# Two Kinds of Laplacian Spectra and Degree Kirchhoff Index of the Weighted Corona Networks

Authors: Haiqin Liu; Yanling Shao
Journal: Journal of Mathematics
 (2022)

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## Abstract

Recently, the study related to network has aroused wide attention of the scientific community. Many problems can be usefully represented by corona graphs or networks. Meanwhile, the weight is a vital factor in characterizing some properties of real networks. In this paper, we give complete information about the signless Laplacian spectra of the weighted corona of a graphG1 and a regular graph G2 and the complete information about the normalized Laplacian spectra of the weighted corona of two regular graphs. The corresponding linearly independent eigenvectors of all these eigenvalues are also obtained. The spanning trees’ total number and the degree Kirchhoff index of the weighted corona graph are computed.


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## Body

## 1. Introduction

In recent years, the network research has attracted more and more attention from scientific communities [1–5], for example, chemical networks, social networks, disease transmission networks, and product networks. Especially, product networks can result some larger networks than before if through some given rules. We can synthesize new compounds through product networks and so on. However, previous researchers pay more attention to the binary networks [6–8]. However, the fact is that the weighted networks which are the extension of the binary networks are more in conformity to the networks in real world. Through the weighted networks, we can model the real world much better.The definition of the corona and the edge corona about two graphs were given in [9] and some results about the corona have been illustrated in [10]. Azari [11–13] deduced the eccentric connectivity coindex, the Mostor index, and the edge Wiener index of the corona of two graphs. Barik et al. [14] completely represented the information about the spectrum of two graphs’ corona. Hou and Shiu [15] determined the eigenvalues of the edge corona graphs of two graphs and gave the eigenvectors corresponding to them. More information about the spectra of G1°G2, such as Laplacian spectra and signless Laplacian spectra, were depicted in [16–18]. Chen and Liao [19] gave a complete description of the normalized Laplacian spectra of the corona and edge corona of two graphs.Liu et al. [20] determined the normalized Laplacian spectra of a linear octagonal-quadrilateral network and computed the number of spanning trees of the octagonal-quadrilateral networks. The Kirchhoff index, multiplicative degree Kirchhoff index, and number of spanning trees of the linear crossed hexagonal chains were obtained in [21]. Recently, Dai et al. [22] deduced the complete conclusion of generalized adjacency eigenvalues and generalized Laplacian eigenvalues of the weighted corona graph with different structures. Liu et al. [23] determined the generalized adjacency, Laplacian, and signless Laplacian spectra of the weighted edge corona networks.Motivated by this, we study two kinds of Laplacian spectra (the signless Laplacian spectrum and the normalized Laplacian spectrum) of the weighted corona of two graphs. We calculate the spanning trees’ number and the degree Kirchhoff index of the weighted corona graph as the application of the normalized Laplacian spectrum.In this paper, we only pay attention to simple graphs. LetG be a simple graph with vertex set V=v1,v2,…,vn. Let DG denote the diagonal degree matrix of G. Let AG be the adjacency matrix of G. It is defined as AG=aijn, where aij=1 if vertices i and j are adjacent in G and 0 otherwise.The signless Laplacian matrix ofG is QG=DG+AG. The signless Laplacian spectrum of the graph G is actually the spectrum of QG. It is denoted by(1)SG=ξ1G,ξ2G,…,ξnG,where ξ1G≤ξ2G≤⋯≤ξnG are the eigenvalues of QG.The normalized Laplacian matrix ofG is defined as(2)ℒG=DG−1/2DG−AGDG−1/2,which is closely correlated with spectrum geometry and the simple random walk on G. Given a simple graph G, let PG=DG−1AG be the matrix about the simple random walk on G (which is named the probability transition matrix); then,(3)ℒG=DG1/2I−DG−1AGDG−1/2=DG1/2I−PGDG−1/2.So, if the spectrum ofℒG is denoted by(4)NG=η1G,η2G,…,ηnG,where 0=η1G≤η2G≤⋯≤ηnG≤2 and ηiG,i=1,2,…,n, are the eigenvalues of ℒG. The spectrum of PG is denoted by(5)NpG=μ1G,μ2G,…,μnG,where 1=μ1G≥μ2G≥⋯≥μnG≥−1 and μiG,i=1,2,…,n, are the eigenvalues of PG. Then,(6)ηiG=1−μiG,i=1,2,…,n.More results of the normalized Laplacian spectra are depicted in [24, 25].Definition 1.
(see [9]). The corona of two graphs G1 and G2, which is denoted by G1°G2, is defined as the graph obtained by taking one copy of G1 and VG1 copies of G2 and then joining the ith vertex of G1 to every vertex in the ith copy of G2.Definition 2.
(see [9]). The edge corona of two graphs G1 and G2, which is denoted by G1⋄G2, is defined as the graph obtained by taking one copy of G1 and EG1 copies of G2 and then joining the two vertices of each edge of graph G1 with all vertices of corresponding copy of graph G2 (see Figure 1).Figure 1
Edge coronaG1⋄G2 of G1 and.G2.Dai et al. have defined the weighted corona graphs of two graphs [22]. Suppose that the initial graph G1 and copy graph G2 are simple graphs. In the weighted corona of G1 and G2 (denoted by G=G1°˜G2), each edge of G1 has a unit weight and each edge in the copy of G2 has a weight factor r, 0<r≤1, and the edges which are newly generated have the weight r (see Figure 2). Let ωij be the weight of edge connecting vertex i and j and WG be the generalized adjacency matrix of G, where its element is ωij if i and j are adjacent, 0 otherwise. Let A=aij and B be matrices. The partition matrix aijB is defined as the Kronecker product of A and B, which are denoted by A⊗B [26]. If every multiplication makes sense, we can obtain(7)A1⊗A2A3⊗A4=A1A3⊗A2A4.Figure 2
The weighted corona graph,G1°˜G2.LetJn denote the row vector of order n with all entries equal to 1, In denote the identity matrix of order n, 0n denote the zero matrix of order n, and ei be the ith unit column vector.The paper is organized as below. In Section2, the signless Laplacian spectrum of weighted corona graph is given and the corresponding linearly independent eigenvectors of all these eigenvalues are also obtained. In Section 3, the analytical formulas of the normalized Laplacian eigenvalues are determined and the corresponding linearly independent eigenvectors of all these eigenvalues are depicted. In Section 4, we give the number of spanning trees and the degree Kirchhoff index of the weighted corona graph. Finally, we draw the conclusion.


