# Some New Results on Various Graph Energies of the Splitting Graph

Authors: Zheng-Qing Chu; Saima Nazeer; Tariq Javed Zia; Imran Ahmed; Sana Shahid
Journal: Journal of Chemistry
 (2019)

---

## Abstract

The energy of a simple connected graphG is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix AG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph S′G is a multiple of corresponding energy of a given graph G.


---

## Body

## 1. Introduction

LetG be a simple, finite, and undirected graph and its vertex set and edge set are denoted by VG=v1,v2,v3,…,vp and EG=e1,e2,e3,…,eq, respectively. Number of edges finishing at a vertex v of a graph G is named as degree of vertex v and is denoted by dv or dv.The adjacency matrix ofG, denoted by AG, is a square matrix aij such that aij is equal to unity if vivj∈EG and is equal to zero otherwise. The eigenvalues of the adjacency matrix AG are known as the eigenvalues of the graph G. Collection of eigenvalues of the graph G together with their multiplicities is called spectrum of the graph G.Letμ1,μ2,μ3,…,μp be eigenvalues of G and are assumed in nonincreasing order; then, Ivan Gutman in 1978 [1] defined the energy of the graph G as the sum of the absolute values of all eigenvalues of the graph G:(1)EG=∑j=1pμj.The inspiration of description energy of graph happened from quantum Chemistry. During 1930s, E. Hückel presented chemical applications of graph theory in his molecular orbital theory where eigenvalues of graphs take place. In quantum chemistry, the skeleton of nonsaturated hydrocarbon is represented by a graph. The energy levels of electrons in such a molecule are eigenvalues of graph. The strength of particles is closely identified with the spectrum of its graph. The carbon atoms and chemical bond between them in a hydrocarbon system denote vertices and edges, respectively, in a molecular graph. A lot of work has been done on graph theory, special graph labeling [2–10], chemical graph theory and graph energies. In the thesis of Siraj [11], certain elementary results on the energy of the graph are also described.The present work is considered to relate several energies of a graph to bigger graph acquired from the given graph with the help of some graph operations, namely, the splitting graph which is defined in [12]. For a graph G, the splitting graph S′G is obtained by taking a new vertex v′ corresponding to each vertex v of the graph G and then join v′ to all vertices of G adjacent to v. In [13], it has been proven that ES′G=5EG.LetA=aij and B=bij be two matrices of order a×m and b×n, respectively. Then, their tensor product, A⊗B is obtained from A when every element aij is replaced by the block aijB and is of order ab×mn.Proposition 1 (see [14]).
Let A∈Mq and B∈Mp. Also, let α be an eigenvalue of the matrix A with corresponding eigenvector y and β be an eigenvalue of the matrix B with corresponding eigenvector z. Then, αβ is an eigenvalue of A⊗B with corresponding eigenvector yz.In recent times, comparable energies are being considered, based on eigenvalues of a variety of other graph matrices. Numerous matrices can be related to a graph, and their spectrums provide certain helpful information about the graph [15–18].


## 2. Maximum Degree Energy

The maximum degree energyEM of a simple connected graph G in [19] is defined as the sum of the absolute values of eigenvalues of the maximum degree matrix MG of a graph G. Then, MG=Mij where(2)Mij=maxdi,dj,ifvi,vjϵEG,0,otherwise,where di and dj are the degrees of vertices vi and vj, respectively.Theorem 1.
For a graphG,(3)EMS′G=2EMG.Proof.
LetG be a graph with vertices v1,v2,v3,…,vp. Then the maximum degree matrix MG is(4)where dkij=maxdi,dj and di and dj are the degrees of vertices vi and vj, respectively, for i=1,2,3,…,p and j=1,2,3,…,p.
Letv1′,v2′,v3′,…,vp′ be the vertices corresponding to v1,v2,v3,…,vp which are added in G to obtain S′G such that Nvj=Nvj′ for j=1,2,3,…,p. Then, the maximum degree matrix of S′G is denoted by MS′G and can be written as a block matrix:(5)
That is(6)MS′G=2MG2MG2MG0or=2220⊗MG.
Here, the maximum degree spectrum ofS′G is(7)1+5μj1−5μjpp,where μj for j=1,2,3,…,p are the eigenvalues of MGand 1±5 are the eigenvalues of 2220.
Here,(8)EMS′G=∑j=1j=p1±5μj=∑j=1j=pμj1+5+1−5=2∑j=1j=pμj=2EMG,which completes the proof.


