Journal of Chemistry
Volume 2019 (2019), Article ID 7214047, 12 pages
https://doi.org/10.1155/2019/7214047

Research Article

Some New Results on Various Graph Energies of the Splitting Graph

Zheng-Qing Chu,¹ Saima Nazeer,² Tariq Javed Zia,³ Imran Ahmed,³ and Sana Shahid²

¹Department of Mathematics and Physics, Anhui Xinhua University, 230088 Hefei, China
²Lahore College for Women University, Lahore, Pakistan
³COMSATS University of Islamabad, Lahore Campus, Lahore 54000, Pakistan

Correspondence should be addressed to Saima Nazeer; saimanazeer123@yahoo.com

Received 20 July 2019; Accepted 1 October 2019; Published 19 November 2019

Guest Editor: Shaohui Wang

Copyright © 2019 Zheng-Qing Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix . Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph is a multiple of corresponding energy of a given graph G.

1. Introduction

Let G be a simple, finite, and undirected graph and its vertex set and edge set are denoted by and , respectively. Number of edges finishing at a vertex of a graph is named as degree of vertex and is denoted by or .

The adjacency matrix of G, denoted by , is a square matrix such that is equal to unity if and is equal to zero otherwise. The eigenvalues of the adjacency matrix are known as the eigenvalues of the graph G. Collection of eigenvalues of the graph G together with their multiplicities is called spectrum of the graph G.

Let be eigenvalues of G and are assumed in nonincreasing order; then, Ivan Gutman in 1978 [1] defined the energy of the graph G as the sum of the absolute values of all eigenvalues of the graph G:

The inspiration of description energy of graph happened from quantum Chemistry. During 1930s, E. Hückel presented chemical applications of graph theory in his molecular orbital theory where eigenvalues of graphs take place. In quantum chemistry, the skeleton of nonsaturated hydrocarbon is represented by a graph. The energy levels of electrons in such a molecule are eigenvalues of graph. The strength of particles is closely identified with the spectrum of its graph. The carbon atoms and chemical bond between them in a hydrocarbon system denote vertices and edges, respectively, in a molecular graph. A lot of work has been done on graph theory, special graph labeling [2–10], chemical graph theory and graph energies. In the thesis of Siraj [11], certain elementary results on the energy of the graph are also described.

The present work is considered to relate several energies of a graph to bigger graph acquired from the given graph with the help of some graph operations, namely, the splitting graph which is defined in [12]. For a graph G, the splitting graph is obtained by taking a new vertex corresponding to each vertex of the graph G and then join to all vertices of G adjacent to . In [13], it has been proven that

Let and be two matrices of order and , respectively. Then, their tensor product, is obtained from A when every element is replaced by the block and is of order .

Proposition 1 (see [14]). Let and . Also, let α be an eigenvalue of the matrix A with corresponding eigenvector y and β be an eigenvalue of the matrix B with corresponding eigenvector z. Then, is an eigenvalue of with corresponding eigenvector .

In recent times, comparable energies are being considered, based on eigenvalues of a variety of other graph matrices. Numerous matrices can be related to a graph, and their spectrums provide certain helpful information about the graph [15–18].

2. Maximum Degree Energy

The maximum degree energy of a simple connected graph G in [19] is defined as the sum of the absolute values of eigenvalues of the maximum degree matrix of a graph G. Then, where

where and are the degrees of vertices and , respectively.

Theorem 1. For a graph G,

Proof. Let G be a graph with vertices . Then the maximum degree matrix is

where and and are the degrees of vertices and , respectively, for and
Let be the vertices corresponding to which are added in G to obtain such that for Then, the maximum degree matrix of is denoted by and can be written as a block matrix:

That is

Here, the maximum degree spectrum of is

where for are the eigenvalues of and are the eigenvalues of .
Here,

which completes the proof.

3. Minimum Degree Energy

In [20], the minimum degree energy of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of minimum degree matrix of a graph G. Here, where

where and are the degrees of vertices and , respectively.

Theorem 2. For a graph G,

Proof. Let G be a graph with vertices . Then, the minimum degree matrix is

where and and are the degrees of vertices and , respectively, for and Let be the vertices corresponding to which are added in G to obtain such that for . Then, the minimum degree matrix of splitting graph of G, denoted by , can be defined as a block matrix as follows:

That is

Here, the minimum degree spectrum of is

where for are the eigenvalues of m(G) and are the eigenvalues of .
Here,

which is the required result.

4. Randić Energy

The randić energy of a simple connected graph G in [21] is the sum of the absolute values of eigenvalues of the randić matrix . Here, where

Here, and are the degrees of vertices and , respectively.

