Numerous graph energies of regular subdivision graph and complete graph

: The graph energy E ( G ) of a simple graph G is sum of its absolute eigenvalues where eigenvalues of adjacency matrix A ( G ) are referred as eigenvalues of graph G . Depends upon eigenvalues of di ﬀ erent graph matrices, several graph energies has been observed recently such as maximum degree energy, Randi´ c energy, sum-connectivity energy etc. Depending on the deﬁnition of a graph matrix, the graph energy can be easily determined. This article contains upper bounds of several graph energies of s -regular subdivision graph S ( G ). Also various graph energies of complete graph are mentioned in this article.


Introduction
Consider a simple connected graph G = (V(G), E(G)) having |V(G)| = p vertices and E(G) = q edges. Number of edges in the neighborhood of a vertex x in a graph G is named as degree of that vertex and is denoted by d x or d(x). If the number of edges in the neighborhood of each vertex in a graph are same say s then graph is said to be a s-regular graph.
Adjacency matrix is a p × p matrix having entries a xy such that The eigenvalues of a graph are actually eigenvalues of its A(G). The set which is constructed from eigenvalues of G with their multiplicities is known as spectrum of G.
In Mathematics, the graph energy was firstly introduced by Ivan Gutman in 1978. Graph energy is built upon eigenvalues of A(G). It is sum of absolute values of elements of spectrum of G. For a p-vertex graph G with eigenvalues β k in non-increasing order for k = 1, 2, 3, ..., p, (1.1) The impression which is stated in Eq (1.1) is associated with computational chemistry. If, in a conjugated hydrocarbon system, the eigenvalues of a molecular graph are α 1 , α 2 , α 3 , ..., α p and are in non-increasing order. Then Hückel molecular orbital approximation calculated the total Π-electron energy E Π as for p is even and for p is odd with γ and δ are constants. A large number of research papers have been published on graph energy. The thesis of Siraj [1] contains some elementary determinations of graph energy. This paper include upper bounds of different graph energies of subdivision graph S (G) of s-regular graph G containing p vertices and q edges. Also various graph energies of complete graph are explored in this paper.
Based on eigenvalues of different graph matrices, several energies of a graph have been such as maximum degree energy, seidel energy, sum-connectivity energy etc. These energies depends upon eigenvalues of their corresponding energy matrices, see [2][3][4].

Energies of s-regular subdivision graph
First we define subdivision graph.

Degree energies
In this section, we present bounds of maximum degree energy, minimum degree energy, Randić energy, sum-connectivity energy and first and second Zagreb energies. Firstly, we define these energies.
Definition 2.2 (Maximum degree energy). [5] The maximum degree energy E M of a simple graph G is define as the sum of the absolute eigenvalues of its maximum degree matrix  Definition 2.5 (Sum-connectivity energy). [8] The sum-connectivity energy E S C of a simple connected graph G is define as the sum of the absolute eigenvalues of the sum-connectivity matrix Definition 2.6 (First Zagreb energy). [9] The first Zagreb energy ZE 1 of a simple connected graph G is define as the sum of the absolute eigenvalues of first Zagreb matrix Definition 2.7 (Second Zagreb energy). [9] The second Zagreb energy ZE 2 of a simple connected graph G is define as the sum of the absolute eigenvalues of second Zagreb matrix Z (2) In the following theorem, we give bounds of all above defined degree energies; Theorem 2.8. Let p and q be vertices and edges of a regular graph G. Then 1. for maximum degree energy, we have E M (S (G)) ≤ 2s 2pq; 2. for minimum degree energy, we have E m (S (G)) ≤ 4 2pq, 3. for Randić energy, we have E R (S (G)) ≤ 4pq s , 4. for sum-connectivity energy, we have E S C (S (G)) ≤ 2 2pq 2+s , 5. for first Zagreb energy, we have ZE 1 (S (G)) ≤ 2(s + 2) 2pq, 6. for second Zagreb energy, we have ZE 2 (S (G)) ≤ 4s 2pq.
Proof. Let the incidence matrix of G is C(G). Note that the degree matrix of the subdivision graph S (G) can be stated as: 1. By taking t = s in Eq (2.1), we have following computations for the maximum degree energy of the subdivision graph S (G); As in [12] where L + (G) is signless Laplacian matrix and ν + j are eigenvalues of L + (G). Thus by Cauchy Schawaz inequality Hence, 2. By taking t = 2 in Eq (2.1), we have following computations for the minimum degree energy of the subdivision graph S (G); Therefore, E m (S (G)) ≤ 4 2pq.
3. By taking t = 1 √ 2s in Eq (2.1), we have following computations for the Randić energy of the subdivision graph S (G); Therefore,

