Resistance Distance and Kirchhoff Index of Two Edge-subdivision Corona Graphs

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Resistance Distance and Kirchhoff Index of Two

Edge-subdivision Corona Graphs

Qun Liu

a,b

Abstract—In this work we first obtain the group inverse of the edge-subdivision-vertex corona and edge-subdivision-edge corona in terms of the group inverse of the factor graphs. Then the resistance distance and Kirchhoff index of these graphs can be derived from the resistance distance and Kirchhoff index of the factor graphs.

Index Terms—Kirchhoff index; Resistance distance; Edge-subdivision-vertex corona; Edge-subdivision-edge corona

I. INTRODUCTION

L

ET G = (V(G), E(G)) be a graph with vertex set V(G)and edge setE(G). Letdibe the degree of vertex

i in G and DG = diag(d1, d2,· · ·d|V(G)|) the diagonal

matrix with all vertex degrees of G as its diagonal entries. For a graphG, letAG andBG denote the adjacency matrix

and vertex-edge incidence matrix of G, respectively. The matrix LG = DG−AG is called the Laplacian matrix of

G, where DG is the diagonal matrix of vertex degrees of

G. We use µ1(G)≥u2(G)≥ · · · ≥µn(G) = 0 to denote

the eigenvalues ofLG. The{1}-inverse ofM is a matrixX

such thatM XM =M. IfM is singular, then it has infinite

{1}-inverse [1]. We useM(1) to denote any{1}-inverse of a matrixM, and let(M)uv denote the(u, v)-entry ofM.

Klein and Randi´c[2] introduced a new distance function named resistance distance on the basis of electrical network theory. The resistance distance between any two vertices u andv inGis defined to be the effective resistance between them when unit resistors are placed on every edge of G. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices of G. Let ruv(G) denote the

resistance distance betweenuandvinGandKf(G)denote the Kirchhoff index of G. The resistance distance and the Kirchhoff index have attracted extensive attention due to its wide applications in physics, chemistry and others. Up till now, many results on the resistance distance and the Kirchhoff index are obtained. See ([4], [5], [7]), [13]-[19]) and the references therein to know more.

The computation of resistance distance and Kirchhoff index is a hot topic in mathematics, computer science and so on. However, the computation of the effective resistances is difficult, as they are highly sensitive to small perturbations on the network, so this has prompted researchers try to find some techniques to compute the resistance distance

Manuscript received August 18, 2018, revised November 23, 2018. This work was supported by the National Natural Science Foundation of China (Nos. 11461020), the Research Foundation of the Higher Education Institu-tions of Gansu Province, China (2018A-093), the Science and Technology Plan of Gansu Province (18JR3RG206) and Research and Innovation Fund Project of President of Hexi University (XZZD2018003).

Q. Liu is with(a)School of Computer Science, Fudan University, Shang-hai 200433, China and(b)School of Mathematics and Statistics, Hexi Uni-versity, Gansu, Zhangye, 734000, P.R. China. Email: liuqun@fudan.edu.cn.

and Kirchhoff index of a given graph and obtained its closed formula. The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. The set of such new vertices is denoted by I(G). In [6], a new graph operation: edge-subdivision-vertex and edge-subdivision-edge corona are introduced, and theirA-spectra(resp.,L-spectra) are investigated. This paper considers the resistance distance and Kirchhoff index of the graph operations below, which come from [6].

Definition 1 [6]The edge-subdivision-vertex corona of two vertex-disjoint graphsG1andG2, denoted byG1∨G2, is the graph obtained fromG1and|E(G1)|copies ofS(G2) with each edge ofG1 corresponding to one copy ofS(G2) and all vertex-disjoint, by joining end-vertex of theith edge ofE(G1)to each vertex ofV(G2)in theith copy ofS(G2).

