − p1 , . . . , tr − pr ]. Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

R.B. Bapat et al. / Discrete Applied Mathematics (

)

–

3

In view of Lemma 2.1, we have

( det(B − xI) =

1+

r ∑ k=1

)

tk − pk dk + pk − tk − x

r ∏

(dk + pk − tk − x).

k=1

So the characteristic polynomial of L(A|A) is

( 1+

r ∑ k=1

)

tk − pk dk + pk − tk − x

r ∏

(dk + pk − tk − x)(dk − x)pk −tk −1 .

k=1

That completes the proof. ■ Remark 2.3. Let n := p1 +· · ·+ pr be the number of vertices of Kp1 ,p2 ,...,pr . Then di = n − pi , so we can rewrite det(L(A|A) − xIn ) as

( 1−

r ∑ pk − tk k=1

)

n − tk

r ∏

(n − tk )(n − pk )pk −tk −1 .

k=1

Corollary 2.4. Let Kp1 ,p2 ,...,pr be a complete r-partite graph with n vertices. For distinct vertices u and v we have

det(L(u, v|u, v )) =

⎧ r ′ ∏ ⎪ r −3 (n − 1)(2n − pl − pl ) ⎪ ⎪ (n − pi )pi −1 n ⎪ ⎨ (n − pl )(n − pl′ )

if u ∈ Vl , v ∈ Vl′ , l ̸ = l′

r ⎪ 2 ∏ ⎪ r −2 ⎪ ⎪ n (n − pi )pi −1 ⎩ n−p

if u, v ∈ Vl .

i=1

l

Moreover, det(L(u|u)) = nr −2

∏r

i=1 (n

i=1

− pi )pi −1 is the number of spanning trees.

Proof. Let u and v be vertices of Kp1 ,p2 ,...,pr and let A := {u, v}. If i and j are chosen from different parts, say Vl and Vl′ with l ̸ = l′ , then by Theorem 2.2 we have tl = tl′ = 1 and tk = 0 for k ̸ = l, l′ . Evaluating (2) at x = 0 gives det(L(u, v|u, v )) = nr −3

r (n − 1)(2n − pl − pl′ ) ∏ (n − pi )pi −1 . (n − pl )(n − pl′ ) i=1

Now let u and v belong to the same part, say Vl . Then, tl = 2 and tk = 0 for k ̸ = l. Again Evaluating (2) at x = 0 gives det(L(u, v|u, v )) = nr −2

2 n − pl

r ∏

(n − pi )pi −1 .

i=1

The expression for det(L(u|u)) is obtained by setting A := {u} in (2) and putting x = 0, and ti = 1 when u ∈ Vi , otherwise ti = 0 for i = 1, . . . , r .

■

Definition 2.5. Let G be a connected graph. The resistance distance between vertices u and v is defined as Ω (u, v ) =: det(L(u,v|u,v )) . det(L(u|u))

Corollary 2.6 ([5], Theorem 5.1). Let u ∈ Vi and v ∈ Vj . Then the resistance distance between u and v is given by

Ω (u, v ) =

⎧ (n − 1)(2n − pi − pj ) ⎪ ⎪ ⎪ ⎪ ⎨ n(n − pi )(n − pj ) 2

⎪ ⎪ ⎪ n − pi ⎪ ⎩ 0

if u ∈ Vi , v ∈ Vj , i ̸ = j if u, v ∈ Vi , u ̸ = v. if u = v.

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

4

R.B. Bapat et al. / Discrete Applied Mathematics (

)

–

Example 2.7. The resistance distance matrix of K2,3,4 is as follows

⎡ 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 2 ⎢ ⎢ 7 ⎢ ⎢ ⎢ ⎢ ⎢ 52 ⎢ ⎢ 189 ⎢ ⎢ ⎢ ⎢ ⎢ 52 ⎢ ⎢ 189 ⎢ ⎢ ⎢ ⎢ ⎢ 52 ⎢ ⎢ 189 ⎢ ⎢ ⎢ ⎢ ⎢ 32 ⎢ ⎢ 105 ⎢ ⎢ ⎢ ⎢ ⎢ 32 ⎢ ⎢ ⎢ 105 ⎢ ⎢ ⎢ ⎢ 32 ⎢ ⎢ ⎢ 105 ⎢ ⎢ ⎢ ⎣ 32 105