## 2. The Signless Spectrum of Weighted Corona GraphG1°˜G2

The strength of vertexi is d˜i=∑j∈Gωiji=1,2,…,n. Let D˜G=diagd˜1,d˜2,…,d˜n denote the diagonal intensity matrix of G. Then, the signless Laplacian matrix of the weighted graph G is defined by QG=D˜G+WG. The spectrum of QG is said to be the signless Laplacian spectrum of G. In this section, we will give the signless Laplacian eigenvalues of the weighted corona graph G=G1°˜G2.LetG1 be a graph with n1 vertices and G2 be a k-regular graph with n2 vertices. Then, the signless Laplacian matrix of the weighted corona graph G1°˜G2 is as follows:(8)QG1°˜G2=QG1+rn2In1rJn2⊗In1rJn2T⊗In1rQG2+In2⊗In1,where QG1 and QG2 are the signless Laplacian matrices of the graphs G1 and G2, respectively.Theorem 1.
LetG1 be a graph with n1 vertices and G2 be a k-regular graph with n2 vertices, and SG1=μ1,μ2,…,μn1 and SG2=η1,η2,…,ηn2=2k. Then, the signless Laplacian spectra of the weighted corona G1°˜G2 are as follows.(i)
μi+rn2+2k+1r±μi+rn2+2k+1r2−42k+1rμi−8r2n2k/2∈SG1°˜G2 with multiplicity 1 for i=1,2,…,n1(ii)
rηj+1∈SG1°˜G2 with multiplicity n1 for j=1,2,…,n2−1Proof.
Letλ be the eigenvalue of QG1°˜G2, X=X1X2⋯Xn2+1T, and Xi∈Rn1i=1,2,…,n2+1 be the eigenvector which corresponds to λ. Then,(9)QG1°˜G2X=λX.
From equations (8) and (9), we can conclude that(10)QG1+rn2In1X1T+rX2T+X3T+⋯+Xn2+1T=λX1T,and(11)lrX1T+rE1QG2+In2⊗In1X2X3…Xn2+1T=λX2T,rX1T+rE2QG2+In2⊗In1X2X3…Xn2+1T=λX3T,⋮rX1T+rEn2QG2+In2⊗In1X2X3…Xn2+1T=λXn2+1T,where Ei=0n1,…,0n1︷i−1,In1,0n1,…,0n1︷n2−i.
By adding the above equations in equation (11), we have(12)rn2X1T+2k+1rX2T+X3T+⋯+Xn2+1T=λX2T+X3T+⋯+Xn2+1T,which is(13)rn2X1T=λ−2k+1rX2T+X3T+⋯+Xn2+1T.Case 1:X1 is the nonzero vector.In this case, we immediately deduce thatλ−2k+1r≠0. From equations (10) and (13), it gives(14)QG1+rn2In1X1T+r2n2λ−2k+1rX1T=λX1T,so(15)QG1X1T=λ−rn2−r2n2λ−2k+1rX1T.Noticing thatSG1=μ1,μ2,…,μn1 and according to equation (15), one obtains(16)λ−rn2−r2n2λ−2k+1r=μi,i=1,2,…,n1,which is(17)λ2−2k+1r+rn2+μiλ+2r2n2k+2k+1rμi=0,i=1,2,…,n1.By solving this equation aboutλ, we obtain(18)λi,λ¯i=μi+rn2+2k+1r±μi+rn2+2k+1r2−42k+1rμi−8r2n2k2.Case 2:X1 is the zero vector.From equations (10) and (11), it gives(19)X1T+X2T+⋯+Xn2+1T=0,and(20)rQG2+In2⊗In1X2X3…Xn2+1T=λX2X3…Xn2+1T.