Definition 2 [6]The edge-subdivision-edge corona of two vertex-disjoint graphs G1 andG2, denoted by G1∀G2, is the graph obtained fromG1and|E(G1)|copies ofS(G2) with each edge ofG1 corresponding to one copy ofS(G2) and all vertex-disjoint, by joining end-vertex of theith edge ofE(G1)to each vertex ofI(G2)in theith copy ofS(G2). Bu et al. investigated resistance distance in subdivision-vertex join and subdivision-edge join of graphs [7]. Liu et al. [8] gave the resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs. Liu et al. [9] gave the resistance distance and Kirchhoff index of corona and neighborhood corona of two graphs. Lu et al. [10] computed the resistance distance and Kirchhoff index of two corona graphs. Motivated by the results, in this paper, we further explore the generalized inverse of the edge-subdivision-vertex corona and edge-subdivision-edge corona in terms of the generalized inverse of the factor graphs. Thus the effective resistances and Kirchhoff index of the edge-subdivision-vertex corona and edge-subdivision-edge corona can be derived from the resistance distance and Kirchhoff index of the factor graphs.

II. PRELIMINARIES

For a square matrix M, the group inverse of M, denoted by M#, is the unique matrix X such that M XM = M, XM X=X andM X=XM. It is known thatM# exists if and only ifrank(M) =rank(M2)([1],[11]). IfM is real symmetric, then M# exists and M# is a symmetric {1} -inverse ofM. Actually, M# is equal to the Moore-Penrose inverse of M since M is symmetric [11]. It is known that resistance distances in a connected graphGcan be obtained from any{1}-inverse ofG([4]).

Lemma 2.1([11]) LetGbe a connected graph. Then ruv(G) = (L

(1)

G )uu+ (L

(1)

G )vv−(L

(1)

G )uv−(L

(1)

G )vu = (L#G)uu+ (LG#)vv−2(L#G)uv.

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Let1n denote the column vector of dimensionnwith all the

entries equal one. We will often use 1 to denote an all-one column vector if the dimension can be read from the context.

Lemma 2.2 ([7]) For any graph, we haveL#G1 = 0.

Lemma 2.3 ([12]) Let

M =

A B

C D

be a nonsingular matrix. IfA andD are nonsingular, then

M−1 =

A−1+A−1BS−1CA−1 A−1BS−1

−S−1CA−1 S−1

=

(A−BD−1C)−1 A−1BS−1

−S−1CA−1 S−1

,

whereS=D−CA−1B.

For a square matrixM, let tr(M)denote the trace of M.

Lemma 2.5([16]) LetGbe a connected graph onnvertices. Then

Kf(G) =ntr(L(1)G )−1TL(1)G 1 =ntr(L#G).

Lemma 2.6 ([15]) LetGbe an r-regular graph withn ver-tices andmedges,µG(x)denote the Laplacian characteristic

polynomial ofG,l(G)be the line graph of G. Then

µl(G)(x) = (x−2r)m−nµG(x).

Lemma 2.7([3]) Let G be a connected graph of order n with edge setE. Then

X

u<v,uv∈E

ruv(G) =n−1.

For a vertex i of a graph G, let T(i) denote the set of all neighbors ofiinG.

Lemma 2.8 ([17]) Let L=

A B

BT D

be the Laplacian matrix of a connected graph. If D is nonsingular, then

X =

H# H#BD−1

−D−1BTH# D−1+D−1BTH#BD−1

is a symmetric{1}-inverse ofL, whereH =A−BD−1BT.

III. RESISTANCE DISTANCE ANDKIRCHHOFF INDEX OF EDGE-SUBDIVISION-VERTEX CORONA FOR GRAPHS

In this section, we focus on determing the resistance dis-tance and Kirchhoff index of edge-subdivision-vertex corona whenever G1 is anr1-regular graph.

Theorem 3.1 LetG1be anr1-regular graph withn1vertices andm1edges andG2anr2-regular graphs onn2vertices and m2 edges. ThenG=G1∨G2 have the resistance distance and Kirchhoff index

(i) For any i, j∈V(G1), we have

rij(G) = 2 n2+ 2

L#1

ii

+ 2

n2+ 2

L#1

jj

− 4

n2+ 2

L#1

ij

= 2

n2+ 2

rij(G1).

(ii) For anyi, j∈V(G2), we have

rij(G) =

(2In2+

1 2L2)

−1

⊗Im1

ii +

(2In2+

1 2L2)

−1

⊗Im1)jj−2

(2In2+

1 2L2)

−1

⊗Im1

ij

.