By taking t = 1
√ 2+s in Eq (2.1), we have following computations for the sum-connectivity energy of the subdivision graph S (G); Hence,
6. By taking t = 2s in Eq (2.1), we have following computations for the second Zagreb energy energy of the subdivision graph S (G); Hence, ZE 2 (S (G)) ≤ 4s 2pq.  Proof. Let u 1 , u 2 , u 3 , ..., u p be vertices of an s-regular graph G and let u j for 1 ≤ j ≤ q be vertices that are added at all edges of G to gain S (G). Note that S E(S (G)) is given as:

Seidel energy
where E p×q = [e pq ] such that Hence,

Degree sum energy
Definition 2.11 (Degree sum energy). [11] The degree sum energy E DS of a simple connected graph G is define as the sum of the absolute eigenvalues of the degree sum matrix DS (G) of G where DS (G) has (i, j)th entry d i + d j if i j and 0 otherwise.
Theorem 2.12. For a s-regular graph G having p and q as order and size respectively, Proof. Let G has vertices w 1 , w 2 , w 3 , ..., w p and w j for 1 ≤ j ≤ q be the vertices that are added at every edge of G to acquire S (G). Then Therefore Hence,

Degree square sum energy
Definition 2.13 (Degree square sum energy). [12] The degree square sum energy E DS S of a simple connected graph G is define as the sum of the absolute eigenvalues of the degree square sum matrix DS S (G) of G where DS S (G) has (i, j)th entry d 2 i + d 2 j if i j and 0 otherwise. Theorem 2.14. For an s-regular graph G E DS S (S (G)) ≤ 4s 2 (p − 1) + 2(s 2 + 4) √ pq + 16(q − 1).
Proof. Let for 1 ≤ j ≤ p, u j be vertices of G and u k for 1 ≤ k ≤ q be the vertices that are added in G to get S (G). Note that the degree square sum matrix of S (G) is denoted by DS S (S (G)) and is given as: Therefore Hence,

Energies of complete graph
A complete graph denoted by K p is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. K 5 is shown in Figure 2. We have following trivial results about energies of complete graphs.   Theorem 3.4. For the complete graph K p , the Seidel energy is E S E (K p ) = 2(p − 1) = 2(p − 1)). Theorem 3.5. For the complete graph K p , the sum-connectivity energy is E S C (K p ) = 2(p − 1). Theorem 3.6. For the complete graph K p , the degree sum energy is E DS (K p ) = 4(p − 1) 2 .
Theorem 3.7. For the complete graph K p , the degree square sum energy is E DS S (K p ) = 4(p − 1) 3 .
Theorem 3.8. For the complete graph K p , the first Zagreb energy is Theorem 3.9. For the complete graph K p , the second Zagreb energy is ZE 2 (K p ) = 2(p − 1) 3 .

Conclusions
In this paper we gave bounds of maximum degree energy, Randić energy, sum-connectivity energy etc of s-regular subdivision graph S (G). Also various graph energies of complete graph are mentioned in this article. In future, we aim to study graph energies for the other families of graphs.