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Research Article

Open Access Peer-reviewed

K. N. Prakasha, P. Siva Kota Reddy, Ismail Naci Cangul^{ }

Received July 28, 2017; Revised August 29, 2017; Accepted September 22, 2017

The purpose of this paper is to introduce and investigate the symmetric division deg energy *SDDE*(*G*) of a graph. We establish upper and lower bounds for *SDDE*(*G*). Also the symmetric division deg energy for certain graphs with one edge deleted are calculated.

Let be a simple graph and let be the set of its vertices. Let If two vertices and of are adjacent, then we use the notation For a vertex the degree of will be denoted by or briefly by

In mathematical chemistry, topological indices play an important role due to their countless applications. There are many topological indices such as Randić index, sum-connectivity index, atom bond connectivity index, Zagreb indices, etc. One of those numerical descriptors, the symmetric division deg index, is included in the list of 148 discrete Adriatic indices and is a very good predictor of total surface area of polychlorobiphenyls (PCB).

The symmetric division deg index of a graph *G* is defined by

The concept of the symmetric division deg index motivates one to associate a symmetric square matrix *SDD*(*G*) to a graph *G*. The symmetric division deg matrix is, by this reason, defined as

Let *G* be a simple, finite, undirected graph. The classical energy *E*(G) is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. For more details on energy of a graph, see ^{ 2, 3}.

Let *SDD*(*G*) be the symmetric division deg matrix. The characteristic polynomial of *SDD*(*G*) will be denoted by and defined as

Since the symmetric division deg matrix is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order The symmetric division deg energy of is similarly defined by

(1) |

This paper is organized as follows. In Section 3, we give some basic properties of symmetric division deg energy of a graph. In Section 4, symmetric division deg energy of some specific graphs are obtained. In Section 5, we find symmetric division deg energy of some complements of some specific graphs. In Section 6, the symmetric division deg energy for certain graphs with one edge deleted are calculated and finally in Section 7, some open problems are given.

Let us define the number* P* as

Then we have

**Proposition 3.1.** *The **fi**rst three coef**fi**cients of the polynomial** ** **are as follows*:

(i)

(ii)

(iii)

*Proof. *(*i*) By the definition of the polynomial

we get a0 = 1.

(*ii*) The sum of determinants of all principal submatrices of is equal to the trace of implying that

(*iii*) By the definition, we have

**Proposition 3.2.** *If** ** **are** **the symmetric division deg eigenvalues** **of** *,* then*

*Proof*. It follows as

Using this result, we now obtain lower and upper bounds for the symmetric division deg energy of a graph:

**Theorem 3.3**.* **Let **G **be a graph with** **n **vertices*. *Then*

*Proof.* Let be the eigenvalues of By the Cauchy-Schwartz inequality we have

Let Then

implying that

and hence we get

as an upper bound.

**Theorem 3.4**. *Let **G **be a graph with** **n **vertices*. *If ** then*

*Proof*. By definition, we have

Using arithmetic-geometric mean inequality, we have

Therefore,

Thus,

Let and are the minimum and maximum values of all Then the following results can easily be proven by means of the above results:

**Theorem 3.5**. *For a graph G of order n*,

**Theorem 3.6**. *For a graph **G **of order** **n **with non-zero eigenvalues*, *we have*

**Theorem 3.7. ***Let **G **be a graph of order **n*. *Let** ** **be the eigenvalues in increasing order*. *Then*

In this section, we calculate the symmetric division deg energy of some well-known and frequently used graph types including complete, cycle, star, friendship, cocktail party, double star, Dutch windmill, crown and complete bipartite graphs.

**Theorem 4.1.** *The symmetric division deg energy of a complete graph** ** **is*

*Proof*. Let be the complete graph with vertex set For this graph, the symmetric division deg matrix is

The characteristic equation then becomes

and the spectrum would be

Therefore,

**T****heorem 4.2**. *The symmetric division deg energy of the cycle graph** ** **is*

*Proof*. The symmetric division deg matrix corresponding to the cycle graph is

This is a circullant matrix of order 2*n*. Its eigenvalues are

Therefore the symmetric division deg energy is

**Theorem 4.3.*** **The symmetric division deg energy of the star graph** ** **is*

*Proof.* Let be the star graph with vertex set with denotes the central vertex. The symmetric division deg matrix is

The characteristic equation becomes

and therefore, the spectrum would have an and times 0. Therefore,

**Definition 4.4.** *The friendship graph, denoted by** ** **is defined as the graph obtained by taking n copies of the cycle graph** ** **with a vertex in common*.

