On Randic, Seidel, and Laplacian Energy of NEPS Graph

Let Z be the simple graph; then, we can obtain the energy E(Z) 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 baseB. 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.


Introduction
e NEPS of the graphs is a graph Z whose vertex set is equal to the simple Cartesian product of the vertices' sets of the graphs [1]. If Z 1 , Z 2 , Z 3 , . . . , Z K are K graphs having v(Z 1 ), v(Z 2 ), v(Z 3 ), . . . , v(Z K ) vertices sets, respectively, then the vertex set of NEPS of graphs is defined as e existence of the edges of NEPS graph is depending on the base elements: whereas some β i � 0 and some β i � 1. When β i � 0, it means there is no edge because in this case u i � v i . However, when β i � 1, then there exists an edge between (u 1 , u 2 , . . . , u K ) and (v 1 , v 2 , . . . , v K ) iff Z i has an edge between u i and v i . e 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]. e energy of graph is the summation of |λ i | of adjacency matrix A(G) of graph G [2,3]. Energy of graphs, first introduced by Ivan Gutman, has remarkable chemical application; see [4][5][6], for details.
is 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 |ρ i |, where ρ i are the eigenvalues of the Randic matrix R(G) [9,10]. It has endless applications in chemistry and became an attractive field of research; for further details, see [11][12][13]. In [7], Haemers defined the Seidel energy; let the Seidel matrix of a graph G be represented as SE(G), and θ i are the eigenvalues of this matrix; then, the energy of graph is the summation of |θ i |, where θ − i is of the Seidel matrix. More results on this energy can be found in [14][15][16]. ese 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][18][19]. en, how this result is used in calculating Randic, Seidel, and Laplacian energy of this graph? Example 1. Consider two graphs P 2 and C 3 ; then, the vertex set of NEPS of P 2 and C 3 is stated as Let V(P 2 ) � (1, 2) and V(C 3 ) � (1 ′ , 2 ′ , 3 ′ ); then, e NEPS graph is dependent on the base elements, and the possible base elements are (1, 1), (1, 0), (0, 1) { }. If we take the base element (1, 1), then the graphical view of NEPS P 2 × C 3 : (1, 1) is expressed as in Figure 1.
If we take three copies of P 2 and (1, 1, 1) as the base element, then its NEPS graph is expressed as in Figure 4. Now, for three copies of P 2 , with base (1, 0, 1), the NEPS graph is shown in Figure 5. e 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 Z 1 and Z 2 be any two graphs; if we take bases (1, 0), (0, 1) { }, then the resulting graph Z is the sum of Z 1 and Z 2 ; if the base element is (1, 1), then the graph Z is the tensor product of Z 1 and Z 2 . Furthermore, the connectedness of the NEPS graph in [2] and NEPS operation on cordial graphs in [8] have been discussed.
e energy of the graph Z is defined as the absolute sum of the eigenvalues of the adjacency matrix of Z. It was first time introduced by Ivan Gutman [5]. e adjacency matrix of the simple graph Z is a matrix whose entries are zeroes and ones. If the vertices A Randic matrix R(Z) is a matrix whose entries are denoted by r ij and is defined as R � [r ij ]. If u i and u j are not adjacent or u i � u j , then r ij � 0, but if u i and u j are adjacent, where d i and d j are degrees of u i and u j , respectively. e 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 μ i , then we can express the Randic energy of Z as [21] Milan Randic is the first person who has introduced the Randic index [22] such as Gutman et al. have explained that the Randic index is used to form the Randic matrix [5]. eye 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].
Let Z be a graph having vertices v 1 , v 2 , v 3 , . . . , v n ; then, the Seidel matrix of Z is a n × n matrix including the entries −1, 0, and 1. e Seidel matrix is expressed as and there exist an edge between v i and v j in Z, then s ij � −1, but if v i ≠ v j and there does not exist an edge between v i and v j in Z, then s ij � 1. Let the eigenvalues of the Seidel matrix be denoted by θ i , then the Seidel energy is expressed as the absolute sum of eigenvalues and written as 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 As D(Z) is a diagonal matrix whose entries are the degree of the vertices of Z and A(Z) is the adjacency matrix, then the Laplacian matrix is stated as L(Z) � D(Z) − A(Z). If the eigenvalues of the Laplacian matrix is denoted by μ i , then the Laplacian energy of the graph Z is expressed as where d(Z) � 2|E(Z)|/|V(Z)|. e Laplacian matrix has been introduced by Grone and Merris [23] such as 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 of Z is the nonincreasing sequence of the distinct eigenvalues μ 1 , μ 2 , μ 3 , . . . , μ n of the adjacency matrix A(Z) 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 (β 1 , β 2 , β 3 , . . . , β K ) ∈ B and number of copies of the path graph, respectively. Also, the dimensions of the base elements are based on i.