(iii) For anyi∈V(G1), j∈V(G2), we have

rij(G) = 2 n2+ 2

L#1

ii +

(2In2+

1 2L2)

−1

⊗Im1)jj− 4 n2+ 2

L#1

ij.

(iv)Kf(LG)

= (n1+n2+ 2m1) 2 +r

1(n2+m2) n1(n2+ 2)

Kf(G1)

+m1

n2

X

i=1

1

1

2µi(G2) + 2

+ (r2+ 2)m1

n1

X

i=1

1 4 +µi(G2)

+m1(r2+ 2)(n2−m2) 4 + 2r2

−(n1−1)(n2+m2) 2(n2+ 2)

−6m1m2+m2(r2+ 2)

4 ,

whereµi(G2)is the Laplacian eigenvalues ofG2.

Proof LetRi(i= 1,2)be the incidence matrix ofGi. Then

with a proper labeling of vertices, the Laplacian matrix of G1∨G2can be written as

L(G1∨G2)

= 

L1+r1n2In1 −1 T

n2⊗R1 0n1×m1m2 −1n2⊗R

T

1 (2 +r2)In2⊗Im1 −R2⊗Im1

0m1m2×n1 −R T

2 ⊗Im1 2Im1m2

.

LetA =L1+r1n2In1,B = −1 T

n2⊗R1 0n1×m1m2

,

BT =

−1n2⊗R T

1

0m1m2×n1

, and

D=

(2 +r2)In2⊗Im1 −R2⊗Im1 −RT2 ⊗Im1 2Im1m2

.

First we compute theD−1. By Lemma 2.3, we have A1−B1D1−1C1

= (2 +r2)In2⊗Im1−

1

2(R2⊗Im1)(R T

2 ⊗Im1)

= [(2 +r2)In2−

1

2(r2In2+A(G2))]⊗Im1

= (2In2+

1

2L2)⊗Im1,

so(A1−B1D1−1C1)−1= (2In2+

1 2L2)

−1I

m1.

By Lemma 2.3, we have

S = (D1−C1A−11B1)

= 2Im1m2−

1

r2+2(R T

2 ⊗Im1)(In2⊗Im1)(R2⊗Im1)

= 2Im1m2−

1

r2+2(R T

2R2⊗Im1)

= (2Im1m2−

1

r2+2(2Im2+A(l(G2)))⊗Im1

= r1

2+2(4Im2+Ll(G2))⊗Im1.

SoS−1= (r2+ 2)(4Im2+Ll(G2))

−1I

m1.

By Lemma 2.3, we have

−A−11B1S−1

= − 1

r2+2(In2⊗Im1)(−R2⊗Im1)(r2+ 2)

((4Im2+Ll(G2))

−1I

m1)

= R2(4Im2+Ll(G2))

−1I

m1.

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Similarly, −S−1C

1A−11= (4Im2+Ll(G2))

−1RT

2 ⊗Im1.

Let V = (2In2 +

1 2L2)

−1I

m1, T = (r2+ 2)(4Im2 +

Ll(G2))

−1I

m1,M =R2(4Im2+Ll(G2))

−1I

m1.

So

D−1 =

V M

MT T

.

Now we are ready to calculate H. Let P = (2In2 +

1 2L2)

−1I

m1, Q = (r2+ 2)(4Im2 +

Ll(G2))

−1I

m1,M =R2(4Im2+Ll(G2))

−1I

m1, then

H = L1+r1n2In1− −1 T

n2⊗R1 0

V M

MT T

−1n2⊗R T

1

0

= L1+r1n2In1− n2

2 R1R

T

1 =

n2+2

2 L1, ,

By Lemma 2.8, we haveH#= 2

n2+2L

# 1.

Next according to Lemma 2.8, we calculate−H#BD−1and

−D−1BTH#.

−H#BD−1

= − 2

n2+2L

#

1 −1Tn2⊗R1 0

!

P M

MT Q

!

= n2

2+2L

# 1

1 21

T

n2⊗R1 1 T n2M

!

.