It is clear that

**Theorem 4.5.** *The symmetric division deg energy of the friendship graph** ** *is

*Proof*. Let be the friendship graph with 2*n* + 1 vertices and let *v*_{0} be the common vertex. The symmetric division deg matrix is

The characteristic equation becomes

implying that the spectrum has n times , times 2, a and a Therefore, we get

**Theorem 4.6.*** **The symmetric division deg energy of the cocktail party graph** ** **is*

*Proof*. Let be the cocktail party graph of order 2*n* having vertex set The symmetric division deg matrix is

In that case, the characteristic equation is

and hence the spectrum becomes

Therefore we arrive at the required result:

**Theorem 4.7.** *The symmetric division deg energy of the double star graph** ** **is*

*Proof. *The symmetric division deg matrix is

Hence, the spectrum would have times and Therefore, we get

**Definition 4.8.** *A graph obtained by joining **n **copies of the cycle graph** ** **of** **length *4* **at a common vertex is called a Dutch windmill graph. It will be denoted** **by** *

It is clear that the Dutch windmill graph has 3*n* + 1 vertices and 4*n* edges.

**Theorem 4.9.** *The symmetric division deg energy of the Dutch windmill graph** ** **is*

*Proof*. Recall that has 3*n* + 1 vertices. Then the symmetric division deg matrix is

Hence the characteristic equation will be

and therefore the spectrum would have times times times 0, and Therefore, it is directly seen that

**Theorem 4.10.** *The symmetric division deg energy of crown graph** ** **is*

*Proof*. Let be the crown graph of order 2*n* and let the vertex set of this graph be

The symmetric division deg matrix of is

Therefore the characteristic equation is

implying that the spectrum has a a times 2 and times . Therefore we obtain

**Theorem 4.11**. *The symmetric division deg energy of the complete bipartite graph** ** **of order** ** **with** **vertex set** ** **is*

*Proof*. The symmetric division deg matrix of the complete bipartite graph is

Then the characteristic equation is

and therefore the spectrum has a times 0 and a Therefore, we get

**Definition 5.1.** ^{ 5} *Let **G **be a graph** **and** ** **be a partition of its vertex set **V**. Then the **k**-complement of **G **is denoted by** ** **and obtained as follows: For all** ** **and** ** **in** ** ** **remove the edges between** ** **and ** **and add the edges between the vertices of** ** **and** ** **which** **are not in **G**.*

**Definition 5.2. **^{ 5} *Let G be a graph and** ** **be a partition of its vertex set **V**. Then the **-complement of G is denoted by** ** **and obtained as follows: For each set** ** **in** ** **remove the edges of G joining the vertices within** ** **and add the edges of** ** **(complement of G) joining the vertices of** ** *

There is usually a nice relation between some properties of a graph and its complement. Here we investigate the relation between some special graph classes and their complements in terms of the symmetric division deg energy.

**Theorem 5.3. ***The symmetric division deg energy of the complement** ** **of the complete graph** ** **is*

*Proof*. Let be the complete graph with vertex set The symmetric division deg connectivity matrix of the complement of the complete graph is

Clearly, the characteristic equation is implying

**Theorem 5.4.*** The symmetric division deg energy of the complement** ** **of the star graph** ** **is*

*Proof*. Let be the complement of the star graph with vertex set where is the central vertex. The symmetric division deg matrix is

The corresponding characteristic equation is

and therefore the spectrum is

Therefore,

**Theorem 5.5.** *The symmetric division deg energy of the complement** ** **of the cocktail party graph** ** **of order **2**n **is*

*Proof*. Let be the cocktail party graph of order 2*n* having the vertex set The corresponding symmetric division deg matrix is

and the characteristic equation becomes

implying that the spectrum would be

Therefore,

**Theorem 5.6.** *The symmetric division deg energy of *2(*i*)*-complement of** **double star graph** ** **is*

*where * *and *

*Proof*. The symmetric division deg matrix for 2(i)-complement of double star graph is

Therefore the spectrum has times a a a and a Therefore we obtain the required result.

**Theorem 5.7.** *The symmetric division deg energy of **2-complement of cocktail party graph** ** **is*

*Proof.* Consider the 2-complement of the cocktail party graph The symmetric division deg matrix is

The characteristic polynomial is

and therefore the symmetric division deg spectrum has times times 0, a and a implying that the symmetric division deg energy is

Edge deletion is very important in combinatorial calculations with graphs. In this section, we obtain the symmetric division deg energy for certain graphs with one edge deleted. This can be used recursively to calculate the symmetric division deg energy of a given graph.

**Theorem 6.1**. *Let e be an edge of the complete graph** ** **Then** ** **is equal to*

*Proof*. The symmetric division deg matrix for is

Therefore the spectrum would have a

a times and a 0, implying the result.

**Theorem 6.2.** *Let e be an edge of the complete bipartite graph** ** **The symmetric division deg energy of** ** **is equal to*

*Proof*. The symmetric division deg matrix for is

Hence, the spectrum would have a a a a and times 0 implying the result

The following result can easily be proven as above:

**Lemma 6.3.*** Let** ** **be the star graph with n vertices and let e be an edge of it. Then** ** ** **for** *

**Open problem 7.1.** With respect to symmetric division deg, determine the class of graphs which are co-spectral and characterize them.

