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In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy , , NCN-energy for some standard graphs is obtained and an upper bound for is found when G is a strongly regular graph. Also the relation between common neigh-bourhood energy and non-common neighbourhood energy of a graph is established.

Let G be a simple graph with n vertices, and let

The energy of the graph G is defined [

Details on the theory of graph energy can be found in the reviews [

For

vertices, different from

The common-neighborhood energy (or, shorter, CN-energy) of the graph G is

where

Theorem 1. [

Theorem 1 immediately implies that almost all graphs are hyperenergetic, making any further search for them pointless.

In what follows we shall need a few auxiliary results.

Lemma 1. [

Lemma 2. [

Corollary 1. [

1) The common-neighborhood eigenvalues of the complement of G are

2) The common-neighborhood eigenvalues of the line graph

where the CN-eigenvalue

Definition. A strongly regular graph with parameters

Definition. Let G be simple graph with vertex set

borhood set of the the vertices

According to the above definition, the non-common neighborhood matrix is a real symmetric

Definition. The non-common neighborhood energy (or, shorter, CNC-energy) of the graph G is

We will Denote by

Proposition 2.

Proof. Observing that

Proposition 3.

Proof. First observe that if the vertices of

Observing that

Which implies

Corollary 2.

Proposition 4.

Proposition 5. For any totally disconnected graph

Proof. Observing that for any two vertices u and v in

both vertices u and v. Therefore

complete graph with n vertices. Hence,

The complete multipartite graph

partitioned into parts of cardinalities

the multipartite graph

Proposition 6. Let

Proof. Let

of complete multipartite graph we observe for any two distinct vertices

where

Corollary 3. For graph

Corollary 4. For any cocktail party graph G which is the complement of

The proof of the following result is straightforward.

Proposition 7. If the graph G consists of (disconnected) components

Theorem 8. Let G be a graph on n vertices, and let

and Let

Proof. Since

Lemma 3. Let

Then

Proof. Observing that if

Therefore,

Hence

Proposition 9. For any k-regular graph G,

Proof. By Theorem 8 and Lemma 3, we have

Hence

Theorem 10. For any graph G,

Proof. Since

equal to the number of vertices which adjacent to both

Theorem 11. Let G be a connected k-regular graph with eigenvalues

of G are

Proof. Theorem 11 follows from Proposition 8 and Lemma 2 or by applying Theorem 10 and Corollary 1

Theorem 12. Let G be a connected k-regular graph and let

Proof. Theorem 12 follows from Proposition 8 and Lemma 2 or by applying Theorem 10 and Corollary 1

Theorem 13. For any connected graph G,

Proof. Let

Conversely, if G is connected graph and

for some positive integer

Lemma 4. If G is a strongly regular graph with parameters

Proof. If

tices of G, then

non-adjacent vertices,

from which Equation (1) follows straightforwardly.

Theorem 14. If G is a strongly regular graph with parameters

Proof. follows an idea first used by Koolen and Moulton [

The Cauchy-Schwarz inequality states that if

Now, by setting

Therefore

i.e.,

i.e.,

By using Lemma 4,

We thank the Editor and the referee for their comments.