Note that1T

n2R2= 2·1 T

m2, then1 T

n2R2(4Im2+Ll(G2))

−1

R1= 2·1Tm2(4Im2+Ll(G2))

−1R

1= 121Tm2⊗R1, so

−H#BD−1 = 1

n2+2L

#

1 1Tn2⊗R1 1 T m2⊗R1

!

and

−D−1BTH#

= −

P M

MT Q

−1n2⊗R T

1

0

1

n2+2L

# 1

= 1

n2+2

1n2⊗R T

1

1m2⊗R T

1

L#1.

We are ready to compute the D−1BTH#BD−1. Let1n2⊗R

T

1 =H,K= 1Tm2⊗R1, then

D−1BTH#BD−1

= 1 2H 1 2K T 1

n2+2L

# 1 H

T K

!

= 2(n1

2+2)

HL#1HT HL#1K KTL#

1HT KTL # 1K

.

Let1n2⊗R T

1 =H,K= 1Tm2⊗R1, then based on Lemma

2.3 and 2.7, the following matrix

N =

2

n2 +2L

#

1 n2 +21 L

#

1HT n2 +21 L

# 1K 1

n2 +2HL #

1 P+2(n2 +2)1 HL #

1HT M+2(n2 +2)1 HL # 1K 1

n2 +2KT L #

1 M T+2(n2 +2)1 KT L #

1HT Q+2(n2 +2)1 KT L # 1HT

!

is a symmetric{1}-inverse ofLG1∨G2, where P= (2In2+

1 2L2)

−1I

m1, Q = (r2+ 2)(4Im2 +Ll(G2))

−1 I

m1,

M =R2(4Im2+Ll(G2))

−1I

m1. LetN be Equation(3.1).

For anyi, j∈V(G1), by Lemma 2.1 and the Equation(3.1), we have

rij(G1∨G2)

= 2

n2+ 2

L#1

ii

+ 2

n2+ 2

L#1

jj

− 4

n2+ 2

L#1

ij = 2 n2+ 2

rij(G1).

For anyi, j∈V(G2), by Lemma 2.1 and the Equation(3.1), we have

rij(G1∨G2)

=

(2In2+

1 2L2)

−1I

m1

ii +

(2In2+

1 2L2)

−1

⊗Im1)jj−2

(2In2+

1 2L2)

−1I

m1

ij

.

For any i ∈ V(G1), j ∈ V(G2), by Lemma 2.1 and the Equation(3.1), we have

rij(G1∨G2)

= 2

n2+ 2

L#1

ii +

(2In2+

1 2L2)

−1I

m1

jj

− 4

n2+ 2

L#1

ij

.

By Lemma 2.5, we have Kf(LG1∨G2)

= (n1+n2+ 2m1)tr(N)−1TN1

= (n1+n2+ 2m1)

2

n2+ 2

trL#1

+tr

(2In2+

1 2L2)

−1

⊗Im1

+(r2+ 2)tr (4Im2+Ll(G2))

−1I

m1

+ 1

2(n2+ 2)

tr(1n2⊗R T

1)L # 1(1

T n2⊗R1)

+ 1

2(n2+ 2)

tr(1m2⊗R T

1)L # 1(1

T

m2⊗R1)

−1TN1.

Note that the eigenvalues of (2In2 +

1

2L2)are 12µ1(G2) +

2,12µ2(G2) + 2, ...,12µn2(G2) + 2. Then

tr((2In2+

1 2L2)

−1I

m1)

−1 = m 1P

n2 i=1(

1

2µi(G2) + 2)

−1

= m1P

n2 i=1

1

1

2µi(G2)+2.

By Lemma 2.6, then

tr((4Im2+Ll(G2))

−1I

m1) = m1

Pn1 i=1

1 4+µi(G2)+ m1(n2−m2)

4+2r2 .

By Lemma 2.7, we have tr(1n2⊗R

T

1)L #

1(1Tn2⊗R1)

= n2tr

RT

1L # 1R1

= n2Pi<j,ijE(G)

L#ii +L#jj+ 2L#ij

= n2Pi<j,ij∈E(G)

2L#ii + 2L#jj−rij(G1)

= 2n2r1tr(L#G1)−n2(n1−1).