⎤

2

52

52

52

32

32

32

32

7

189

189

189

105

105

105

105 ⎥ ⎥

0

52

52

52

32

32

32

189

189

189

105

105

105

1

1

44

44

44

3

3

135

135

135

52

0

189 52

1

189

3

0

52

1

1

189

3

3

1

44

44

44

3

135

135

135

0

32

44

44

44

105

135

135

135

44

44

44

135

135

135

2

2

5

5

0

32

44

44

44

2

105

135

135

135

5

32

44

44

44

2

2

105

135

135

135

5

5

32

44

44

44

2

2

2

105

135

135

135

5

5

5

0

2 5

0

⎥ ⎥ ⎥ 32 ⎥ ⎥ 105 ⎥ ⎥ ⎥ ⎥ ⎥ 44 ⎥ ⎥ 135 ⎥ ⎥ ⎥ ⎥ ⎥ 44 ⎥ ⎥ 135 ⎥ ⎥ ⎥ ⎥ ⎥ 44 ⎥ ⎥ 135 ⎥ ⎥ ⎥ ⎥ ⎥ 2 ⎥ ⎥ ⎥ 5 ⎥ ⎥ ⎥ ⎥ 2 ⎥ ⎥ ⎥ 5 ⎥ ⎥ ⎥ ⎥ 2 ⎥ ⎥ ⎥ 5 ⎥ ⎥ ⎥ ⎥ ⎦ 0

3. Kirchhoff index In the following, we compute the Kirchhoff index of a complete multipartite graph. Definition 3.1. The Kirchhoff index of a connected graph G, denoted by Kf(G), is defined as half of the summation of all resistance distances between any two vertices, i.e., Kf(G) =

1 2

∑

Ω (u, v ).

u,v∈V (G)

In this regard, there are two applicable relations: Kf(G) = ntrace(L+ (G))

and

Kf(G) = n

n−1 ∑ 1 , λi

(3)

i=1

where L+ (G) is the Moore–Penrose inverse of the Laplacian of G and λ1 ≥ · · · ≥ λn−1 > λn = 0 are the Laplacian eigenvalues of G. Remark 3.2. Let G be a connected graph. Then the following facts hold (1) L(G) + L(G) = nI − J. (2) (L(G) + J)−1 = L+ (G) + (3) (−L(G) + nI)

−1

+

1 J. n2

= L (G) +

1 J. n2

(4) If α, β are scalars in such a way that α I + β J is nonsingular, then (α I + β J)−1 =

1

α

I−

β J. α (α+nβ )

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

R.B. Bapat et al. / Discrete Applied Mathematics (

)

–

5

Proposition 3.3. Let Gi be a connected graph of order pi , for i = 1, . . . , r. Then L+ (∨ri=1 Gi ) = −

1 n2

J + ⊕ri=1 (nIpi − L(Gi ))−1 .

In particular, If λi1 ≥ · · · ≥ λip −1 > λipi = 0 are the Laplacian eigenvalues of Gi , then i

0, [n]

r −1

, n − pi + λ

j i

for i = 1, . . . , pj − 1 and j = 1, . . . , r are the Laplacian eigenvalues of ∨ri=1 Gi , where n = p1 + · · · + pr . Proof. We see that L(∨ri=1 Gi ) + J = ⊕ri=1 (Jpi + (n − pi )Ipi + L(Gi )) = ⊕ri=1 (nI − L(Gi )). In view of Remark 3.2, (L(∨ri=1 Gi ) + J)−1 = L+ (∨ri=1 Gi ) +

1 n2

J = ⊕ri=1 (nIpi − L(Gi ))−1 .

Therefore, L+ (∨ri=1 Gi ) = − n12 J + ⊕ri=1 (nIpi − L(Gi ))−1 .

In particular, in view of Remark 3.2(1) and [1, Lemma 4.5], the Laplacian eigenvalues of Gi are λ = 0 along with pi − λij ,

for j = 1, . . . , pi . Thus, the eigenvalues of L(∨ri=1 Gi ) + J comprise of the eigenvalues of nI − L(Gi ), for i = 1, . . . , r, that is, {n}r , n − pi + λij , for j = 1, . . . , pi and i = 1 . . . , r. Again, in view of [1, Lemma 4.5], the eigenvalues of L(∨ri=1 Gi ) are j

0, [n]r −1 , n − pi +λi for i = 1, . . . , pj − 1 and j = 1, . . . , r are the Laplacian eigenvalues of ∨ri=1 Gi , where n = p1 +· · ·+ pr . ■ Corollary 3.4. Under the assumptions of Proposition 3.3, we have Kf((∨ri=1 Gi )) = r − 1 + n

r pi −1 ∑ ∑

1

i=1 j=1

n − pi + λij

.