Note that2k+1r is not the eigenvalue of QG1°˜G2. Otherwise, we can deduce that the eigenvector X=X1X2…Xk+1T of the eigenvalue 2k+1r fits XT=Jn1⊗Jn2+1. Thus,(21)X1+X2+⋯+Xn2+1=n2+1Jn1,which contradicts with equation (19).
Note that the spectrum ofQG2 is SG2=η1,η2,…,ηn2=2k, so according to the equation (20), we can directly obtain(22)λ=rηj+1,j=1,2,…,n2−1,and the multiplicity of λ=rηj+1 is n1.
According to equations (18) and (22), we have n11+n2 eigenvalues of QG1°˜G2. The eigenvectors of QG1°˜G2 corresponding to these eigenvalues are presented as follows.
From equation (15), we obtain that X1T is the eigenvector set corresponding to the eigenvalues of QG1. Let X1=Yi,X2=X3=⋯=Xn2+1=rYi/λi−2k+1r, and X2′=X3′=⋯=Xn2+1′=rYi/λi¯−2k+1r and Yii=1,2,…,n1 be the nonzero eigenvectors corresponding to the eigenvalues μ1,μ2,…,μn1. Let Fi=X1X2…Xn2+1T and Fi′=X1X2′…Xn2+1′T, for i=1,2,…,n1. We need to prove QG1°˜G2Fi=λiFi and QG1°˜G2Fi′=λi¯Fi′ in order to confirm that Fi and Fi′ are 2n1 eigenvectors corresponding to λi and λ¯i, respectively.
Given a vertexu in G1°˜G2. If u∈VG1, one has Fiu=Yiu, and it obtains(23)QG1+rn2In1rJn2⊗In1Fiu=QG1Yiu+rn2Yiu+r2n2Yiuλi−2k+1r=μiYiu+rn2Yiu+r2n2Yiuλi−2k+1r=λiYiu.=λiFiu.
Ifu is a vertex in the tth copy of G2, one has Fiu=rYit/λi−2k+1r, and it obtains(24)rIn1rEtQG2+In2⊗In1Fiu=rYit+2k+1r2λi−2k+1rYit=λiFiu.
For each vertex inG1°˜G2, it follows that QG1°˜G2Fi=λiFi. Similarly, for each vertex in G1°˜G2, we have QG1°˜G2Fi′=λi¯Fi′.
LetZjj=1,2,…,n2 be the orthogonal eigenvectors of QG2 corresponding to η1,η2,…,ηn2=2k. Noticing that Zn2=aJn2T and Zj⊥Jn2T, for j=1,2,…,n2−1. So, for i=1,2,…,n1andj=1,2,…,n2−1, one knows(25)QG1°˜G20n1Zj⊗ei=0n1rQG2+In2⊗In1Zj⊗ei=0n1rQG2+In2Zj⊗In1ei=rηj+10n1Zj⊗ei.
Now, we have obtainedn11+n2 eigenvectors of QG1°˜G2, and they are linearly independent obviously.