Similarly,

tr(1m2⊗R T

1)L # 1(1

T

m2⊗R1)

= 2m2r1tr(L #

G1)−m2(n1−1) .

So

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Kf(LG1∨G2)

= (n1+n2+ 2m1)tr(N)−1TN1

= (n1+n2+ 2m1)

2

n1(n2+ 2)

Kf(G1)

+m1

n2

X

i=1

1

1

2µi(G2) + 2

+(r2+ 2)tr (4Im2+Ll(G2))

−1I

m1

+ 1

2(n2+ 2)

tr(1n2⊗R T

1)L # 1(1

T n2⊗R1)

+ 1

2(n2+ 2)

tr(1m2⊗R T

1)L # 1(1

T

m2⊗R1)

−1TN1

= (n1+n2+ 2m1)

2 n1(n2+ 2)

Kf(G1)

+m1

n2

X

i=1

1

1

2µi(G2) + 2

+(r2+ 2) m1

n1

X

i=1

1 4 +µi(G2)

) +m1(n2−m2) 4 + 2r2

!

+ 1

2(n2+ 2)

2n2r1tr(L #

G1)−n2(n1−1)

+ 1

2(n2+ 2)

2m2r1tr(L #

G1)−m2(n1−1)

−1TN1.

Next, we calculate the1T(L(1)G

1∨G2)1. SinceL

#

G1 = 0, then 1T(L(1)

G1∨G2)1

= 1T((2In2+

1 2L2)

−1I

m1)1 + 1 T((r

2+ 2)

(4Im2+Ll(G2))

−1I

m1)1

+1T(R2(4Im2+Ll(G2))

−1I

m1)1

+1T((4Im2+Ll(G2))

−1RT

2 ⊗Im1)1

+ 1

2(n2+ 2)

1T(1n2⊗R T

1)L # 1(1

T

n2⊗R1)1

+ 1

2(n2+ 2)

1T(1n2⊗R T

1)L # 1(1

T

m2⊗R1)1

+ 1

2(n2+ 2)

1T(1m2⊗R T

1)L # 1(1

T

n2⊗R1)1

+ 1

2(n2+ 2)

1T(1m2⊗R T

1)L # 1(1

T

m2⊗R1)1.

LetT = 1T

m1n2((2In2+

1 2L2)

−1I

m1)1m1n2,F = ((2In2+

1

2L2)⊗Im1), then

T = 1T

n2 1 T

n2 · · · 1 T n2

   

F−1 F−1

. .. F−1

   

  

1n2

1n2 · · · 1n2

  

= m11Tn2(2In2+

1 2L2)

−11

n2 = m1n2

2 .

Similarly,

1T(4I

m2+Ll(G2))

−11 = m2

4 ,1

T(R

2(4Im2 +Ll(G2))

−1

Im1)1 = 1 T((4I

m2+Ll(G2))

−1RT

2 ⊗Im1)1 = m1m2

2 .

By direct computation,

1T(1 n2⊗R

T

1)L #

1(1Tn2⊗R1)1 =n

2 2πTL

#

1π=n22r121TL # 11 =

0, 1T(1 m2⊗R

T

1)L #

1(1Tm2⊗R1)1 =m

2 2πTL

# 1π= 0,

1T(1n2⊗R T

1)L # 1(1

T

m2⊗R1)1 = 1 T(1

m2⊗R T

1)L # 1(1

T n2⊗

R1)1 =n2m2πTL # 1π= 0.

So

1T(L(1)G

1∨G2)1 =

m1m2

2 +

m2(r2+ 2)

4 +m1m2

= 6m1m2+m2(r2+ 2)

4 .

Lemma 2.5 implies that

Kf(G) = (n1+n2+ 2m1)tr(N)−1TN1.

Then pluggingtr(L(1)G

1∨G2)and1 T(L(1)

G1∨G2)1into the

equa-tion above, we obtain the required result.