Proof. It is enough to employ 3 together with the result of Proposition 3.3. We see Kf((∨

r i=1 Gi ))

⎛ ⎞ n−1 r pi −1 ∑ ∑ ∑ 1 r −1 1 ⎠ =n = n⎝ + λi n n − pi + λij i=1

i=1 j=1

and the proof is complete. ■ Corollary 3.5. Let G := Kp1 ,...,pr be a complete multipartite graph. Then Kf((∨ri=1 Gi )) = r − 1 + n

r ∑ pi − 1 i=1

n − pi

.

Proof. In view of Corollary 5.6, we need only to consider Gi as empty graph of order pi . Apparently, they are 0-regular graphs. j So, in this case we have λi = 0. Substitute in Corollary 5.6 to get Kf((∨ri=1 Gi )) = r − 1 + n

r pi −1 ∑ ∑ i=1 j=1

1 n − pi

=r −1+n

r ∑ pi − 1 i=1

n − pi

,

completing the proof. ■ An alternative way to derive the Kirchhoff index is as follows. Since Kf(G) = ntrace(L+ (G)), thus in view of Proposition 3.3,

( Kf(Kp1 ,...,pr ) = ntrace −

1 n

J + ⊕ri=1 (nIpi − L(Kpi ))−1 2

)

(

) 1 = ntrace − 2 J + ⊕ri=1 (nIpi − L(Kpi ))−1 n ) ( 1 = ntrace − 2 J + ⊕ri=1 ((n − pi )Ipi + Jpi )−1 n ( ( )) 1 1 1 = ntrace − 2 J + ⊕ri=1 I− J n n − pi n(n − pi ) ( )) r ( ∑ −1 pi pi =n + − n n − pi n(n − pi ) i=1

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

6

R.B. Bapat et al. / Discrete Applied Mathematics (

= −1 + n

r ∑ i=1

(

pi n − pi r

= −1 + (n − 1)

1

–

)

n

pi

∑ i=1

1−

)

n − pi

= −1 − r(n − 1) + n(n − 1)

r ∑ i=1

6 Example 3.6. Kf(K2,2,2,3,3,5,7 ) = −1 + 23( 22 +

6 21

1 n − pi

+

5 19

.

+

(4)

7 ) 17

≈ 27.367.

4. Extremal Kirchhoff index In this section, motivated by [7], we explore within all complete r-partite graphs with n vertices, the graphs which have the maximal or the minimal Kirchhoff index. By routine and straightforward calculations we obtain the following lemma. Lemma 4.1. Let x ≥ y > α > 0. Then (1) (2)

1 x+α 1 x−α

+ +

1 y−α 1 y+α

≥ ≤

1 x 1 x

+ +

1 y 1 y

if x − y ≥ α .

Theorem 4.2. Let p1 ≥ · · · ≥ pr and n = p1 + · · · + pr . Kf(Kp1 ,...,pr ) has the maximal Kirchhoff index if Kp1 ,...,pr = K(n−r +1),1r −1 Also, Kf(Kp1 ,...,pr ) has the minimal Kirchhoff index if Kp1 ,...,pr = Kpk ,(p−1)r −k where p = ⌊ nr ⌋ + 1 and k = n − r ⌊ nr ⌋. Proof. In view of (4), parameters p1 , . . . , pr with extremal Kirchhoff index can be derived from those which give extremal ∑r 1 values of f (p1 , . . . , pr ) := i=1 n−p where p1 + · · · + pr = n. Employing Lemma 4.1, we have i

f (p1 , . . . , pr ) ≤ f (p1 + pr − 1, p2 . . . , pr −1 , 1)

≤ f (p1 + pr + pr −1 − 2, p2 , . . . , 1, 1) .. . ≤ f (p1 + pr + · · · + p2 − r + 1, 1, . . . , 1) 1 r −1 = f (n − r + 1, 1, . . . , 1) = + . r −1 n−1 So the maximal Kirchhoff index is attained when p1 = n − r + 1 and pi = 1 for i = 2, . . . , r. In this case we have ) ( r −1 1 + . max (Kf(Kp1 ,...,pr )) = −1 − r(n − 1) + n(n − 1) p1 +···+pr =n r −1 n−1 ∑r 1 As to the minimal Kirchhoff index, let f (p1 , . . . , pr ) := i=1 n−pi be minimal. We claim that |pi − pj | ≤ 1 for 1 ≤ i, j ≤ r. In contrast, let there exist pi and pj such that pi − pj ≥ 2. Then, f (p1 , . . . , pi − 1, . . . , pj + 1, . . . , pr ) − f (p1 , . . . , pr ) =

1 n − pi + 1

+

1 n − pj − 1

−

1 pi

−

1 pj

.