## 3. Minimum Degree Energy

In [20], the minimum degree energy Em of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of minimum degree matrix mG of a graph G. Here, mG=mij where(9)mij=mindi,dj,ifvi,vjϵEG,0,otherwise,where di and dj are the degrees of vertices vi and vj, respectively.Theorem 2.
For a graphG,(10)EmS′G=2EmG.Proof.
LetG be a graph with vertices v1,v2,v3,…,vp. Then, the minimum degree matrix mG is(11)where dkij=mindi,dj and di and dj are the degrees of vertices vi and vj, respectively, for i=1,2,3,…,p and j=1,2,3,…,p. Let v1′,v2′,v3′,…,vp′ be the vertices corresponding to v1,v2,v3,…,vp which are added in G to obtain S′G such that Nvj=Nvj′ for j=1,2,3,…,p. Then, the minimum degree matrix of splitting graph of G, denoted by mS′G, can be defined as a block matrix as follows:(12)That is(13)mS′G=2mGmGmG0or=2110⊗mG.
Here, the minimum degree spectrum ofS′G is(14)1+2νj1−2νjpp,where νj for j=1,2,3,…,p are the eigenvalues of m(G) and 1±2 are the eigenvalues of 2110.
Here,(15)EmS′G=∑j=1j=n1±2νj=∑j=1j=pνj1+2+1−2=2∑j=1j=nνj=2EmG,which is the required result.


## 4. Randić Energy

The randić energyER of a simple connected graph G in [21] is the sum of the absolute values of eigenvalues of the randić matrix RG. Here, RG=rij where(16)rij=1didj,ifvi,vjϵEG,0,otherwise.Here,di and dj are the degrees of vertices vi and vj, respectively.Theorem 3.
For a graphG,(17)ERS′G=32ERG.Proof.
Letx1,x2,x3,…,xp be vertices of a graph G. Then, the randić matrix of G is denoted by RG and is given as(18)
Letx1′,x2′,x3′,…,xp′ be the vertices corresponding to x1,x2,x3,…,xp which are added in G to obtain S′G such that Nxj=Nxj′ for j=1,2,3,…,p. Then, the randić matrix of S′G is denoted by RS′G and can be written as a block matrix as follows:(19)
That is,(20)RS′G=12RG12RG12RG0=1212120⊗RG.
Here, the randić spectrum ofS′G is(21)−12ρj1ρjpp,where ρj for j=1,2,3,…,p are the eigenvalues of RGand −1/2 and 1 are the eigenvalues of 1/21/21/20.
Here,(22)ERS′G=∑j=1j=p−12+1ρj=∑j=1j=pρj12+1=32∑j=1j=pρj=32ERG.


## 5. Seidel Energy

In [22], Haemers defined the Seidel energy ESE of a simple connected graph G as the sum of the absolute values of eigenvalues of the seidel matrix SEG of G. Here, SEG=sij where(23)sij=−1,ifviandvjare adjacent  andi≠j,1,ifviandvjare non adjacent andi≠j,0,ifi=j.Theorem 4.
For as-regular graph G,(24)ESES′G≥ESEG.Proof.
LetG be a graph with vertices v1,v2,v3,…,vp. Then, the seidel matrix SEG of G is(25)
Letu1,u2,u3,…,up be the vertices corresponding to v1,v2,v3,…,vp which are added in G to obtain S′G. Then, the seidel matrix of S′G is denoted by SES′G and can be written as a block matrix as follows:(26)
That is,(27)SES′G=SEGSEG+IpSEG+IpJp−Ipor=SEGSEGSEG0+0IpIpIpJp−Ip.
Hence,(28)SES′G≥SEGSEGSEG0=1110⊗SEG.
Here, the seidel spectrum ofS′G is greater than or equal to the spectrum(29)121+5sj121−5sjpp,where sj for j=1,2,3,…,p are the eigenvalues of SEGand 1/21±5 are the eigenvalues of 1110.
Thus,(30)ESES′G≥∑j=1j=p121±5sj=∑j=1j=psj12+52+12−52=12+12∑j=1j=psj=ESEG.
Hence,(31)ESES′G≥ESEG.