## 3. The Normalized Laplacian Spectrum of Weighted Corona GraphG1°˜G2

LetG1 be an r1-regular graph with n1 vertices and G2 an r2-regular graph with n2 vertices. Let G=G1°˜G2. Then,(26)WG=WG1rJn2⊗In1rJn2T⊗In1rWG2⊗In1,(27)D˜G=r1+rn2In100rr2+1In1n2,(28)PG=D˜G−1WG=1r1+rn2WG1rr1+rn2Jn2⊗In11r2+1Jn2T⊗In11r2+1WG2⊗In1=r1r1+rn2PG1rr1+rn2Jn2⊗In11r2+1Jn2T⊗In1r2r2+1PG2⊗In1.Theorem 2.
LetG1 be an r1-regular graph with n1 vertices and G2 an r2-regular graph with n2 vertices. The spectra of PG1 and PG2 are(29)NpG1=β1,β2,…,βn1andNpG2=ρ1,ρ2,…,ρn2,respectively, where −1=β1≤β2≤⋯≤βn1=1 and −1=ρ1≤ρ2≤⋯≤ρn2=1. Then, the spectra of PG1°˜G2 are as follows:(i)
r2r1+rn2+r1βir2+1±σi/2r1+rn2r2+1∈NpG1°˜G2 with multiplicity 1, for i=1,2,…,n1, where σi=r2r1+rn2−r1βir2+12+4rn2r1+rn2r2+1(ii)
r2ρj/r2+1∈NpG1°˜G2 with multiplicity n1, for j=1,2,…,n2−1Proof.
Letλ be the eigenvalue of PG, X=X1X2…Xn2+1T, and Xi∈Rn1i=1,2,…,n2+1 be the eigenvector which corresponds to λ. Then,(30)PGX=λX.
From equations (28) and (30), it follows that(31)r1/r1+rn2PG1X1T+r/r1+rn2X2T+X3T+⋯+Xn2+1T=λX1Tand(32)l1r2+1X1T+r2r2+1E1PG2⊗In1X2X3…Xn2+1T=λX2T,1r2+1X1T+r2r2+1E2PG2⊗In1X2X3…Xn2+1T=λX3T,⋮1r2+1X1T+r2r2+1En2PG2⊗In1X2X3…Xn2+1T=λXn2+1T,where Ei=0n1,…,0n1︷i−1,In1,0n1,…,0n1︷n2−i.
By equation (32), we obtain(33)n2r2+1X1T+r2r2+1X2T+X3T+⋯+Xn2+1T=λX2T+X3T+⋯+Xn2+1T,that is,(34)n2r2+1X1T+r2r2+1−λX2T+X3T+⋯+Xn2+1T=0.Case 1:X1 is the nonzero vector.In this case, we immediately deduce thatr2/r2+1−λ≠0. From equations (31) and (34), we have(35)r1r1+rn2PG1X1T+rr1+rn2n2λr2+λ−r2X1T=λX1T,so(36)PG1X1T=1r1λr1+rn2−rn2λr2+λ−r2X1T.Noting thatNpG1=β1,β2,…,βn1 and according to equation (36), we can obtain(37)λ(r1+rn2−rn2λr2+λ−r2=r1βi,i=1,2,…,n1.By solving this equation aboutλ, we obtain(38)λi,λ¯i=r2r1+rn2+r1βir2+1±σi2r1+rn2r2+1,σi=r2r1+rn2−r1βir2+12+4rn2r1+rn2r2+1.Case 2:X1 is the zero vector.From equations (31) and (32), we obtain(39)X2T+⋯+Xn2+1T=0and(40)r2r2+1PG2⊗InX2X3…Xn2+1T=λX2X3…Xn2+1T.
Notice thatr2/r2+1 is not the eigenvalue of PG1°˜G2. Otherwise, we can deduce that the eigenvector X=X1X2…Xn2+1T of the eigenvalue r2/r2+1 fits XT=Jn1⊗Jn2+1. Thus,(41)X1+X2+⋯+Xn2+1=n2+1Jn,which contradicts with equation (39). Note that the spectrum of PG2 is NpG2=−1=ρ1≤ρ2≤⋯≤ρn2=1, so according to equation (40), we can directly obtain(42)λ=r2ρjr2+1,j=1,2,…,n2−1,and the multiplicity of λ=r2ρj/r2+1 is n1.