This together with Lemma 4.1 implies that f (p1 , . . . , pi − 1, . . . , pj + 1, . . . , pr ) ≤ f (p1 , . . . , pr ). This violates the minimality ∑r 1 of f (p1 , . . . , pr ) := i=1 n−p . So the maximal Kirchhoff index is attained when |pi − pj | ≤ 1, for 1 ≤ i, j ≤ r. In this regard let

∑r

i

1 f (p1 , . . . , pr ) := i=1 n−pi be minimal. In view of the discussion above, for some integer k we assume that p := p1 = · · · = pk and pk+1 = · · · = pr = p − 1. Since p1 + · · · + pr = n, we see that kp + (r − k)(p − 1). So, p = ⌊ nr ⌋ + 1 and k = n − r ⌊ nr ⌋. In this case we have

( min

(Kf(Kp1 ,...,pr )) = −1 − r(n − 1) + n(n − 1)

p1 +···+pr =n

k n−p

+

r −k n−p+1

)

,

and the proof is complete. ■ Example 4.3. Let n = 24 and r = 7. Then we see that k = 3, p = 4, so 25.943 ≈ Kf(K4,4,4,3,3,3,3 ) ≤ Kf(Kp1 , . . . , p7 ) ≤ 74 = Kf(K18,1,1,1,1,1,1 ). Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

R.B. Bapat et al. / Discrete Applied Mathematics (

)

–

7

5. Degree Kirchhoff index −1

−1

Provided that the graph G has no isolated vertices, the normalized Laplacian matrix is defined as L(G) := D 2 L(G)D 2 , where L(G) is the standard Laplacian matrix of G. Motivated by the notion of Kirchhoff index, Chen and Zhang [4] defined another graph invariant, denoted by Kf′ (G), which is called the degree Kirchhoff index. Definition 5.1. The degree Kirchhoff index of the graph G is defined as Kf′ (G) :=

∑

di dj Ω (i, j),

i
where Ω (i, j) is the resistance distance between vertices i and j. We refer to [2,6,8,11,12] for some results about this topological index. Lemma 5.2. For the graph G with m edges, Kf′ (G) = 2mtrace(L+ (G)). Proof. It is known that if λ1 , . . . , λt are the nonzero eigenvalues of a given matrix A, then trace(A+ ) =

1

λ1

[4, Theorem 2.5], we see that if µ1 ≥ · · · ≥ µn−1 > µn = 0 are the eigenvalues of L(G), then Kf (G) = 2m Definition 5.1 gives Kf′ (G) = 2mtrace(L+ (G)). ■ ′

+ ··· + ∑n−1 1 i=1 µi

1

λt

. By

. Thus,

Before proceeding to obtain the degree Kirchhoff index of complete multipartite graphs we need the following auxiliary lemma. Lemma 5.3 ([10, Theorem 6]). For a given m × n real matrix A and two column vectors c and d, if c lies in the column space of A and d′ lies in the row space of A and if furthermore 1 + d′ A+ c = 0, then (A + cd′ )+ = A+ − kk+ A+ − a+ h+ h + (k+ A+ h+ )kh, at which k := A+ c and h := d′ A+ . Theorem 5.4. Let Kp1 ,...,pr be a complete r-partite graph with n vertices and m edges. Then,

( ′

Kf (Kp1 ,...,pr ) = 2m(n − 1) −

r 1∑

2m −

n

) p2i di

,

i=1

where di is the degree of vertices in a part of size pi . Proof. By Lemma 5.2 , we need only to compute trace(L+ (Kp1 ,...,pr )). Since L(G) = ⊕ri=1 (di Ipi + Jpi ) − Jn , we have L(Kp1 ,...,pr ) = D