## 6. Sum-Connectivity Energy

The sum-connectivity energyESC of a simple connected graph G in [23] is defined as the sum of the absolute values of eigenvalues of the sum-connectivity matrix SCG. Here, SCG=scij where(32)scij=1di+dj,ifvi,vjϵEG,0,otherwise.Here,di and dj are the degrees of vertices vi and vj, respectively.Theorem 5.
For a regular graphG,(33)ESCS′G=12ESCG.Proof.
LetG be a graph with vertices z1,z2,z3,…,zp. Then the sum-connectivity matrix of G is denoted by SCG and is defined as(34)where dj is the degree of vertex zj for j=1,2,3,…,p. Let z1′,z2′,z3′,…,zp′ be the vertices that are added in G to acquire S′G such that Nzj=Nzj′. Then the sum-connectivity matrix of S′G is denoted by SCS′G and is defined as a block matrix as follows:(35)where dj′ is the degree of vertex zj′ for j=1,2,3,…,p. Thus,(36)SCS′G=12SCG23SCG23SCG0or⟹SCS′G=1223230⊗SCG.
Here, the sum-connectivity spectrum ofS′G is(37)32+11412βj32−11412βjpp,where βj, for j=1,2,3,…,p are the eigenvalues of SCG and 32±114/12 are the eigenvalues of 1/22/32/30.
Hence,(38)ESCS′G=∑j=1j=p32±11412βj=∑j=1j=pβj32+11412+32−11412=6212∑j=1j=pβj=12ESCG,which completes the proof.


## 7. Degree Sum Energy

In [24], the degree sum energy EDS of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of the degree sum matrix DSG of G. Here, DSG=dsij where(39)dsij=di+dj,ifi≠j,0,otherwise.Here,di and dj are the degrees of vertices vi and vj, respectively.Theorem 6.
For as-regular graph G with order p,(40)EDSS′G≤6s3p−2.Proof.
LetG be a s‐regular graph with vertices w1,w2,w3,…,wp. Then the degree sum matrix of G is denoted by DSG and is defined as(41)where dj is degree of vertex wj for j=1,2,3,…,p. Let w1′,w2′,w3′,…,wp′ be the vertices corresponding to vertices w1,w2,w3,…,wp that are added in G to get the splitting graph S′G. Then the degree sum matrix of S′G is given as(42)where dj′ is the degree of vertex wj′ for j=1,2,3,…,p.
Note that(43)DSS′G=4sJp−Ip3sJp×p3sJp×p2sJp−Ip.
Thus,(44)EDSS′G≤∑j=12pμj4sJp−Ip0Ip0Ip0Ip+∑j=12pμj0Ip3sJp×p3sJp×p0Ip+∑j=12pμj0Ip0Ip0Ip4sJp−Ip.
As(45)∑j=12pμj4sJp−Ip0Ip0Ip0Ip=8sp−1,∑j=12pμj0Ip3sJp×p3sJp×p0Ip=6sp,∑j=12pμj0Ip0Ip0Ip4sJp−Ip=4sp−1.
Hence,(46)EDSS′G≤8sp−1+6sp+4sp−1=12sp−1+6sp=18sp−12s=6s3p−2,which is the required result.


