# On Randic, Seidel, and Laplacian Energy of NEPS Graph

Authors: Kun Han; S. Ahmad; Syed Ajaz K. Kirmani; M. K. Siddiqui; Y. Ali; E. Bashier
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetZ be the simple graph; then, we can obtain the energy EZ of a graph Z by taking the absolute sum of the eigenvalues of the adjacency matrix of Z. In this research, we have computed different energy invariants of the noncompleted extended P-Sum (NEPS) of graph Zi. In particular, we investigate the Randic, Seidel, and Laplacian energies of the NEPS of path graph Pni with any base ℬ. Here, n denotes the number of vertices and i denotes the number of copies of path graph Pn. Some of the results depend on the number of zeroes in base elements, for which we use the notation j.


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## Body

## 1. Introduction

The NEPS of the graphs is a graphZ whose vertex set is equal to the simple Cartesian product of the vertices’ sets of the graphs [1]. If Z1,Z2,Z3,…,ZK are K graphs having vZ1,vZ2,vZ3,…,vZK vertices sets, respectively, then the vertex set of NEPS of graphs is defined as(1)VZ=vZ1×vZ2×vZ3×⋯×vZK.The existence of the edges of NEPS graph is depending on the base elements:(2)β1,β2,β3,…,βK∈B¯,whereas some βi=0 and some βi=1. When βi=0, it means there is no edge because in this case ui=vi. However, when βi=1, then there exists an edge between u1,u2,…,uK and v1,v2,…,vK iff Zi has an edge between ui and vi.The NEPS of the graph is abbreviated as noncomplete extended p-sum of the graph. It has many graph operations as special cases, and names are as follows: the product, the sum, and the strong product of graphs [1]. The energy of graph is the summation of λi of adjacency matrix AG of graph G [2, 3]. Energy of graphs, first introduced by Ivan Gutman, has remarkable chemical application; see [4–6], for details.This energy have applications in image analysis [7, 8], which is been used in investigation of medical fields like in brain activity. Randic energy of graph G is the summation of ρi, where ρi are the eigenvalues of the Randic matrix RG [9, 10]. It has endless applications in chemistry and became an attractive field of research; for further details, see [11–13]. In [7], Haemers defined the Seidel energy; let the Seidel matrix of a graph G be represented as SEG, and θi are the eigenvalues of this matrix; then, the energy of graph is the summation of θi, where θ−i is of the Seidel matrix. More results on this energy can be found in [14–16]. These energies have lot of work in calculating upper and lower bounds of different graphs. In Section 2, we first show NEPS of the cyclic graph and how the energy of NEPS with any other basis do not have any effect on energies, see [17–19]. Then, how this result is used in calculating Randic, Seidel, and Laplacian energy of this graph?Example 1.
Consider two graphsP2 and C3; then, the vertex set of NEPS of P2 and C3 is stated as(3)VZ=VP2×VC3.
LetVP2=1,2 and VC3=1′,2′,3′; then,(4)VZ=1,1′,1,2′,1,3′,2,1′,2,2′,2,3′.
The NEPS graph is dependent on the base elements, and the possible base elements are1,1,1,0,0,1. If we take the base element 1,1, then the graphical view of NEPSP2×C3:1,1 is expressed as in Figure 1.
If we take base element0,1, then its graph is shown in Figure 2.
Now, for1,0, we have Figure 3.
If we take three copies ofP2 and 1,1,1 as the base element, then its NEPS graph is expressed as in Figure 4.
Now, for three copies ofP2, with base 1,0,1, the NEPS graph is shown in Figure 5.
The NEPS graph was first time introduced by D. Cvetkovic and R. Lucic [3], and after some time, it was redefined by S.C. Shee [20]. It has defined various graph operations; some of the results we explain as follows. Let Z1 and Z2 be any two graphs; if we take bases 1,0,0,1, then the resulting graph Z is the sum of Z1 and Z2; if the base element is 1,1, then the graph Z is the tensor product of Z1 and Z2. Furthermore, the connectedness of the NEPS graph in [2] and NEPS operation on cordial graphs in [8] have been discussed.