**Open problem 7.2. **With respect to symmetric division deg, determine the class of graphs which are hyperenergetic and characterize them.

**Open problem 7.3.** With respect to symmetric division deg, determine the class of graphs whose symmetric division deg energy and symmetric division deg energy of their complements are equal.

**Open problem 7.4.** With respect to symmetric division deg, determine the class of non-co-spectral graphs which are equienergetic.

**Open problem 7.5.** Determine the class of graphs whose symmetric division deg energy is equal to usual energy.

[1] | Alexander, V., Upper and lower bounds of symmetric division deg index, Iranian Journal of Mathematical Chemistry, 5 (2) (2014), 91-98. | ||

In article | View Article | ||

[2] | Gutman, I., The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz, 103 (1978), 1-22. | ||

In article | |||

[3] | Gutman, I., The energy of a graph: old and new results, Combinatorics and applications, A. Betten, A. Khoner, R. Laue and A. Wassermann, (Eds.), Springer, Berlin, (2001), 196-211. | ||

In article | View Article | ||

[4] | Randić, M., On characterization of molecular branching, J. Am. Chem. Soc., 97 (1975), 6609-6615. | ||

In article | View Article | ||

[5] | Sampathkumar, E., Pushpalatha, L., Venkatachalam, C. V. and Bhat, P., Generalized complements of a graph, Indian J. Pure Appl. Math., 29(6) (1998), 625-639. | ||

In article | View Article | ||

[6] | Todeschini, R., Consonni, V., Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, (2000), 84-90. | ||

In article | View Article | ||

[7] | Todeschini, R., Consonni, V., Molecular Descriptors for Chemoinformatics, Wiley-VCH, Weinheim, (2009), 161-172. | ||

In article | View Article | ||

[8] | Zhou, B., Trinajstic, N., On Sum-Connectivity Matrix and Sum-Connectivity Energy of (Molecular) Graphs, Acta Chim. Slov, 57 (2010), 518-523. | ||

In article | PubMed | ||

[9] | Zhou, B., Trinajstic, N., On a novel connectivity index, J. Math. Chem., 46 (2009), 1252-1270. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2017 K. N. Prakasha, P. Siva Kota Reddy and Ismail Naci Cangul

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

K. N. Prakasha, P. Siva Kota Reddy, Ismail Naci Cangul. Symmetric Division Deg Energy of a Graph. *Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 6, 2017, pp 202-209. http://pubs.sciepub.com/tjant/5/6/2

Prakasha, K. N., P. Siva Kota Reddy, and Ismail Naci Cangul. "Symmetric Division Deg Energy of a Graph." *Turkish Journal of Analysis and Number Theory* 5.6 (2017): 202-209.

Prakasha, K. N. , Reddy, P. S. K. , & Cangul, I. N. (2017). Symmetric Division Deg Energy of a Graph. *Turkish Journal of Analysis and Number Theory*, *5*(6), 202-209.

Prakasha, K. N., P. Siva Kota Reddy, and Ismail Naci Cangul. "Symmetric Division Deg Energy of a Graph." *Turkish Journal of Analysis and Number Theory* 5, no. 6 (2017): 202-209.

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[1] | Alexander, V., Upper and lower bounds of symmetric division deg index, Iranian Journal of Mathematical Chemistry, 5 (2) (2014), 91-98. | ||

In article | View Article | ||

[2] | Gutman, I., The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz, 103 (1978), 1-22. | ||

In article | |||

[3] | Gutman, I., The energy of a graph: old and new results, Combinatorics and applications, A. Betten, A. Khoner, R. Laue and A. Wassermann, (Eds.), Springer, Berlin, (2001), 196-211. | ||

In article | View Article | ||

[4] | Randić, M., On characterization of molecular branching, J. Am. Chem. Soc., 97 (1975), 6609-6615. | ||

In article | View Article | ||

[5] | Sampathkumar, E., Pushpalatha, L., Venkatachalam, C. V. and Bhat, P., Generalized complements of a graph, Indian J. Pure Appl. Math., 29(6) (1998), 625-639. | ||

In article | View Article | ||

[6] | Todeschini, R., Consonni, V., Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, (2000), 84-90. | ||

In article | View Article | ||

[7] | Todeschini, R., Consonni, V., Molecular Descriptors for Chemoinformatics, Wiley-VCH, Weinheim, (2009), 161-172. | ||

In article | View Article | ||

[8] | Zhou, B., Trinajstic, N., On Sum-Connectivity Matrix and Sum-Connectivity Energy of (Molecular) Graphs, Acta Chim. Slov, 57 (2010), 518-523. | ||

In article | PubMed | ||

[9] | Zhou, B., Trinajstic, N., On a novel connectivity index, J. Math. Chem., 46 (2009), 1252-1270. | ||

In article | View Article | ||