−1 2

L(Kp1 ,...,pr )D

) ( −1 1 −1 = ⊕ri=1 Ipi + Jpi − d 2 d′ 2

−1 2

di

where d is the column vector that consists of degrees of vertices. −1 Let A := ⊕ri=1 (Ipi + d1 Jpi ). Then by Remark 3.2(4) we get A−1 = ⊕ri=1 (Ipi − 1n Jpi ). Adopting Lemma 5.3, letting c := d 2 i

−1

1

2

2

n n and d := d′ 2 , we get k = −n1 d 2 and h = −k′ , and so k+ = 2m k′ and h+ = −2m k. On the other hand, the nonsingularity of A implies that c and d′ , respectively, lie in the column and row spaces of A. Furthermore, owing to the fact −1 ′ −1 that L(Kp1 ,...,pr ) = A − d 2 d 2 is not invertible, we get 1 + d′ A−1 c = 0. Up to here all necessary conditions of Lemma 5.3 are fulfilled. So we conclude that,

L+ (Kp1 ,...,pr ) = A−1 − kk+ A−1 − A−1 h+ h + (k+ A−1 h+ )kh

= A−1 − = A−1 − 1 n −1

where X := (2m − Having trace(A 1

1

1

2m 1 2m

1

d 2 d′ 2 A−1 − 1

1

1

2m

1

A−1 d 2 d′ 2 +

1

1

1

(d 2 d′ 2 A−1 + A−1 d 2 d′ 2 ) +

1 n2 X

1

1

1

1

(d′ 2 A−1 d 2 )d 2 d′ 2

(2m)2

1

1

d 2 d′ 2

∑r

2 i=1 pi di ).

1

1

1

1

)=n-1 and trace(d 2 d′ 2 ) = 2m, and that

1

1

1

trace(d 2 d′ 2 A−1 ) = trace(A−1 d 2 d′ 2 ) = d′ 2 A−1 d 2 = X , we obtain trace(L+ (Kp1 ,...,pr )) = (n − 1) −

1 X, 2m

and then

Kf (Kp1 ,...,pr ) = 2mtrace(L (Kp1 ,...,pr )) = 2m(n − 1) − X . ′

+

So Kf′ (Kp1 ,...,pr ) = 2m(n − 1) − (2m −

1 n

∑r

2 i=1 pi di )

as desired. ■

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

8

R.B. Bapat et al. / Discrete Applied Mathematics (

)

–

Table 1 The Kirchhoff indices and the degree Kirchhoff indices of all complete 3-partite graphs with 9 vertices. K(9, 3)

m = |E | Kf′ Kf

K1,1,7

K1,2,6

K1,3,5

K1,4,4

K2,2,5

K2,3,4

K3,3,3

15

20

23

24

24

26

227.33 ↗ 29 ↘

304 ↗ 18.286 ↘

350 ↗ 14 ↘

365.33 ↗ 12.800 ↗

366.67 ↗ 13.571 ↘

398. ↗ 11.686 ↘

27 414. 11

Example 5.5. In these examples we explicitly compute the degree Kirchhoff index of some specific families of complete multipartite graphs. (i) The complete r-partite graph Kq,...,q . So n = qr and m = qr(r − 1). Thus we have Kf′ (Kq,...,q ) = 2(r − 1)(rq − 1)2 . In the case of q = 2 we obtain Cocktail party graph with Kf′ (K2,...,2 ) = 2(r − 1)(2r − 1)2 . 1 (ii) Complete bipartite graph Kp,q . We have p + q = n and m = pq. Thus Kf′ (Kp,q ) = 2pq(p + q) − (2pq − p+ (p2 q + q2 p)). q ′ After simplifying we have Kf (Kp,q ) = pq(2(p + q) − 3) = m(2n − 3). This implies that for star graph Sn = K1,n−1 we have Kf′ = (n − 1)(2n − 3). We introduce some notation. If x ∈ Rn , then we denote by x[1] ≥ · · · ≥ x[n] the components of x arranged in nonincreasing order. If x, y ∈ Rn , then x is said to be majorized by y, denote x ≺ y, if k ∑

x[k] ≤

i=1

and

k ∑

y[k] ,

k = 1, . . . , n − 1

i=1

∑k

i=1 x[n]

=

∑k

i=1 y[n] .

For more information on majorization refer to [9].