The energy of the graphZ is defined as the absolute sum of the eigenvalues of the adjacency matrix of Z. It was first time introduced by Ivan Gutman [5]. The adjacency matrix of the simple graph Z is a matrix whose entries are zeroes and ones. If the vertices vivj∈E, then Aij=1; otherwise, Aij=0. It is denoted by AZ.
A Randic matrixRZ is a matrix whose entries are denoted by rij and is defined as R=rij. If ui and uj are not adjacent or ui=uj, then rij=0, but if ui and uj are adjacent, then rij=1/didj, where di and dj are degrees of ui and uj, respectively. The Randic energy of a graph Z can be evaluated by taking the absolute sum of the eigenvalues of the Randic matrix. If the eigenvalues of the Randic matrix are denoted by μi, then we can express the Randic energy of Z as [21](5)REZ=∑i=1nμi.
Milan Randic is the first person who has introduced the Randic index [22] such as(6)RZ=∑i∼j1didj.
Gutman et al. have explained that the Randic index is used to form the Randic matrix [5]. Theye also introduced the energy of the graphs [9, 10]; then, they extended this topic and defined the Randic energy [15]. Dasa et al. have discussed the upper and lower bounds on the Randic energy of the graphs [4].
LetZ be a graph having vertices v1,v2,v3,…,vn; then, the Seidel matrix of Z is a n×n matrix including the entries −1,0, and 1. The Seidel matrix is expressed as SZ=sij. If vi=vj, then sij=0; if vi≠vj and there exist an edge between vi and vj in Z, then sij=−1, but if vi≠vj and there does not exist an edge between vi and vj in Z, then sij=1. Let the eigenvalues of the Seidel matrix be denoted by θi, then the Seidel energy is expressed as the absolute sum of eigenvalues and written as(7)SEZ=∑i=1nθi.
Liu Jian-ping and Liu Bo-lian have explained [14] the seidel energy. Seidel energy and its bounds have been calculated by a sharp method which is made by P. Nageswari and P. B. Sarasija [11].
Now, we discuss on Laplacian energy AsDZ is a diagonal matrix whose entries are the degree of the vertices of Z and AZ is the adjacency matrix, then the Laplacian matrix is stated as LZ=DZ−AZ. If the eigenvalues of the Laplacian matrix is denoted by μi, then the Laplacian energy of the graph Z is expressed as(8)LEZ=∑i=1nμi−dZ,where dZ=2EZ/VZ. The Laplacian matrix has been introduced by Grone and Merris [23] such as LZ=DZ−AZ. Gutman and Zhou, in 2006 [6], defined a result that energy of graph cannot be exceeded from the Laplacian energy of that graph; also, they explained some properties of Laplacian energy in [24]. Zhou has been working on energy and Laplacian energy [25] and gave some useful results. Dragan Stevanovica, Ivan Stankovicb, and Marko Milosevicb have explained some positive and negative results between the relation of energy and Laplacian energy.
We discuss some important results that are made on the path graphs by defining its spectrum, whereas the spectrum ofZ is the nonincreasing sequence of the distinct eigenvalues μ1,μ2,μ3,…,μn of the adjacency matrix AZ of Z. In particular, we explained some results regarding Randic, seidel, and Laplacian energies of NEPS of path graphs. Some results of the Randic, seidel, and Laplacian energies depend on j and i, for i>j. Whereas, j and i denote the number of zeroes in base elements β1,β2,β3,…,βK∈ℬ and number of copies of the path graph, respectively. Also, the dimensions of the base elements are based on i.
Such as, if we takej=1 and i=3, then, in this case, possible base elements are
1,1,0,1,0,1,0,1,1. One more important point is that we will discuss on all base elements but will not includ the ones such as 1,1,…,1.Figure 1
NEPSP2×C3:1,1.Figure 2
NEPSP2×C3:0,1.Figure 3
NEPSP2×C3:1,0.Figure 4
NEPSP2×P2×P2:1,1,1.Figure 5
NEPSP2×P2×P2:1,0,1.