IV. RESISTANCE DISTANCE ANDKIRCHHOFF INDEX OF EDGE-SUBDIVISION-EDGE CORONA FOR GRAPHS

In this section, we focus on determing the resistance dis-tance and Kirchhoff index of edge-subdivision-edge corona wheneverG1 is an r1-regular graph.

Theorem 4.1 LetG1be anr1-regular graph withn1vertices andm1 edges andG2 anr2-regular graphs with n2vertices andm2edges. ThenG1∀G2have the resistance distance and Kirchhoff index

(i) For anyi, j∈V(G1), we have rij(G1∀G2)

= 2

m2+ 2

(L#1)ii+ 2 m2+ 2

(L#1)jj − 4 m2+ 2

(L#1)ij

= 2

m2+ 2

rij(G1).

(ii) For anyi, j∈V(G2), we have rij(G1∀G2)

= (In1⊗(LG2+In2)

−1)

ii+ (In1⊗(LG2+In2)

−1)

jj −2(In1⊗(LG2+In2)

−1

)ij.

(iii) For anyi∈V(G1), j∈V(G2), we have rij(G1∀G2)

= 2

m2+ 2

L#1

ii

+ In1⊗(LG2+In2)

−1

jj

− 4

m2+ 2

L#1

ij

.

(iv)Kf(G1∀G2)

= (n1+n2+ 2m1) 4r

2+ 2n1r31+m1r2

2n1r2(m2+ 2)

Kf(G1)

+m1

n2

X

i=1

1

1

4µi(G2) +

r2

2

+r2m1

n2

X

i=1

1 2r2+µi(G2)

+m1(n2−m2)

4 −

(n1−1)(n1r12+m2r2)

2r2(m2+ 2)

−4m1n2+m2r2+ 4m1m2 2r2

.

Proof LetRi(i= 1,2)be the incidence matrix ofGi. Then

with a proper labeling of vertices, the Laplacian matrix of G1∀G2 can be written as

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L(G1∀G2) =

L1+r1m2In1 0n1×m1n2 −1 T m2⊗R1

0m1n2×n1 r2In2⊗Im1 −R2⊗Im1 −1m2⊗R

T

1 −R

T

2 ⊗Im1 4Im1m2

.

LetA=L1+r1m1In1,B = 0n1×m1n2 −1 T m2⊗R1

,

BT =

0m1n2×n1 −1m2⊗R T

1

and

D=

r2In2⊗Im1 −R2⊗Im1 −RT2 ⊗Im1 4Im1m2

.

First we compute the D−1. By Lemma 2.3, we have A1−B1D−11C1

= r2In2⊗Im1−

1

4(R2⊗Im1)(R T

2 ⊗Im1)

= (r2In2−

1

4(r2In2+A(G2)))⊗Im1

= (r2

2In2+

1

4L2)⊗Im1,

so (A1−B1D1−1C1)−1= (r22In2+

1 4L2)

−1I

m1.

By Lemma 2.3, we have

S = D1−C1A−11B1

= 4Im1m2−

1

r2(R T

2 ⊗Im1)(In2⊗Im1)(R2⊗Im1)

= 4Im1m2−

1

r2(R T

2R2⊗Im1)

= 4Im1m2−

1

r2((2Im2+A(l(G2))⊗Im1)

= r1

2(2r2Im2+Ll(G2))⊗Im1,

so S−1=r

2(2r2Im2+Ll(G2))

−1I

m1.

By Lemma 2.3, we have

−A−11B1S−1 = −r12(In2⊗Im1)(−R2⊗Im1)

r2((2r2Im2+Ll(G2))

−1I

m1)

= R2(2r2Im2+Ll(G2))

−1I

m1.

Similarly, −S−1C

1A−11= (2r2Im2+Ll(G2))

−1RT

2 ⊗Im1.

Let P = (r2

2In2 +

1

4L2)−1 ⊗Im1, Q = r2(2r2Im2 +

Ll(G2))

−1I

m1,M =R2(2r2Im2+Ll(G2))

−1I

m1, then

D−1 =

P M

MT Q

.

Now we are ready to calculate H.