According to equations (38) and (42), we have n11+n2 eigenvalues of PG1°˜G2. The eigenvectors of PG1°˜G2 which corresponding to these eigenvalues are presented as follows.
From equation (36), we obtain that X1T is the eigenvector set corresponding to the eigenvalues of PG1. Let X1=Yi,X2=X3=⋯=Xn2+1=rYi/λir2+1−r2, and X2′=X3′=⋯=Xn2+1′=rYi/λi¯r2+1−r2, and Yii=1,2,…,n1 be the nonzero eigenvectors which correspond to the eigenvalues μ1,μ2,…,μn1. Let Fi=X1X2…Xn2+1T and Fi′=X1X2′…Xn2+1′T, for i=1,2,…,n1. We need to prove PG1°˜G2Fi=λiFi and PG1°˜G2Fi′=λi¯Fi′ in order to confirm that Fi and Fi′ are 2n1 eigenvectors corresponding to λi and λ¯i, respectively.
Given a vertexu in G1°˜G2. If u∈VG1, it obtain(43)r1r1+rn2PG1rr1+rn2Jn2⊗In1Fiu=r1r1+rn2PG1Yiu+rn2r1+rn2λir2+1−r2Yiu=r1r1+rn2βiYiu+rn2r1+rn2λir2+1−r2Yiu=λiFiu.
Ifu is a vertex in one copy of G2, one has Fiu=Yit/λir2+1−r2, and it obtains(44)1r2+1In1r2r2+1EtPG2⊗In1Fiu=1r2+1Yit+r2r2+1λir2+1−r2Yit=λiFiu.
For each vertex inG1°˜G2, it follows that PG1°˜G2Fiu=λiFiu. Similarly, for each vertex in G1°˜G2, we have PG1°˜G2Fiu=λ¯iFiu.
LetZjj=1,2,…,n2 be the orthogonal eigenvectors of PG2 corresponding to ρ1,ρ2,…,ρn2=2k. Noticing that Zn2=aJn2T and Zj⊥Jn2T, for j=1,2,…,n2−1. So, for i=1,2,…,n1andj=1,2,…,n2−1, one knows(45)PG1°˜G20n1Zj⊗ei=0n1r2/r2+1PG2⊗In1Zj⊗ei=0n1r2/r2+1PG2Zj⊗In1ei=r2ρjr2+10n1Zj⊗ei.
Now, we have obtainedn11+n2 eigenvectors of PG1°˜G2, and they are linearly independent obviously. We easily deduce the following result according to relation (6).Theorem 3.
LetG1 be an r1-regular graph with n1 vertices and G2 an r2-regular graph with n2 vertices. The spectra of ℒG1 and ℒG2 are(46)NG1=0=γ1,γ2,…,γn1andNG2=0=δ1,δ2,…,δn2,respectively. Then, the normalized Laplacian spectra of G1°˜G2 are as follows:(i)
rn2r2+2+r1+γir1r2+1±τi/2r1+rn2r2+1∈NG1°˜G2 with multiplicity 1 for i=1,2,…,n1, where τi=rr2n2−r1+r1γir2+12+4rn2r1+rn2r2+1(ii)
1+r2δj/r2+1∈NG1°˜G2 with multiplicity n1 for j=1,2,…,n2−1


