# On Zagreb Energy and edge-Zagreb energy

Document Type : Original paper

Author

Vidyavardhaka College of Engineering

Abstract

In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $\delta \ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we give some methods to construct (edge) Zagreb equienergetic graphs and show that there are (edge) Zagreb equienergetic graphs of order $n\ge 9$.

Keywords

Main Subjects

#### References

[1] C. Adiga and B.R. Rakshith, Upper bounds for the extended energy of graphs and some extended equienergetic graphs, Opuscula Math. 38 (2018), no. 1, 5–13.
[2] G. Arizmendi and O. Arizmendi, Energy of a graph and Randić index, Lin. Algebra. Appl. 609 (2021), 332–338.
[3] D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Academic Press, New York, 1980.
[4] K.C. Das, On the Zagreb energy and Zagreb Estrada index of graphs, MATCH Commun. Math. Comput. Chem. 82 (2019), no. 2, 529–542.
[5] K.C. Das, I. Gutman, and B. Furtula, On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116–123.
[6] K.C. Das, I. Gutman, I. Milovanović, E. Milovanović, and B. Furtula, Degreebased energies of graphs, Lin. Algebra. Appl. 554 (2018), 185–204.
[7] I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013), no. 4, 351–361.
[8] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), no. 1, 83–92.
[9] I. Gutman, S. Filipovski, and R. Jajcay, Variations on McClelland’s bound for graph energy, Discrete Math. Lett. 3 (2020), 57–60.
[10] I. Gutman and H. Ramane, Research on graph energies in 2019, MATCH Commun. Math. Comput. Chem. 84 (2020), no. 2, 277–292.
[11] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, New York, 1990.
[12] W. Imrich and S. Klavžar, Product of Graphs - Structure and Recognition, Wiley, New York, 2000.
[13] N. Jafari Rad, A. Jahanbani, and I. Gutman, Zagreb energy and Zagreb Estrada index of graphs, Match. Commun. Math. Comput. Chem. 79 (2018), 371–386.
[14] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008), no. 1, 127–156.
[15] X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012.
[16] A.W. Marshall, I. Olkin, and B.C. Arnold, Inequalities: Theory of Majorization and its Applications, Springer, New York, 2011.
[17] F. Zhan, Y. Qiao, and J. Cai, On edge-Zagreb spectral radius and edge-Zagreb energy of graphs, Linear and Multilinear Algebra 66 (2018), no. 12, 2512–2523.