## 2. Main Results

Our main focus is to compute Randic, seidel, and Laplacian energy. The broad generation of energies that is consisted on different graph matrices was the first to categorize the Laplacian energy. This is defined in the form of spectrum. The spectrum of the Laplacian matrix consists of the eigenvalues.Theorem 1.
Letn and m be the number of vertices and edges of NEPSZi=P2i:ℬ, respectively, for i=2,3,…,K, where i denotes the number of copies of P2. Then,(9)RENEPSZi=P2i:ℬ=2m.Proof.
LetZ=NEPSZi=P2i:ℬ, for i=2,3,…,K, be a graph. Here, i denotes the number of copies of P2 (which is a path graph heaving 2 vertices and 1 edge) and ℬ denotes the base element which depend on the number of copies of P2.
In that case, the spectrum of the Randic matrix ofZ is defined as(10)2mnn2time,−2mnn2time,REZ=n2×2mn+n2×−2mn=m+m=2m.
Thus, we conclude the proof:(11)RENEPSZi=P2i:ℬ=2m.Theorem 2.
Letn and m be the number of vertices and edges of NEPSZi=P3i:ℬ, respectively, where number of copies of P3 and number of zeroes in the base element are designated by i and j, respectively. Then, for any j and i>j, we have(12)RENEPSZi=P3i:ℬ=3j2i−j.Proof.
LetZ=NEPSZi=P3i:ℬ, for i=2,3,…,K, be a graph. For any j and i>j, we have the spectrum of the Randic matrix of Z:(13)13j2i−j−1time,0n−3j2i−jtime,−13j2i−j−1time.
Thus, we follow the result:(14)REZ=3j2i−j−11+n−3j2i−j0+3j2i−j−1−1=3j2i−j−1+3j2i−j−1,REZ=RENEPSZi=P3i:ℬ=3j2i−j.Theorem 3.
Letn=4i and m be the number of vertices and edges of NEPSZi=P4i:ℬ, respectively. Then, for any j and i>j, we have(15)RENEPSZi=P4i:ℬ=∑t=0i−jCi−jt×2i+j−t.Proof.
LetZ=NEPSZi=P4i:ℬ be a graph. Then, for any j≥1 and i>j, we have the spectrum of the Randic matrix as given below:(16)−12tCi−jt×n2i−j+1time,12tCi−jt×n2i−j+1time,where 0≤t≤i−j,(17)REZ=∑t=0i−jCi−jt×n2i−j+1×−12t+∑t=0i−jCi−jt×n2i−j+1×12t=2×∑t=0i−jCi−jt×n2i−j+1×12t.
Forn=4i, we have(18)REZ=RENEPSZi=P4i:ℬ=∑t=0i−jCi−jt×2i+j−t.
Hence, it is done.Theorem 4.
Forn number of vertices and m=n/2 number of edges, we have(19)SENEPSGi=P2i:ℬ=alog2n−1+2n−3,where alog2n−1=alog2n−2+2log2n−1, for a1=1.Proof.
LetZ=NEPSZi=P2i:ℬ, for i=2,3,…,K, be a graph. Then, for n vertices of Z, we have m=n/2 edges in Z; we have the spectrum of the seidel matrix as(20)alog2n−1,2mnn2time,−6mnn2−1time,where alog2n−1=alog2n−2+2log2n−1, for a1=1.
Thus, we have(21)SEZ=alog2n−1+n22mn+n2−1−6mn=alog2n−1+4m−6mn=alog2n−1+2m2−3n,for m=n/2; we have(22)=alog2n−1+2n−3.Theorem 5.
Letn=3i be the number of vertices; we have m=8n/9 as edges of NEPSZi=P3i:ℬ, where i is the number of copies of P3. Furthermore, suppose that j is the number of zeroes in the base element; then, for any j≥1 and i=j+2, we have(23)SENEPSZi=P3i:ℬ=8aj+1+14n−189,where a1=1 and aj+1=3aj+2.Proof.
LetZ=NEPSZi=P3i:ℬ be the graph. Then, for j≥1 and i=j+2, we have the spectrum of the seidel matrix of Z as given below:(24)−45m8naj+1=3aj+2time,−9m8n15n27time,27m8naj+1=3aj+2time,n−2sum of two eigenvalues,where a1=1 and aj+1=3aj+2, for any j≥1.
By using this spectrum, we have(25)SEZ=aj+1−45m8n+15n27−9m8n+aj+127m8n+n−2=aj+145m8n+15n27−9m8n+aj+127m8n+n−2=9mnaj+1+5m8+n−2.
Form=8n/9, we have(26)SEZ=SENEPSZi=P3i:ℬ=8aj+1+14n−189.Theorem 6.
Letn and m=n/2 be the number of vertices and edges of the graph Z=NEPSZi=P2i:ℬ, respectively, for i=2,3,…,K. Then,(27)LENEPSZi=P2i:ℬ=n.Proof.
We haven and m=n/2 be the number of vertices and edges of the graph Z=NEPSZi=P2i:ℬ, respectively, for i=2,3,…,K. Then, the average degree of Z is dZ=2m/n.
Now, the spectrum of the Laplacian matrix can be expressed as(28)2−2mnn2time,0−2mnn2time.
By using this spectrum, we get the required result:(29)LEZ=n2×2−2mn+n2×0−2mn=n−m+m=n.Theorem 7.