## 4. Application

In this section, the spanning trees’ number and the degree Kirchhoff index of the weighted corona graph are obtained as applications of Theorem2.Given a graphG, the spanning tree is the tree which contains all vertices of G. Therefore, there are many spanning trees of graph G due to the complex structure of the tree. Let the graph G be a connected graph, the number of vertices be n, and the number of edges be m. The normalized Laplacian eigenvalues of G is 0=λ1≤λ2≤λ3≤…≤λn. Then, the number of spanning tree of G [21] can be expressed by(47)tG=∏i∈VGdi∏j=2nλj2m,where di is the degree of vertex i.In [25], Chen and Zhang proposed a new graph invariant (called the degree Kirchhoff index) which is defined as KfG=∑i<jdidjrij. They have proved that this graph invariant is associated with the normalized Laplacian eigenvalues of G, that is,(48)KfG=2m∑j=2n1λj.In the following theorem, we will give the number of spanning trees of the weighted corona graphG1°˜G2 and its degree Kirchhoff index, by using the normalized Laplacian spectrum of graph G1°˜G2.Theorem 4.
LetG1 be an r1-regular graph with n1 vertices and G2 an r2-regular graph with n2 vertices. Then, for the weighted corona G=G1°˜G2, we have(i)
tG=rn1n2rr2n2+2rn2+r1/r1+r2n2+2n2tG1∏j=1n2−1r2δj+1n1(ii)
KfG=rr2n2+2rn2+r1r2n2+2n2+r1/r12KfG1+2EGr2+1r1+rn2/rr2n2+2rn2+r1+n1−1+∑j=1n2−1n1/r2δj+1, where δj is the normalized Laplacian eigenvalue of G2Proof.
Let the normalized Laplacian spectra ofG1 and G2 be as depicted in Theorem 2. Then, by Theorem 2, γ1=0 corresponds two eigenvalues 0 and rr2n2+2rn2+r1/r1+rn2r2+1 of ℒG; for each γi≠0, the product of the two corresponding eigenvalues is(49)rn2r2+2+r1+γir1r2+12−τi24r1+rn22r2+12=r1γir1+rn2r2+1.
So,(50)∏λ∈LG,λ≠0λ=rr2n2+2rn2+r1r1+rn2r2+1∏i=2n1r1γir1+rn2r2+1∏j=1n2−1r2δj+1r2+1n1=rr2n2+2rn2+r1r1n1−1∏i=2n1γi∏j=1n2−1r2δj+1n1r1+rn2n1r2+1n1n2.
Hence,(51)tG=r1+rn2n1rr2+1n1n2∏λ∈LG,λ≠0λn1r1+r2n2+2n2=rn1n2rr2n2+2rn2+r1r1+r2n2+2n2r1n1∏i=2n1γir1n1∏j=1n2−1r2δj+1n1=rn1n2rr2n2+2rn2+r1r1+r2n2+2n2tG1∏j=1n2−1r2δj+1n1.
For the Kirchhoff index, we have(52)KfG=2EG∑λ∈LG,λ≠01λ=2EGr1+rn2r2+1rr2n2+2rn2+r1+∑i=2n1rn2r2+2+r1+γir1r2+1γir1+∑j=1n2−1n1r2+1r2δj+1=rr2n2+2rn2+r1r2n2+2n2+r1r12KfG1+2EGr2+1r1+rn2rr2n2+2rn2+r1+n1−1+∑j=1n2−1n1r2δj+1.


## 5. Conclusion

In recent decades, the study of the normalized Laplacian spectrum has more practical significance, for some dynamical processes and structural aspects of various networks are related to it. This study explores the signless Laplacian spectrum and the normalized Laplacian spectrum of the weighted corona graphs.



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