Theorem 3. For a graph G,

Proof. Let be vertices of a graph G. Then, the randić matrix of G is denoted by and is given as

Let be the vertices corresponding to which are added in G to obtain such that for Then, the randić matrix of is denoted by and can be written as a block matrix as follows:

That is,

Here, the randić spectrum of is

where for are the eigenvalues of and and 1 are the eigenvalues of .
Here,

5. Seidel Energy

In [22], Haemers defined the Seidel energy of a simple connected graph G as the sum of the absolute values of eigenvalues of the seidel matrix of G. Here, where

Theorem 4. For a s-regular graph G,

Proof. Let G be a graph with vertices . Then, the seidel matrix of G is

Let be the vertices corresponding to which are added in G to obtain . Then, the seidel matrix of is denoted by and can be written as a block matrix as follows:

That is,

Hence,

Here, the seidel spectrum of is greater than or equal to the spectrum

where for are the eigenvalues of and are the eigenvalues of .
Thus,

Hence,

6. Sum-Connectivity Energy

The sum-connectivity energy of a simple connected graph G in [23] is defined as the sum of the absolute values of eigenvalues of the sum-connectivity matrix . Here, where

Here, and are the degrees of vertices and , respectively.

Theorem 5. For a regular graph G,

Proof. Let G be a graph with vertices . Then the sum-connectivity matrix of G is denoted by and is defined as

where is the degree of vertex for . Let be the vertices that are added in G to acquire such that . Then the sum-connectivity matrix of is denoted by and is defined as a block matrix as follows:

where is the degree of vertex for . Thus,

Here, the sum-connectivity spectrum of is

where , for are the eigenvalues of and are the eigenvalues of .
Hence,

which completes the proof.

7. Degree Sum Energy

In [24], the degree sum energy of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of the degree sum matrix of G. Here, where

Here, and are the degrees of vertices and , respectively.

Theorem 6. For a s-regular graph G with order p,

Proof. Let G be a graph with vertices . Then the degree sum matrix of G is denoted by and is defined as

where is degree of vertex for . Let be the vertices corresponding to vertices that are added in G to get the splitting graph . Then the degree sum matrix of is given as

where is the degree of vertex for .
Note that

Thus,

As

Hence,

which is the required result.

8. Degree Square Sum Energy

The degree square sum energy of a simple connected graph G in [25] is defined as the sum of the absolute values of eigenvalues of the degree square sum matrix . Here, where

Here, and are degrees of vertices and , respectively.

Theorem 7. For a s-regular graph G with p vertices,

Proof. Let be vertices of a graph G and be the vertices added in G corresponding to to get the splitting graph such that for , . Then, the degree square sum matrix of G and are given as

where is the degree of vertex and is the degree of vertex , for .
Note that

Thus,

As

Hence,

which completes the proof.

9. First Zagreb Energy

In [26], the First zagreb energy of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of first zagreb matrix of G where where

Here, and are degrees of vertices and , respectively.

Theorem 8. For a regular graph G,

Proof. Let G be a graph with vertices . Then, the first zagreb matrix of G is denoted by and is defined as

where is the degree of vertex for . Let be vertices added in G corresponding to to get such that . Then, the first zagreb matrix of can be written as a block matrix as follows:

where is the degree of vertex for .
Here,

Here, the first zagreb spectrum of is

where for are the eigenvalues of and are the eigenvalues of . Hence,

10. Second Zagreb Energy

The second zagreb energy of a simple connected graph G is defined in [26] as the sum of the absolute values of eigenvalues of the second zagreb matrix of G where , where

Here, and are the degrees of vertices and , respectively.

Theorem 9. For a regular graph G,

Proof. Let G be a graph with vertices . Then, the second zagreb matrix of G is denoted by and is defined as

where is the degree of vertex for . Let be vertices added in G corresponding to to get such that . Then, the second Zagreb matrix of is denoted by and can be written as a square matrix as follows:

where is the degree of vertex for .
Here,

Here, the second zagreb spectrum of is

where for are the eigenvalues of and are the eigenvalues of . Hence,

11. Conclusion

The energy of a graph is one of the important idea of spectral graph theory. This idea links organic chemistry to mathematics. Numerous graph energies established from the eigenvalues of a variety of graph matrices and their bounds has been discovered. In this paper, we give a relation of various graph energies between the regular graph and its splitting graph. It is interesting to compute graph energies for the families of graphs considered in [27–31].

Data Availability

All data are included within this paper.

Conflicts of Interest

The authors declare no conflicts of interests

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

The authors would like to express their sincere gratitude to the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (nos. KJ2019A0875 and KJ2019A0876) and the Quality Engineering Projects of Anhui Xinhua University of China (no. 2017jxtdx05).

References

I. Gutman, “The energy of a graph,” Berichte der Mathematisch–Statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1–22, 1978.

M. A. Umar, N. Ali, A. Tabassum, and B. R. Ali, “Book graphs are cycle antimagic,” Open Journal of Mathematical Sciences, vol. 3, no. 1, pp. 184–190, 2019.

M. A. Umar, “Cyclic-antimagic construction of ladders,” Engineering and Applied Science Letters, vol. 2, no. 2, pp. 43–47, 2019.

L. Chen, H. Jiang, Z. Shao, and M. Ivanović, “Dominator and total dominator colorings in vague graphs,” Engineering and Applied Science Letters, vol. 2, no. 2, pp. 10–17, 2019.