H = L1+r1m2In1− 0n1×m1n2 −1 T m2⊗R1

P M

MT Q

0m1n2×n1 −1m2⊗R T

1

= L1+r1m2In1− m2

2 R1R

T

1 =

m2+2

2 L1.

By Lemma 2.8, we haveH#= m2

2+2L

# 1.

Next according to Lemma 2.8, we calculate−H#BD−1and

−D−1BTH#.

Note thatR(G)1=π, whereπ= (d1, d2, ..., dn)T, then −H#BD−1

= − 2

m2+2L

#

1 0 −1Tm2⊗R1

!

P M

MT Q

!

= m2

2+2L

# 1

1 21

T m2R

T

2 ⊗R1 121Tm2⊗R1

!

= m1

2+2L

# 1 π

T R

1 1Tm2⊗R1

!

and

−D−1BTH#

= −

P M

MT Q

0 −1m2⊗R

T

1

1

m2+2L

# 1

= 1

m2+2

π⊗RT

1

1m2⊗R T

1

L#1.

We are ready to compute theD−1BTH#BD−1. Let W = π⊗RT

1,R= 1m2⊗R T

1, then D−1BTH#BD−1

=

1

2r2π⊗R T

1 1

2(1m2⊗R T

1)

1

m2+2L

#

1 πT ⊗R1 1Tm2⊗R1

= 1

m2+2

1 2r2W L

# 1W

T 1 2r2W L

# 1R

T

1 2RL

# 1W

T 1 2RL

# 1R

T

!

.

Based on Lemmas 2.3 and 2.8, the following matrix

N =

2

m2 +2L

#

1 m2 +21 L

#

1W m2 +21 L

# 1RT 1

m2 +2W T P+ 1 2r2 (m2 +2)W L

#

1W T M+2r2 (m12 +2)W L # 1RT 1

m2 +2R M T+ 1 2(m2 +2)RL

#

1W T Q+2(m12 +2)RL # 1RT

!

is a symmetric{1}-inverse ofLG1∀G2, whereP = ( r2

2In2+

1 4L2)

−1I

m1, Q =r2(2r2Im2 +Ll(G2))

−1I

m1, M =

R2(2r2Im2 +Ll(G2))

−1 I

m1. Let the above N be the

Equation(4.1).

For anyi, j∈V(G1), by Lemma 2.1 and the Equation(4.1), we have

rij(G1∀G2)

= 2

m2+ 2

(L#1)ii+ 2 m2+ 2

(L#1)jj − 4 m2+ 2

(L#1)ij

= 2

m2+ 2

rij(G1).

For anyi, j∈V(G2), by Lemma 2.1 and the Equation(4.1), we have

rij(G1∀G2)

=

(r2 2In2+

1 4L2)

−1I

m1

ii +

(r2

2In2+

1 4L2)

−1I

m1

jj

−2

(r2 2 In2+

1 4L2)

−1I

m1

ij

.

For any i ∈ V(G1), j ∈ V(G2), by Lemma 2.1 and the Equation(4.1), we have

rij(G1∀G2)

= 2

m2+ 2

L#1

ii+

(r2

2In2+

1 4L2)

−1I

m1

jj

− 4

m2+ 2

L#1

ij.

By Lemma 2.5, we have

IAENG International Journal of Applied Mathematics, 49:1, IJAM_49_1_17

(6)

Kf(LG1∀G2)

= (n1+n2+ 2m1)tr(N)−1TN1

= (n1+n2+ 2m1)

2

m2+ 2 tr(L#1)

+tr

(r2 2In2+

1 4L2)

−1

⊗Im1

+r2tr (2r2Im2+Ll(G2))

−1I

m1

+ 1

2r2(m2+ 2)

tr(π⊗RT1)L # 1(π

T ⊗R1)

+ 1

2(m2+ 2)

tr(1m2⊗R T

1)L # 1(1

T

m2⊗R1))

−1TN1.

Note that the Laplacian eigenvalues of (r2

2In2 +

1 4L2) are 1

4µ1(G2) +

r2

2, 1

4µ2(G2) +

r2

2, ..., 1

4µn2(G2) + r2

2. Then

tr((r2

2In2+

1 4L2)

−1I

m1)

−1

= m1Pni=12 (41µi(G2) +r22)

= m1Pni=12 1 1 4µi(G2)+r22

.