Letn and m be the number of vertices and edges of NEPSZi=P3i:ℬ, respectively, where number of copies of P3 and number of zeroes in the base element are designated by i and j, respectively. Then, for j≥1, we have the following.Case I: fori=j+1, we have(30)LENEPSZi=P3i:ℬ=10n9.Case II: fori=j+2, we have(31)LENEPSZi=P3i:ℬ=106n81.Case III: fori=j+3, we have(32)LENEPSZi=P3i:ℬ=1210n729.Proof.
Letn and m be the number of vertices and edges of Z=NEPSZi=P3i:ℬ, respectively, whose average degree is 2m/n, where the number of copies of P3 and number of zeroes in the base element are designated by i and j, respectively.
Case I: for anyj and i=j+1, then the spectrum of the Laplacian matrix is(33)3−2mnn3time,1−2mnn3time,0−2mnn3time,LEZ=n3×3−2mn+n3×1−2mn+n3×0−2mn=n3×3−2mn+n3×2mn−1+n3×2mn=23n+m.
In this case, forn=3i, number of vertices, we have m=2n/3 edges. Using this condition, we follow the result:(34)LEZ=LENEPSZi=P3i:B=10n9.Case II: for anyj and i=j+2, we have the spectrum of the Laplacian matrix as given below:(35)5−2mnn9time,4−2mnn9time,2−2mn2n9time,1−2mnn3time,0−2mn2n9time.
By following this spectrum, we have calculated the Laplacian energy:(36)LEZ=n9×5−2mn+n9×4−2mn+2n9×2−2mn+n3×1−2mn+2n9×0−2mn=n9×5−2mn+n9×4−2mn+2n9×2−2mn+n3×2mn−1+2n9×2mn.
After some simplification, we obtain(37)LEZ=295n+m.
In this case, forn=3i, number of vertices, we have m=8n/9 edges. The condition has carried out this case:(38)LEZ=LENEPSZi=P3i:ℬ=106n81.
Case III: for anyj and i=j+3, we have the spectrum of the Laplacian matrix as given below:(39)9−2mnn27time,6−2mnn9time,4−2mnn9time,2−2mnn3time,1−2mn7n27time,0−2mn4n27time.
Using this spectrum, we have(40)LEZ=n27×9−2mn+n9×6−2mn+n9×4−2mn+n3×2−2mn+7n27×1−2mn+4n27×0−2mn=n27×9−2mn+n9×6−2mn+n9×4−2mn+n3×2mn−2+7n27×2mn−1+4n27×2mn.
After some calculations, we obtain(41)LEZ=2277n+13m.
In this case, forn=3i, number of vertices, we have m=32n/27 edges. By this condition, we have(42)LEZ=LENEPSZi=P3i:ℬ=1210n729.Theorem 8.
Letn=4i and m are the number of vertices and edges of NEPSZi=P4i:ℬ, respectively; then, for j≥1, we have the following.Case I: fori=j+1, we have(43)LENEPSZi=P4i:ℬ=n×12071×10−4.Case II: fori=j+2, we have(44)LENEPSZi=P4i:ℬ=n×15731×10−4.Proof.
LetZ=NEPSZi=P4i:ℬ be a graph that depends on j and i. We discuss the cases for different conditions of i and j.Case I: forj≥1 and i=j+1, we have the spectrum for the Laplacian energy as(45)0−2mnn4time,0.5858−2mnn4time,2−2mnn4time3.4142−2mnn4time,wheredZ=2m/n is the degree of the graph, and in this case, we have m=3n/4 edges in graph Z:(46)LEZ=n4×0−2mn+n4×0.5858−2mn+n4×2−2mn+n4×3.4142−2mn=n4×2mn+n4×2mn−0.5858+n4×2−2mn+n4×3.4142−2mn.After the simplification, we obtain(47)LEZ=LENEPSZi=P4i:ℬ=n×12071×10−4.Case II: forj≥1 and i=j+2, we have the spectrum for the Laplacian energy as(48)0−2mnn8time,0.7076−2mnn8time,1−2mnn4time,2.3973−2mnn8time,3−2mnn8time4−2mnn8time,5.8951−2mnn8time
Now, in this case, we havem=9n/8 edges in graph Z:(49)LEZ=n8×0−2mn+n8×0.7076−2mn+n4×1−2mn+n8×2.3973−2mn+n8×3−2mn+n8×4−2mn+n8×5.8951−2mn=n8×0−2mn+n8×0.7076−2mn+n4×1−2mn+n8×2.3973−2mn+n8×3−2mn+n8×4−2mn+n8×5.8951−2mn.
After some calculations, we get the result(50)LEZ=LENEPSZi=P4i:ℬ=n×15731×10−4.


## 3. Conclusion

In this study, we investigated the Randic, seidel, and Laplacian energies of the NEPS of path graphPni with any base ℬ. Here, n denotes the number of vertices and i denotes the number of copies of path graph Pn. Some of the results depend on the number of zeroes in base elements, for which we use the notation j.



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