Main Results
Our main focus is to compute Randic, seidel, and Laplacian energy. e broad generation of energies that is consisted on different graph matrices was the first to categorize the Laplacian energy. is is defined in the form of spectrum. e spectrum of the Laplacian matrix consists of the eigenvalues.

Theorem 1. Let n and m be the number of vertices and edges
of NEPS Z i � P 2 i : B , respectively, for i � 2, 3, . . . , K, where i denotes the number of copies of P 2 . en, Proof. Let Z � NEPS Z i � P 2 i : B , for i � 2, 3, . . . , K, be a graph. Here, i denotes the number of copies of P 2 (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 P 2 .
In that case, the spectrum of the Randic matrix of Z is defined as 2m n us, we conclude the proof: Proof. Let Z � NEPS Z i � P 3 i : 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: us, we follow the result: □ Theorem 3. Let n � 4 i and m be the number of vertices and edges of NEPS Z i � P 4 i : B , respectively. en, for any j and i > j, we have Proof. Let Z � NEPS Z i � P 4 i : B be a graph. en, for any j ≥ 1 and i > j, we have the spectrum of the Randic matrix as given below: For n � 4 i , we have Hence, it is done.

Theorem 4. For n number of vertices and m � n/2 number of edges, we have
where a log n 2 −1 � a log n 2 −2 + 2 log n 2 − 1 , for a 1 � 1.
Proof. Let Z � NEPS Z i � P 2 i : B , for i � 2, 3, . . . , K, be a graph. en, for n vertices of Z, we have m � n/2 edges in Z; we have the spectrum of the seidel matrix as where a log n 2 −1 � a log n 2 −2 + 2 log n 2 − 1 , for a 1 � 1. us, we have for m � n/2; we have � a log n 2 −1 +(2n − 3).
Proof. Let Z � NEPS Z i � P 3 i : B be the graph. en, for j ≥ 1 and i � j + 2, we have the spectrum of the seidel matrix of Z as given below: where a 1 � 1 and a j+1 � 3a j + 2, for any j ≥ 1. By using this spectrum, we have For m � 8n/9, we have (26) □ Theorem 6. Let n and m � n/2 be the number of vertices and edges of the graph Z � NEPS Z i � P 2 i : B , respectively, for i � 2, 3, . . . , K. en, Proof. We have n and m � n/2 be the number of vertices and edges of the graph Z � NEPS Z i � P 2 i : B , respectively, for i � 2, 3, . . . , K. en, the average degree of Z is d(Z) � 2m/n. Now, the spectrum of the Laplacian matrix can be expressed as By using this spectrum, we get the required result: In this case, for n � 3 i , number of vertices, we have m � 8n/9 edges. e condition has carried out this case: Case III: for any j and i � j + 3, we have the spectrum of the Laplacian matrix as given below: (39) Using this spectrum, we have After some calculations, we obtain LE(Z) � 2 27 (7n + 13m).
In this case, for n � 3 i , number of vertices, we have m � 32n/27 edges. By this condition, we have