By Lemma 2.6, then

tr((2r2Im2+Ll(G2))

−1I

m1) =

m1P

n2 i=1

1 2r2+µi(G2)+

m1(n2−m2)

4r2 . .

By direct computation and Lemma 2.7, tr(π⊗RT1)L

# 1(π

T R

1)

= (Pn1 i=1d

2

i)tr(RT1L # 1R1)

= (Pn1 i=1d

2

i)

P

i<j,ij∈E(G)

L#ii +L#jj+ 2L#ij

= (Pn1 i=1d

2

i)

P

i<j,ij∈E(G)

2L#ii+ 2L#jj−rij(G1)

= 2(Pn1 i=1d

2

i)tr(DG1L

#

G1)−(

Pn1 i=1d

2

i)(n1−1)

= 2n1r31tr(L #

G1)−n1r

2

1(n1−1),

.

and

tr(1m2⊗R T

1)L # 1(1

T

m2⊗R1)

= m2tr(RT1L # 1R1)

= m2Pi<j,ij∈E(G) h

L#ii +L#jj+ 2L#iji

= m2tr(2L#ii + 2L

#

jj−rij(G1))

= m2(tr(DG1L

#

G1)−(n1−1))

= m2(r1tr(LG#1)−(n1−1)).

.

Next, we calculate the1T(L(1)

G1∨G2)1. SinceL

#

G1 = 0, then 1T(L(1)G

1∀G2)1

= 1T((r2 2In2+

1 4L2)

−1

⊗Im1)1 +r21 T(2r

2Im2

+Ll(G2))

−1I

m11

+1T(R2(2r2Im2+Ll(G2))

−1I

m1)1

+1T((2r2Im2+Ll(G2)R T

2)− 1

⊗Im1)1

+ 1

2r2(n2+ 2)

1T(π⊗RT1)L#1(πT ⊗R1)1

+ 1

2r2(n2+ 2)

1T(π⊗RT1)L#1(1Tm2⊗R1)1

+ 1

2(m2+ 2)

1T(1m2⊗R T

1)L # 1(π

T R

1)1

+ 1

2(m2+ 2)

1T(1m2⊗R T

1)L # 1(1

T

m2⊗R1)1.

Similarly, 1T((r2

2In2 +

1 4L2)

−1 I

m1)

−11 = 2m1n2 r2 ,

1T(2r

2Im2 + Ll(G2))

−11 = m2

2r2, 1 T(R

2(2r2Im2 +

Ll(G2))

−1I

m1)1 = 1 T((2r

2Im2+Ll(G2))

−1RT

2⊗Im1)1 = m1m2

r2 .

By direct computation, 1T(π ⊗ RT1)L#1(πT ⊗ R1)1 = n2

11TRT1L #

1R11 =n21r221TL # 11 = 0. Similarly, 1T(π ⊗ RT1)L

# 1(1

T

m2 ⊗ R1)1 = 1 T(1

m2 ⊗

RT

1)L #

1(πT ⊗R1)1 = 1T(1m2⊗R T

1)L #

1(1Tm2⊗R1)1 = 0.

So

1T(G1∀G2)1 =

2m1n2 r2

+m2

2 +

2m1m2 r2

= 4m1n2+m2r2+ 4m1m2 2r2

.

Lemma 2.5 implies that

Kf(G1∀G2) = (n1+n2+ 2m1)tr(N)−1TN1.

Then pluggingtr(L(1)G

1∀G2)and1 T(L(1)

G1∀G2)1into the

equa-tion above, we obtain the required result.

V. CONCLUSION

In this paper, we give the closed-form formulas for resistance distance and Kirchhoff index of the edge-subdivision-vertex and edge-subdivision-edge corona. This method is a general method. The resistance distance and Kirchhoff index of the the vertex corona and edge-subdivision-edge corona can obtain in terms of the resistance distance and Kirchhoff index of the factor graph.

ACKNOWLEDGMENT

The author is very grateful to the anonymous referees for their carefully reading the paper and for constructive com-ments and suggestions which have improved this paper.

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References

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