F. Aslam, Z. Zahid, and S. Zafar, “3-total edge mean cordial labeling of some standard graphs,” Open Journal of Mathematical Sciences, vol. 3, no. 1, pp. 129–138, 2019.

N. Ali, M. A. Umar, A. Tabassum, and A. Raheem, “Super (a, d)-C₃-antimagicness of a corona graph,” Open Journal of Mathematical Sciences, vol. 2, no. 1, pp. 371–378, 2018.

M. C. M. Kumar and H. M. Nagesh, “Directed pathos total digraph of an arborescence,” Engineering and Applied Science Letters, vol. 1, no. 1, pp. 29–42, 2018.

W. Gao, M. Asif, and W. Nazeer, “The study of honey comb derived network via topological indices,” Open Journal of Mathematical Analysis, vol. 2, no. 2, pp. 10–26, 2018.

H. M. Nagesh and M. C. M. Kumar, “Block digraph of a directed graph,” Open Journal of Mathematical Sciences, vol. 2, no. 1, pp. 202–208, 2018.

G. V. Rajasekharaiah and U. P. Murthy, “Secure domination in lict graphs,” Open Journal of Mathematical Sciences, vol. 2, no. 1, pp. 134–145, 2018.

M. A. Sriraj, Some Studies on Energy of Graphs, University of Mysore, Mysore, India, 2014, Ph.D. thesis.

E. Sampathkumar and H. B. Walikar, “On the splitting graph of a graph,” Journal of the Karnatak University Science, vol. 25, no. 13, pp. 13–16, 1980.

S. K. Vaidya and K. M. Popat, “Some new results on energy of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 77, pp. 589–594, 2017.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.

J.-B. Liu, C. Wang, S. Wang, and B. Wei, “Zagreb indices and multiplicative zagreb indices of eulerian graphs,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 42, no. 1, pp. 67–78, 2019.

J.-B. Liu, X.-F. Pan, F.-T. Hu, and F.-F. Hu, “Asymptotic Laplacian-energy-like invariant of lattices,” Applied Mathematics and Computation, vol. 253, pp. 205–214, 2015.

J.-B. Liu and X.-F. Pan, “Minimizing Kirchhoff index among graphs with a given vertex bipartiteness,” Applied Mathematics and Computation, vol. 291, pp. 84–88, 2016.

J.-B. Liu, J. Zhao, and Z. Zhu, “On the number of spanning trees and normalized Laplacian of linear octagonal-quadrilateral networks,” International Journal of Quantum Chemistry, vol. 119, no. 17, Article ID e25971, 2019.

C. Adiga and M. Smitha, “On maximum degree energy of a graph,” International Journal of Contemporary Mathematical Sciences, vol. 4, no. 8, pp. 385–396, 2009.

C. Adiga and C. S. Swamy, “Bounds on the largest of minimum degree eigenvalues of graphs,” International Mathematical Forum, vol. 5, no. 37, pp. 1823–1831, 2010.

K. C. Das, S. Sorguna, and K. Xu, “On randic energy of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 72, no. 1, pp. 227–238, 2014.

P. Nageswari and P. B. Sarasija, “Seidel energy and its bounds,” International Journal of Mathematical Analysis, vol. 8, no. 58, pp. 2869–2871, 2014.

B. Zhou and N. Trinajstic, “On the sum-connectivity matrix and sum-connectivity energy of (molecular) graphs,” Acta Chimica Slovenica, vol. 57, no. 3, pp. 513–517, 2010.

S. M. Hosamani and H. S. Ramane, “On degree sum energy of a graph,” European Journal of Pure and Applied Mathematics, vol. 9, no. 3, pp. 340–345, 2016.

B. Basavanagoud and E. Chitra, “Degree square sum energy of graphs,” International Journal of Mathematics and Its Applications, vol. 6, no. 2B, pp. 193–205, 2018.

N. J. Rad, A. Jahanbani, and I. Gutman, “Zagreb energy and Zagreb estrada index of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 79, pp. 371–386, 2018.

H. Siddiqui and M. R. Farahani, “Forgotten polynomial and forgotten index of certain interconnection networks,” Open Journal of Mathematical Analysis, vol. 1, no. 1, pp. 44–59, 2017.

S. Noreen and A. Mahmood, “Zagreb polynomials and redefined Zagreb indices for the line graph of carbon nanocones,” Open Journal of Mathematical Analysis, vol. 2, no. 1, pp. 66–73, 2018.

Z. Tang, L. Liang, and W. Gao, “Wiener polarity index of quasi-tree molecular structures,” Open Journal of Mathematical Sciences, vol. 2, no. 1, pp. 73–83, 2018.

M. Imran, A. Asghar, and A. Q. Baig, “On graph invariants of oxide network,” Engineering and Applied Science Letters, vol. 1, no. 1, pp. 23–28, 2018.

W. Gao, A. Asghar, and W. Nazeer, “Computing degree-based topological indices of Jahangir graph,” Engineering and Applied Science Letters, vol. 1, no. 1, pp. 16–22, 2018.