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
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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
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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
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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
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Then we have
Proposition 3.1. The first three coefficients of the polynomial are as follows:
(i)
(ii)
(iii)
Proof. (i) By the definition of the polynomial
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we get a0 = 1.
(ii) The sum of determinants of all principal submatrices of
is equal to the trace of
implying that
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(iii) By the definition, we have
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Proposition 3.2. If are the symmetric division deg eigenvalues of
, then
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Proof. It follows as
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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
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Proof. Let be the eigenvalues of
By the Cauchy-Schwartz inequality we have
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Let Then
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implying that
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and hence we get
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as an upper bound.
Theorem 3.4. Let G be a graph with n vertices. If then
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Proof. By definition, we have
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Using arithmetic-geometric mean inequality, we have
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Therefore,
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Thus,
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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,
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Theorem 3.6. For a graph G of order n with non-zero eigenvalues, we have
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Theorem 3.7. Let G be a graph of order n. Let be the eigenvalues in increasing order. Then
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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
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Proof. Let be the complete graph with vertex set
For this graph, the symmetric division deg matrix is
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The characteristic equation then becomes
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and the spectrum would be
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Therefore,
Theorem 4.2. The symmetric division deg energy of the cycle graph is
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Proof. The symmetric division deg matrix corresponding to the cycle graph is
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This is a circullant matrix of order 2n. Its eigenvalues are
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Therefore the symmetric division deg energy is
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Theorem 4.3. The symmetric division deg energy of the star graph is
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Proof. Let be the star graph with vertex set
with
denotes the central vertex. The symmetric division deg matrix is
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The characteristic equation becomes
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and therefore, the spectrum would have an
and
times 0. Therefore,
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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
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Proof. Let be the friendship graph with 2n + 1 vertices and let v0 be the common vertex. The symmetric division deg matrix is
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The characteristic equation becomes
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implying that the spectrum has n times ,
times 2, a
and a
Therefore, we get
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Theorem 4.6. The symmetric division deg energy of the cocktail party graph is
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Proof. Let be the cocktail party graph of order 2n having vertex set
The symmetric division deg matrix is
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In that case, the characteristic equation is
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and hence the spectrum becomes
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Therefore we arrive at the required result:
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Theorem 4.7. The symmetric division deg energy of the double star graph is
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Proof. The symmetric division deg matrix is
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Hence, the spectrum would have times
and
Therefore, we get
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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 3n + 1 vertices and 4n edges.
Theorem 4.9. The symmetric division deg energy of the Dutch windmill graph is
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Proof. Recall that has 3n + 1 vertices. Then the symmetric division deg matrix is
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Hence the characteristic equation will be
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and therefore the spectrum would have times
times
times 0,
and
Therefore, it is directly seen that
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Theorem 4.10. The symmetric division deg energy of crown graph is
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Proof. Let be the crown graph of order 2n and let the vertex set of this graph be
The symmetric division deg matrix of is
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Therefore the characteristic equation is
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implying that the spectrum has a a
times 2 and
times
. Therefore we obtain
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Theorem 4.11. The symmetric division deg energy of the complete bipartite graph of order
with vertex set
is
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Proof. The symmetric division deg matrix of the complete bipartite graph is
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Then the characteristic equation is
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and therefore the spectrum has a times 0 and a
Therefore, we get
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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
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Proof. Let be the complete graph with vertex set
The symmetric division deg connectivity matrix of the complement of the complete graph
is
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Clearly, the characteristic equation is implying
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Theorem 5.4. The symmetric division deg energy of the complement of the star graph
is
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Proof. Let be the complement of the star graph with vertex set
where
is the central vertex. The symmetric division deg matrix is
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The corresponding characteristic equation is
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and therefore the spectrum is
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Therefore,
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Theorem 5.5. The symmetric division deg energy of the complement of the cocktail party graph
of order 2n is
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Proof. Let be the cocktail party graph of order 2n having the vertex set
The corresponding symmetric division deg matrix is
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and the characteristic equation becomes
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implying that the spectrum would be
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Therefore,
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Theorem 5.6. The symmetric division deg energy of 2(i)-complement of double star graph is
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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
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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
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Proof. The symmetric division deg matrix for is
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Therefore the spectrum would have a
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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
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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.
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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
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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 | ||