Corollary 5.6. Let pi , qi , i = 1, . . . , r be positive integers such that p1 + · · · + pr = q1 + · · · + qr = n. (i) If (q1 , . . . , qr ) ≺ (p1 , . . . , pr ), then Kf′ (Kp1 ,...,pr ) ≥ Kf′ (Kq1 ,...,qr ). (ii) Kf′ (K1r −1 ,n−r +1 ) ≤ Kf′ (Kp1 ,...,pr ) ≤ Kf′ (Kpk ,(p−1)r −k ) where p = ⌊ nr ⌋ + 1 and k = n − r ⌊ nr ⌋. Proof. (i) Choose i and j such that 1 < pj < pi (if p1 , . . . , pr are all equal, then q1 , . . . , qr must also be all be equal and the result is trivial). Let T denote the transformation from (p1 , . . . , pr ) to (p1 , . . . , pi − 1, . . . , pj + 1, . . . , pr ). Then the difference Kf′ (Kp1 ,pi −1,...,pj +1,...,pr ) − Kf′ (Kp1 ,...,pr ) =

1 n

(pi − pj − 1)(2n2 − 6n + 3pi + 3pj )

is evidently negative. Accordingly, Kf′ (Kp1 ,pi −1,...,pj +1,...,pr ) ≤ Kf′ (Kp1 ,...,pr ). Since (q1 , . . . , qr ) ≺ (p1 , . . . , pr ),then (q1 , . . . , qr ) can be obtained from (p1 , . . . , pr ) by a sequence of such transformations and it follows that (i) holds. (ii) It can be verified that (1, 1 . . . , 1, n − r + 1) ≺ (p1 , . . . , pr ) ≺ (p, . . . , p, p − 1, . . . , p − 1),





rtimes



    rtimes



r −1times



where p = ⌊ nr ⌋ + 1 and k = n − r ⌊ nr ⌋. Thus (ii) is an immediate consequence of (i) and Theorem 5.4. ■ We conclude with an open problem. Problem 5.7. Let K(n, r) denote the family of complete r-partite graphs of order n and let G1 , G2 ∈ K(n, r). Let m1 and m2 be the sizes of the edge-sets of G1 and G2 , respectively. Then is it always true that m1 ≤ m2 if and only if kf ′ (G1 ) ≤ kf ′ (G2 )? In K(n, r), unlike the ordinary Kirchhoff index, we could not find any example where m1 ≤ m2 and Kf ′ (G1 ) > Kf ′ (G2 ). See Table 1. Acknowledgements The first author acknowledges support from the JC Bose Fellowship, Department of Science and Technology, Government of India. The third author was partially supported by National Natural Science Foundation of China Grant No. 11601006. References [1] R.B. Bapat, Graphs and Matrices, Vol. 27, second ed., Springer, London, 2010. [2] S. Bozkurt, D. Bozkurt, On the sum of powers of normalized Laplacian eigenvalues of graphs, MATCH Commun. Math. Comput. Chem. 68 (2013) 917–930. [3] A.E. Brouwer, W. Haemers, Spectra of Graphs, Springer Science & Business Media, 2012.

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.

R.B. Bapat et al. / Discrete Applied Mathematics ( [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

)

–

9

H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math. 155 (5) (2007) 654–661. S.V. Gervacio, Resistance distance in complete n-partite graphs, Discrete Appl. Math. 203 (2016) 53–61. M. Hakimi-Nezhaad, A.R. Ashrafi, I. Gutman, Note on degree Kirchhoff index of graphs, Trans. Combin. 2 (3) (2013) 43–52. J.B. Liu, X.F. Pan, L. Yu, D. Li, Complete characterization of bicyclic graphs with minimal Kirchhoff index, Discrete Appl. Math. 200 (2016) 95–107. J.B. Liu, W.R. Wang, Y.M. Zhang, X.F. Pan, On degree resistance distance of cacti, Discrete Appl. Math. 203 (2016) 217–225. A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Vol. 143, Academic press, New York, 2016. C.D. Meyer Jr., Generalized inversion of modified matrices, SIAM J. Appl. Math. 24 (3) (1973) 315–323. J.L. Palacios, A new lower bound for the multiplicative degree-Kirchhoff index, MATCH Commun. Math. Comput. Chem 76 (1) (2016) 251–254. J.L. Palacios, J.M. Renom, Another look at the degree-Kirchhoff index, Int. J. Quantum Chem. 111 (14) (2011) 3453–3455. D.B. West, et al., Introduction to Graph Theory, Vol. 2, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.

Please cite this article in press as: R.B. Bapat, et al., Kirchhoff index and degree Kirchhoff index of complete multipartite graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.040.