## 8. Degree Square Sum Energy

The degree square sum energyEDSSG of a simple connected graph G in [25] is defined as the sum of the absolute values of eigenvalues of the degree square sum matrix DSSG. Here, DSSG=dssij where(47)dssij=di2+dj2,ifi≠j,0,otherwise.Here,di and dj are degrees of vertices vi and vj, respectively.Theorem 7.
For a s-regular graphG with p vertices,(48)EDSSS′G≤2s214p−9.Proof.
Letx1,x2,x3,…,xp be vertices of a s‐regular graph G and x1′,x2′,x3′,…,xp′ be the vertices added in G corresponding to x1,x2,x3,…,xp to get the splitting graph S′G such that for j=1,2,3,…,p, Nxj=Nxj′. Then, the degree square sum matrix of G and S′G are given as(49)where di is the degree of vertex xi and di′ is the degree of vertex xi′, for i=1,2,3,…,p.
Note that(50)DSSS′G=8s2Jp−Ip5s2Jp×p5s2Jp×p2s2Jp−Ip.
Thus,(51)EDSSS′G≤∑j=12pμj8s2Jp−Ip0Ip0Ip0Ip+∑j=12pμj0Ip5s2Jp×p5s2Jp×p0Ip+∑j=12pμj0Ip0Ip0Ip2s2Jp−Ip.
As(52)∑j=12pμj8s2Jp−Ip0Ip0Ip0Ip=16s2p−1,∑j=12pμj0Ip5s2Jp×p5s2Jp×p0Ip=10s2p,∑j=12pμj0Ip0Ip0Ip2s2Jp−Ip=4s2p−1.
Hence,(53)EDSSS′G≤16s2p−1+10s2p+4s2p−1=20s2p−1+10s2p=30s2p−20s2=10s23p−2,which completes the proof.


## 9. First Zagreb Energy

In [26], the First zagreb energy ZE1 of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of first zagreb matrix Z1G of G where Z1G=zij1 where(54)zij1=di+dj,ifvi,vjϵEG,0,otherwise.Here,di and dj are degrees of vertices vi and vj, respectively.Theorem 8.
For a regular graphG,(55)ZE1S′G=2ZE1G.Proof.
LetG be a graph with vertices z1,z2,z3,…,zp. Then, the first zagreb matrix of G is denoted by Z1G and is defined as(56)where dj is the degree of vertex zj for j=1,2,3,…,p. Let z1′,z2′,z3′,…,zp″ be vertices added in G corresponding to z1,z2,z3,…,zp to get S′G such that Nzi=Nzi′. Then, the first zagreb matrix of S′G can be written as a block matrix as follows:(57)where dj′ is the degree of vertex zj′ for j=1,2,3,…,p.
Here,(58)Z1S′G=2Z1G32Z1G32Z1G0orZ1S′G=232320⊗Z1G.
Here, the first zagreb spectrum ofS′G is(59)2−132ζj2+132ζjpp,where ζj for j=1,2,3,…,p are the eigenvalues of Z1Gand 2±13/2 are the eigenvalues of 23/23/20. Hence,(60)ZE1S′G=∑j=1j=p2±132ζj=∑j=1j=pζj2−132+2+132=2ZE1G.


## 10. Second Zagreb Energy

The second zagreb energyZE2 of a simple connected graph G is defined in [26] as the sum of the absolute values of eigenvalues of the second zagreb matrix Z2G of G where Z2G=zij2, where(61)zij2=di⋅dj,ifvi,vjϵEG,0,otherwise.Here,di and dj are the degrees of vertices vi and vj, respectively.Theorem 9.
For a regular graph G,(62)ZE2S′G=4ZE2G.Proof.
LetG be a graph with vertices z1,z2,z3,…,zp. Then, the second zagreb matrix of G is denoted by Z2G and is defined as(63)where dj is the degree of vertex zj for j=1,2,3,…,p. Let z1′,z2′,z3′,…,zp′ be vertices added in G corresponding to z1,z2,z3,…,zp to get S′G such that Nzj=Nzj′. Then, the second Zagreb matrix of S′G is denoted by Z2S′G and can be written as a square matrix as follows:(64)where dj′ is the degree of vertex zj′ for j=1,2,3,…,p.
Here,(65)Z2S′G=4Z2G2Z2G2Z2G0,Z2S′G=4220⊗Z2G.
Here, the second zagreb spectrum ofS′G is(66)2+22ηj2−22ηjpp,where ηj for j=1,2,3,…,p are the eigenvalues of Z2Gand 2±22 are the eigenvalues of 4220. Hence,(67)ZE2S′G=∑j=1j=p2±22ηj=∑j=1j=pηi2+22+2−22=∑j=1j=pηi4=4ZE2G.


## 11. Conclusion

The energy of a graph is one of the important idea of spectral graph theory. This idea links organic chemistry to mathematics. Numerous graph energies established from the eigenvalues of a variety of graph matrices and their bounds has been discovered. In this paper, we give a relation of various graph energies between the regular graph and its splitting graph. It is interesting to compute graph energies for the families of graphs considered in [27–31].



---

*Source: 7214047-2019-11-19.xml*